Theory of Interacting Quantum Fields 9783110250633, 9783110250626

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De Gruyter Studies in Mathematics 39 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, Missouri, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Alexei L. Rebenko

Theory of Interacting Quantum Fields

De Gruyter

0DWKHPDWLFV6XEMHFW&ODVVL¿FDWLRQPrimary: 81S05, 81U20, 81T05, 81T13, 81T15, 81T18; Secondary: 22E43,30H20, 81S40, 81V10.

ISBN 978-3-11-025062-6 e-ISBN 978-3-11-025063-3 ISSN 0179-0986 /LEUDU\RI&RQJUHVV&DWDORJLQJLQ3XEOLFDWLRQ'DWD A CIP catalog record for this book has been applied for at the Library of Congress. %LEOLRJUDSKLFLQIRUPDWLRQSXEOLVKHGE\WKH'HXWVFKH1DWLRQDOELEOLRWKHN 7KH'HXWVFKH1DWLRQDOELEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD¿H detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Translation: Peter V. Malyshev, Kiev Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ˆ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

The extensive development of the theory of gauge fields, rapid progress in the physics of elementary particles, and significant achievements in the solution of the mathematical problems of quantum field theory (both in the constructive and Euclidean approaches) require the availability of rigorous, logically complete, and comprehensive (from the mathematical and physical points of view) description of various issues related to the construction of contemporary quantum field theory. Our monograph is an extended version of the lectures delivered by the author for many years at the Taras Shevchenko Kiev National University and can be regarded as, possibly, the first attempt to combine the contemporary approach to the theory of quantum fields with a more profound investigation of the mathematical problems of this theory and the available constructive methods aimed at the solution of these problems. The idea to write this book was formed by the necessity to answer the questions of inquisitive students who demanded the logical substantiation of various items (axioms) fundamental for the construction of the quantum field theory but sometimes presented without proper explanations. The major part of available works dealing with the problems of quantum field theory are characterized by the professional (expert) level of presentation of the material and, hence, often remain hardly comprehensible for the novices. Especially many questions arise in the mathematically educated audience. Thus, the main aim of our presentation is to avoid these drawbacks of the previous publications. Clearly, it is impossible to consider all problems of the quantum field theory within a single short course. Hence, we sometimes only formulate a problem and indicate the main sources, where it is discussed in more detail. Our aim is not only to present the required body of knowledge in the field of quantum field theory but also to show the reader how to apply this knowledge in various branches of the theoretical physics. The book contains the introduction and seven parts. Each part is preceded by a short introductory part containing a brief presentation of its contents and explanation of its importance for the construction of the quantum field theory. At the end of each part, we propose some problems for the independent analysis of the reader. In most cases, these problems are supplied with guidelines required for their solution. In the introduction, we present a brief historical survey of the appearance and evolution of the fundamental ideas used for the construction of the quantum field theory. In the first part “Symmetry Groups of Elementary Particles”, we present the analysis of various symmetry groups connected with the most important laws of the physics of interacting elementary particles.

vi

Preface

In the second part “Classical Theory of Free Fields”, the Lagrangian formalism of the classical field theory is presented and the Klein–Fock–Gordon, Dirac, and Maxwell equations of free fields, as well as the corresponding equations for the vector fields, are investigated. A brief presentation of the elementary theory of fields with higher spins can also be found in this part. The third part “Classical Theory of Interacting Fields” is devoted to the substantiation of the concept of interacting fields, construction of the theory of gauge fields, and investigation of some basic problems related to the general principles of symmetry and its breaking. Some Lagrangians corresponding to the main types of interactions and their unification within the framework of gauge theories are presented. The solutions of the classical field equations are briefly discussed. In the fourth part “Second Quantization of Fields”, we consider the general principles of quantization and their realization by analyzing an example of free fields. The canonical commutation and anticommutation relations and their representations in the Fock space are constructed. The problems connected with the introduction of interaction between quantum fields are discussed. The equations for interacting fields are deduced, the Hamiltonians of interaction are investigated, and the problems of their rigorous mathematical definition in the Fock space are analyzed. In the fifth part “Quantum Theory of Interacting Fields. General Problems”, we study the mathematical problems encountered in the construction of interacting fields and the properties of the scattering operator. The scattering matrix is constructed by using perturbation theory. The scattering operator and its matrix elements are graphically represented in the form of Feynman diagrams. A brief presentation of the renormalization theory can also be found in this part. The method of functional integrals and its application to the theory of quantum fields are described in detail. In the sixth part “Axiomatic and Euclidean Field Theory”, we describe the existing axiomatic approaches to the quantum field theory and propose a more rigorous (from the mathematical viewpoint) procedure of investigation of the scattering matrix and Green’s functions in the Euclidean space–time metric. The Euclidean formulation enables us to construct, under certain restrictions, the scattering matrix and Green’s functions beyond the limits of applicability of perturbation theory. The seventh part “Quantum Theory of Gauge Fields” is devoted to the construction of quantum gauge fields by the method of functional integration. The QED and QCD models are discussed in detail. We also present a brief description of the models of weak and electroweak interactions. The major part of the presented material can be found in other publications. First of all, we can mention the works [2, 12, 16, 17, 22, 23, 26, 40, 41, 49, 101, 105, 157, 158, 164, 175, 181, 191, 208, 212, 213, 225] included in the list of references. It is also necessary to add to this list a large number of new publications containing not only the material presented in our book but also the contemporary approaches to the theory of gauge fields, supersymmetry, gravitation theory, etc. We cite some of these

Preface

vii

publications that became available after the appearance of the first edition of this book published in Ukrainian, namely [1, 37, 38, 15, 8, 28, 80, 117, 173, 179, 214]. I am very grateful to my colleagues V. P. Gusynin, V. M. Gorkavenko, and D. Ya. Petryna for numerous critical remarks and valuable advice aimed at the improvement of the book. Kiev, July 2011

Alexei L. Rebenko

Notation

We use double numbering of formulas, theorems, definitions, and remarks, where the first number is the number of the chapter and the second number is the number of the corresponding formula, theorem, definition, or remark inside this chapter. 1. Abbreviations BMP stands for Bogolyubov–Medvedev–Polivanov; BRST-symmetry stands for the Becchi–Rouet–Stora–Tyutin-symmetry; CQFT stands for the constructive quantum field theory; HRST stands for the Haag–Ruelle scattering theory; LSZ stands for Lehmann–Symanzik–Zimmermann; QCD stands for quantum chromodynamics; QED stands for quantum electrodynamics; QFT stands for the quantum field theory; RT stands for the renormalization theory; WOF is a Wick ordered functional; a WD b means that the quantity a is equal to b by definition; 1; : : : ; n WD 1; nI ŒA; B WD AB  BA is the commutator; ŒA; BC WD AB C BA is the anticommutator;   P @F @G @F @G  ¹F; GºP WD ¹F; Gº D niD1 @q are the Poisson brackets; @pi @qi i @pi PS ¹F; GºD WD ¹F; Gº  s;s 0 ¹F; 's º.C 1 /s;s 0 ¹'s 0 ; Gº are the Dirac brackets; P pO WD  p  D 3D0  p  I 'P WD @' is the time derivative in the classical mechanics; @t @ WD @x@ ; @ WD @x@ I $

a @  b WD a@ b  .@ a/bI –conjugation: for scalars, it coincides with the operation of complex conjugation: '   '; for vector columns or spinors, this is the complex conjugation of components and the operation of transposition, i.e., a column is transformed into a row [see, e.g., relations (5.17) and (5.18)], for matrices, this is the complex conjugation of components and the operation of transposition: .A /ij D Aj i I .a/n WD .a1 ; : : : ; an /I

Notation

.˛; a/n WD ..˛1 ; a1 /; : : : ; .˛n ; an //I .a/n n ¹ai1 ; : : : ; aik º is a sequence .a/n without elements ai1 ; : : : ; aik : 2. Notation of Basic Spaces or Sets Es  Rs , s D 1; 2; 3; is the s-dimensional Euclidean space; E3  EI Md , d D 1; 2; 3; 4; is the d -dimensional Minkowski space; M4  MI MC  MC 4 is the complex Minkowski space; C WD ¹x 2 M j x 2 D 0º is a light cone; V˙ (V ˙ ) are open (closed) cones of the future (C) or of the past (); H is the Hilbert space of states; HB is the Hilbert space of states of bosons; HF is the Hilbert space of states of fermions; F is the Fock space; FB is the Fock space of bosons; FF is the Fock space of fermions; D is the space of test functions with finite support; S is the space of test functions vanishing faster than a polynomial of any degree; D 0 is the space of distributions over D; S 0 is the space of distributions over S ; a  b is the scalar product in Rs , s D 1; 2; 3; a  b is the scalar product in Md , d D s C 1. 3. Notation and Values of Some Basic Constants c D 2:998  108 m=sec is the velocity of electromagnetic waves in vacuum;  h 34 erg  c I h is the Planck constant „ D 2 D 1:055  10 2

e 1  137 is the constant of fine structure; ˛ D 4„c e D jqel jI qel D 1:60219  1019 K is the charge of electron.

4. Notation of Operators, Matrices, and Tensors r D ¹@k º3kD1 is the gradient operator;  D r  r is the Laplace operator;  D   @20 is the d’Alembert operator; b is the operator of charge conjugation; C b is the operator of space reflection; P b T is the operator of time reversal; Q is the operator of charge; Y is the operator of hypercharge; Jk are the generators of isotopic spin;

ix

x

Notation

k ; k D 1; 2; 3; are the Pauli matrices; j , j D 1; : : : ; 8; are the Gell-Mann matrices (generators of group SU(3));   ,  D 0; 1; 2; 3; 5 are the Dirac matrices; "ij k are the structural constants of the group SU.2/; fij k are the structural constants of the group S U.N /; N  3; F is the tensor of the electromagnetic (or gauge) field. 5. Groups, Subgroups, and Some Transformations e is the general Lorentz group; L ƒ D ..ƒ // are transformations of the Lorentz group; "

LC are proper orthochronous Lorentz transformations .ƒ00 > 1, det ƒ D C1/; #

LC are proper nonorthochronous Lorentz transformations .ƒ00 6 1, det ƒ D C1/; L" are nonproper orthochronous Lorentz transformations .ƒ00 > 1, det ƒ D 1/; L# are nonproper nonorthochronous Lorentz transformations .ƒ00 6 1; det ƒ D 1/; L .C/ are complex Lorentz transformations; U.1/ is the unitary Abelian group; S U.n/ is a special unitary group; G is a gauge group.

Contents

Preface

v

Notation

viii

0 Introduction

1

I Symmetry Groups of Elementary Particles 1 Lorentz Group 1.1 Euclidean and Minkowski Spaces. Relativistic Notation . . . . . . . 1.2 Homogeneous Lorentz Group . . . . . . . . . . . . . . . . . . . . . 1.3 Inhomogeneous Lorentz Group–Poincaré Group . . . . . . . . . . . 1.4 Complex Lorentz Transformations . . . . . . . . . . . . . . . . . . 1.5 Representations of the Lorentz and Poincaré Groups, Field Functions, and Physical States . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Representation D .0;0/ . . . . . . . . . . . . . . . . . . . . 1 1 1.5.2 Representations D . 2 ;0/ and D .0; 2 / . . . . . . . . . . . . . 1 1 1.5.3 Representation D . 2 ; 2 / . . . . . . . . . . . . . . . . . . . .

8 8 11 14 15

2 Groups of Internal Symmetries 2.1 Abelian Unitary Group U.1/ . . . . . . . . . . 2.2 Charge Conjugation C . . . . . . . . . . . . . 2.3 Special Unitary Group SU.n/ . . . . . . . . . 2.3.1 SU.2/ Symmetry . . . . . . . . . . . . 2.3.2 SU.3/ Symmetry . . . . . . . . . . . . 2.4 Groups of Local Transformations. Gauge Group

23 23 24 24 25 27 29

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3 Problems to Part I

16 19 20 20

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II Classical Theory of the Free Fields 4 Lagrangian and Hamiltonian Formalisms of the Classical Field Theory 4.1 Variational Principle and Canonical Formalism of Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Lagrangian Equations . . . . . . . . . . . . . . . . . . . .

39 39 39

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4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

4.1.2 Canonical Variables. Hamiltonian Equations . . . . . 4.1.3 Poisson Brackets. Integrals of Motion . . . . . . . . . 4.1.4 Canonical Formalism in the Presence of Constraints . From Classical to Quantum Mechanics. Primary Quantization General Requirements to the Lagrangians of the Field Theory . Lagrange–Euler Equations . . . . . . . . . . . . . . . . . . . Noether’s Theorem and Dynamic Invariants . . . . . . . . . . Vector of Energy-Momentum . . . . . . . . . . . . . . . . . . Tensors of Angular Momentum and Spin . . . . . . . . . . . . Charge and the Vector of Current . . . . . . . . . . . . . . . . Canonical Variables . . . . . . . . . . . . . . . . . . . . . . .

5 Classical Theory of Free Scalar Fields 5.1 Klein–Fock–Gordon Equation . . . . . . . . . . . . . . . . . 5.2 Relativistic Invariance of the Klein–Fock–Gordon Equation . . 5.3 Solutions of the Klein–Fock–Gordon Equation . . . . . . . . . 5.4 Interpretation of Solutions. Hilbert Space of States . . . . . . b, P b , and T b Transformations . . . . . . . . . . . . . . . . . . 5.5 C b . . . . . . . 5.5.1 Transformation of Charge Conjugation C b 5.5.2 Space Reflection P . . . . . . . . . . . . . . . . . . . b . . . . . . . . . . . . . . . . . . . . 5.5.3 Time Reversal T bP bT b-Invariance . . . . . . . . . . . . . . . . . . . . 5.5.4 C 5.6 Representations of the Lorentz Group in the Space of States . . 5.7 Lagrangian Formalism of the Scalar Field. Dynamic Invariants 6 Spinor Field 6.1 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Construction of the Dirac Equation . . . . . . . . . 6.1.2 Properties of Dirac Matrices. Conjugate Equation . . 6.2 Relativistic Invariance . . . . . . . . . . . . . . . . . . . . . 6.2.1 Transformation Properties of the Spinor Field . . . . 6.2.2 On Reducible and Irreducible Spinor Representations 6.2.3 Transformation Properties of Bilinear Forms N O . 6.3 Solutions of the Dirac Equation . . . . . . . . . . . . . . . . 6.3.1 Structure of Solutions in the Momentum Space . . . 6.3.2 Classification of Solutions. Helicity . . . . . . . . . 6.3.3 Relations Between Spinors . . . . . . . . . . . . . . 6.3.4 Wave Functions of the Electron and Positron. Charge Conjugation . . . . . . . . . . . . . . . . .

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41 42 43 48 52 53 54 56 57 59 60

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61 61 62 64 66 70 70 72 73 73 74 78

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82 82 82 83 86 87 91 92 94 94 97 102

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6.4 6.5

6.6 6.7

bP bT b-Transformation . . . . . . . . . . . . . . . . . . . . 6.3.5 C Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . Representations of the Lorentz Group . . . . . . . . . . . . . . . . 6.5.1 Hilbert Space of States . . . . . . . . . . . . . . . . . . . . 6.5.2 Representations of the Lorentz Group in the Space of States Applications of the Dirac Equation . . . . . . . . . . . . . . . . . . 6.6.1 Dirac Equation in the Presence of External Fields . . . . . . Massless Spinor Field . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Two-component Massless Spinor Field . . . . . . . . . . . 6.7.2 Relativistic Invariance . . . . . . . . . . . . . . . . . . . . 6.7.3 Are There Actual Particles Corresponding to the Massless Spinor Fields? Physical Interpretation of Solutions. Neutrino . . . . . . . . . . . . . . . . . . . . 6.7.4 Lagrangian and Dynamic Invariants . . . . . . . . . . . . . 6.7.5 On the Mass of Neutrino and Majorana Spinors . . . . . . .

7 Vector Fields 7.1 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . 7.2 Representations in the Momentum Space . . . . . . . . . . . . . . 7.3 Decomposition into the Longitudinal and Transverse Components b; T b; C b -Transformations . . . . . . . . . . . . . . . . . . . . . . 7.4 P

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107 110 115 115 117 118 118 121 121 123

123 125 126 128 128 131 131 133

8 Electromagnetic Field 8.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Potential of the Electromagnetic Field . . . . . . . . . . . . . . . . 8.3 Gradient Transformations and the Lorentz Condition: Transversality Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Lagrangian Formalism for Electromagnetic Fields . . . . . . . . . . 8.5 Transversal, Longitudinal, and Time Components of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 8.6 Quantum-Mechanical Characteristics of Photons . . . . . . . . . . . b; P b; T b-Transformations . . . . . . . . . . . . . . . . . . . . . . . 8.7 C 8.8 Consistency of the Lorentz and Gauge Transformations. Various Types of Gauges . . . . . . . . . . . . . . . . . . . . . . .

135 135 136

9 Equations for Fields with Higher Spins 9.1 Fields with Spin 3=2 . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Particles with Spin 2 . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 151

10 Problems to Part II

152

137 139 141 143 146 146

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Contents

III Classical Theory of Interacting Fields 11 Gauge Theory of the Electromagnetic Interaction 11.1 Principle of Gauge Invariance in the Maxwell Theory . . . . . . . . 11.2 Schrödinger Equation and Gradient (Gauge) Invariance . . . . . . . 11.3 Gauge Principle as the Dynamical Principle of Interaction between the Electromagnetic and Electron-Positron Fields . . . . . . . . . .

158 158 159

12 Classical Theory of Yang–Mills Fields 12.1 Gauge Principle and the Lagrangian of the Yang–Mills Fields . . . . 12.2 Equations of Motion for the Free Yang–Mills fields . . . . . . . . . 12.3 Yang–Mills Fields for Arbitrary Representations of the Group SU.N / . . . . . . . . . . . . . . . . . . . . . . . . .

164 164 167

13 Masses of Particles and Spontaneous Breaking of Symmetry 13.1 Spontaneous Breaking of Symmetry . . . . . . . . . . . . 13.2 Higgs Mechanism for the Local U.1/ Symmetry . . . . . . 13.3 Higgs Mechanism for the Local SU.2/ symmetry . . . . . 13.4 Generation of the Masses of Fermions . . . . . . . . . . .

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171 172 174 176 179

14 On the Construction of the General Lagrangian of Interacting Fields 14.1 Lagrangian of the QCD . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Lagrangian of Weak Interactions . . . . . . . . . . . . . . . . . . . 14.3 On the Electroweak Interactions . . . . . . . . . . . . . . . . . . . 14.4 On the Lagrangian of Great Unification . . . . . . . . . . . . . . .

181 183 184 188 189

15 Solutions of the Equations for Classical Fields: Solitary Waves, Solitons, Instantons

191

16 Problems to Part III

197

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IV Second Quantization of Fields 17 Axioms and General Principles of Quantization 17.1 Why Do We Need the Procedure of Second Quantization? Operator Nature of the Field Functions . . . . . . . . . . . . . . . . . . . . . 17.2 Schrödinger, Heisenberg, and Interaction Pictures . . . . . . . . . . 17.3 Axioms of Quantization . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Relativistic Heisenberg Equation for Quantized Fields . . . . . . . . 17.4.1 Heisenberg Equation for a Free Scalar Field . . . . . . . . . 17.4.2 Heisenberg Equation for a Free Electron-Positron Field . . .

201 201 202 204 213 214 215

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17.5 Physical Content of Positive- and Negative-Frequency Solutions of Equations for Free-Field Operators . . . . . . . . . . . . . . . . . .

217

18 Quantization of the Free Scalar Field 18.1 Commutation Relations. Commutator Functions . . . . . . . . . . . 18.2 Complex Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Operator Relations for Dynamic Invariants . . . . . . . . . . . . . .

218 218 220 221

19 Quantization of the Free Spinor Field 19.1 Commutator Functions of Fermi Fields . . . . . . . . . . . . . . . . 19.2 Dynamic Invariants of a Free Spinor Field . . . . . . . . . . . . . .

222 222 224

20 Quantization of the Vector and Electromagnetic Fields. Specific Features of the Quantization of Gauge Fields 20.1 Quantization of the Complex Vector Field . . . . . . . . . . . 20.2 Quantization of an Electromagnetic Field . . . . . . . . . . . 20.2.1 Specific Features and Difficulties of the Quantization Electromagnetic Field . . . . . . . . . . . . . . . . . 20.2.2 Gupta–Bleuler Formalism . . . . . . . . . . . . . . . 20.2.3 Canonical Method of Quantization . . . . . . . . . . . 20.3 On the Quantization of Gauge Fields . . . . . . . . . . . . . . 21 CPT . Spin and Statistics 21.1 The Transformation of Charge Conjugation . . . . . . . 21.2 The Transformation of Space Reflection . . . . . . . . . 21.3 The Transformation of Time Reversal . . . . . . . . . . 21.4 CP T -Theorem and the Connection of Spin and Statistics 21.5 Proof of the Pauli Theorem . . . . . . . . . . . . . . . .

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22 Representations of Commutation and Anticommutation Relations 22.1 General Structure of the Fock Space . . . . . . . . . . . . . . . . . 22.2 Representations of Commutation Relations for a Free Real Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.1 The Fock Space of Free Scalar Bosons . . . . . . . . . . . . 22.2.2 Operators of Creation and Annihilation in the Fock Space. Momentum Representation . . . . . . . . . . . . . . . . . . 22.2.3 Vacuum State of Free Particle. Cyclicity of Vacuum. Set of Exponential Vectors . . . . . . . . . . . . . . . . . . . . . 22.2.4 Construction of Representations of Commutation Relations for a Complex Scalar Field . . . . . . . . . . . . . . . . . .

225 225 229 229 232 236 238 240 241 242 243 246 248 250 250 252 252 252 256 259

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22.2.5 Construction of Representations of Commutation Relations in the Configuration Space. Relativistic Invariance of a Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Representation of Anticommutation Relations of Spinor Fields . . . 22.3.1 Representation of Anticommutation Relations of the Operators of Creation and Annihilation of Fermions and Antifermions . . . . . . . . . . . . . . . . . . . . . . . . . 22.3.2 Representation of Anticommutation Relations in the Configuration Space . . . . . . . . . . . . . . . . . . . . . 22.4 Space of States of a Free Electromagnetic Field . . . . . . . . . . . 22.5 Space of Occupation Numbers . . . . . . . . . . . . . . . . . . . . 23 Green Functions 23.1 Green Functions of the Scalar Field . . . . . . . . . . . . . . . . 23.2 The Green Functions of Spinor, Vector, and Electromagnetic Fields 23.3 Time-Ordered Product and Green Functions . . . . . . . . . . . . 23.4 Wick Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4.1 Wick Theorem for Normal Products . . . . . . . . . . . . 23.4.2 Wick Theorem for a Time-Ordered Product . . . . . . . . 23.4.3 Generalized Wick Theorem . . . . . . . . . . . . . . . . 23.5 Operation of Multiplication and the Regularization of Distributions 23.6 N -Point Green Functions of Free Fields . . . . . . . . . . . . . .

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24 Problems to Part IV

260 262

262 265 267 271 274 274 277 278 279 279 281 284 284 285 287

V Quantum Theory of Interacting Fields. General Problems 25 Construction of Quantum Interacting Fields and Problems of This Construction 25.1 Formal Construction of a Quantum Field . . . . . . . . . . . . . . . 25.2 Mathematical Problems of Construction of a Quantum Interacting Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Scattering Theory. Scattering Matrix 26.1 Quantum Description of Scattering. Definition of Scattering Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 Formal Construction of the Scattering Operator by the Method of Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . 26.3 Main Properties of the S -Operator . . . . . . . . . . . . . . . . . . 26.3.1 Normal Form of the Operator S . . . . . . . . . . . . . . . 26.3.2 Invariance of the Scattering Matrix under Lorentz Transformations and Transformations of Charge Conjugation

291 291 294 298 298 302 305 305 309

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26.3.3 Unitarity of the Scattering Operator . . . . . . . . . . . . 26.3.4 Law of Conservation of Energy . . . . . . . . . . . . . . 26.3.5 Matrix Elements of the S-Operator and the Scattering Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 26.4 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 26.4.1 Feynman Diagrams for the S-Operator . . . . . . . . . . 26.4.2 Feynman Diagrams for Coefficient Functions of the S-Operator . . . . . . . . . . . . . . . . . . . . . . 26.4.3 Feynman Diagrams for Matrix Elements of the S -Operator 26.5 Effective Cross-Sections and Scattering Matrix . . . . . . . . . . 26.5.1 Classical Picture . . . . . . . . . . . . . . . . . . . . . . 26.5.2 Quantum Picture . . . . . . . . . . . . . . . . . . . . . .

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313 316 317

. . . . .

317 319 322 323 325

27 Equations for Coefficient Functions of the S -Matrix 27.1 Creation and Annihilation Operators of External Lines of Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2 Equations of the Resolvent Type . . . . . . . . . . . . . . . . . . . 27.3 Equations of the Evolution Type . . . . . . . . . . . . . . . . . . .

327

28 Green Functions and Scattering Matrix 28.1 Green Functions and the S-Matrix in the Interaction Picture . . . . . 28.2 Schwinger Equation for Green Functions . . . . . . . . . . . . . . . 28.3 On the Relationship between the Green Functions and the Coefficient Functions of the Scattering S-Operator . . . . . . . . . . . . . . . . 28.4 Equations for Green Functions in Terms of Functional Derivatives . 28.5 Equations for Truncated Green Functions . . . . . . . . . . . . . . 28.6 Equations for One-Particle Irreducible Green Functions. Dyson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.7 Spectral Representation of the 2-Point Green Function (Källén–Lehmann Representation) . . . . . . . . . . . . . . . . . . 29 On Renormalization in Perturbation Theory 29.1 Primitively-Divergent Diagrams. Separation of Divergences by the Pauli–Villars Method . . . . . . . . . . . . . . . . . . . . . . . . . 29.2 Degree of Divergence of Feynman Diagram . . . . . . . . . . . . . 29.3 Elimination of Divergences by the Method of Bogoliubov–Parasiuk R-Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4 R-Operation and Counterterms of a Lagrangian . . . . . . . . . . . 29.5 Classification of Interactions: Renormalizable and Nonrenormalizable Theories . . . . . . . . . . . . . . . . . . . . .

328 331 334 336 336 338 341 342 344 347 353 358 358 365 368 374 379

xviii

Contents

29.6 Relationship between Counterterms and the Renormalization of Main Constants of the Theory . . . . . . . . . . . . . . . . . . . . . . . . 29.7 Equivalent Types of Renormalizations . . . . . . . . . . . . . . . . 30 Method of Functional (Path) Integrals in Quantized Field Theory 30.1 Notion of Path Integration and Main Formulas . . . . . . . . . . 30.2 Formalism of Feynman Integrals (Path Integrals) in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 Formalism of Feynman Integrals for Systems with Constraints . 30.4 Path Integral Representation for Scalar Fields . . . . . . . . . . 30.5 Path Integral Representation for Fermi Fields . . . . . . . . . .

380 387

. .

391 392

. . . .

400 405 409 411

. . . .

31 Problems to Part V

416

VI Axiomatic and Euclidean Field Theories 32 Wightman Axiomatics 32.1 Wightman Axioms for Real Scalar Fields . . . . . . . . . . . . . . 32.2 Wightman Functions and Their Properties . . . . . . . . . . . . . . 32.3 Reconstruction Theorem . . . . . . . . . . . . . . . . . . . . . . .

423 423 425 427

33 Other Axiomatic Approaches 33.1 Haag–Ruelle Scattering Theory (HRST) . . . . . . . . . . . . . . . 33.2 Lehmann–Symanzik–Zimmermann Axiomatics . . . . . . . . . . . 33.3 Bogoliubov–Medvedev–Polivanov (BMP) Axiomatic Approach . .

429 429 432 435

34 Euclidean Field Theory 34.1 Analytic Continuation of Feynman Amplitudes . 34.2 Operators of Free Euclidean Fields . . . . . . . . 34.2.1 Real scalar field . . . . . . . . . . . . . . 34.2.2 Euclidean Fermi fields . . . . . . . . . . 34.3 Euclidean Green Functions of a Free Scalar Field 34.4 Euclidean Green Functions of Interacting Fields .

. . . . . .

439 440 442 442 444 446 447

35 Euclidean Axiomatics 35.1 Analytic Continuation of Generalized Wightman Functions . . . . . 35.2 Euclidean Green Functions. Osterwalder–Schrader Axioms . . . . . 35.3 Reconstruction of the Wightman Theory . . . . . . . . . . . . . . .

453 453 455 457

36 Problems to Part VI

460

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

xix

Contents

VII Quantum Theory of Gauge Fields 37 Quantum Electrodynamics (QED) 37.1 Quantization of Interacting Electromagnetic Fields . . . . . . . . . 37.1.1 Gupta–Bleuler Formalism for Interacting Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . 37.1.2 Quantization of Interacting Electromagnetic Fields in the Coulomb Gauge . . . . . . . . . . . . . . . . . . . . 37.1.3 Photon Propagator and Gauge Conditions . . . . . . . . . . 37.2 S-Matrix in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.2.1 Perturbation Theory. Feynman Diagrams . . . . . . . . . . 37.2.2 Coefficient Functions of the S-Matrix in Terms of Creation and Annihilation Operators of Lines of Feynman Diagrams . 37.2.3 Furry Theorem . . . . . . . . . . . . . . . . . . . . . . . . 37.2.4 Gauge Invariance for Coefficient Functions of the S-Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 37.3 Equations for Green Functions and Coefficient Functions of the S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.3.1 Schwinger Equation . . . . . . . . . . . . . . . . . . . . . 37.3.2 System of Equations for Self-Energy and Vertex Parts of Green Functions . . . . . . . . . . . . . . . . . . . . . . 37.4 Divergences in QED and Methods for Their Elimination . . . . . . 37.4.1 Primitively-Divergent Diagrams and Their Regularization . 37.4.2 Mass and Charge Renormalization of Electron (Positron) . . 37.5 Spectral Representations of 2-Point Green Functions . . . . . . . . 38 Quantization of Gauge Fields 38.1 Path Integral for Green Functions in QED (Coulomb Gauge) . . 38.2 Covariant Gauges: Popov–Faddeev–de Witt Method . . . . . . . 38.3 Covariant Quantization of Electromagnetic Interaction . . . . . 38.3.1 Connection between Different Gauges . . . . . . . . . . 38.3.2 Ward Identity . . . . . . . . . . . . . . . . . . . . . . . 38.4 Quantization of Yang–Mills Fields Interacting with Matter Fields 38.5 Faddeev–Popov Ghosts . . . . . . . . . . . . . . . . . . . . . . 38.6 BRST-Invariance . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

39 Standard Models of Interactions 39.1 Renormalization of Gauge Theories . . . . . . . . . . . . . . . . . 39.2 On the Masses of Gluons and Spontaneous Symmetry Breakdown . 39.2.1 Connection of the Radius of Interaction and the Mass of Exchange Bosons . . . . . . . . . . . . . . . . . . . . . . .

465 466 466 467 469 471 471 474 476 479 480 480 482 485 485 490 493 498 499 502 506 507 508 510 514 516 521 521 530 530

xx

Contents

39.2.2 Are Theories with Nonzero Mass of Exchange Bosons Renormalizable? . . . . . . . . . . . . . . . . . . . . . 39.2.3 Spontaneous Breakdown of the U.1/-Symmetry . . . . . 39.2.4 Spontaneous Breakdown of the Local S U.N /-Symmetry 39.3 Models of Interactions of Elementary Particles . . . . . . . . . . 39.3.1 Strong Interaction. Model of QCD . . . . . . . . . . . . 39.3.2 Weak and Electroweak Interactions . . . . . . . . . . .

. . . . . .

. . . . . .

532 533 535 537 537 539

40 Problems to Part VII

542

Appendix

544

Hints for the Solution of Problems

Bibliography

549

Index

562

Chapter 0

Introduction

Quantum field theory (QFT) was created with the aim to give a mathematical background to the physics of elementary particles and can be regarded, in fact, as relativistic quantum mechanics, i.e., it describes quantum phenomena with regard for special relativity theory. This description is based on the hypothesis of wave–particle duality according to which the fields and particles are not two different types of objects but should be regarded as two different types of description used for the investigation of the same physical object. As bright experimental confirmations of this hypothesis, we can mention the discovery of photoelectric emission in the description of which light (electromagnetic waves) is regarded as a flux of particles (photons) and the discovery of the diffraction (wave properties) of electrons. At the beginning of the 20th century, only two types of elementary particles were known: electrons and protons. At present, their number exceeds two hundred. However, even the notion of elementary particle is relative. Thus, according to the theory of representations of the Lorentz group used as the theoretical foundation of relativistic phenomena (and discussed in the next chapter), any particle with specific values of mass and spin (e.g., an atom) is elementary. However, if we consider the interaction of elementary particles at higher energies, then we observe the decomposition of atoms into more elementary objects—protons and neutrons. Moreover, as the energy increases further, we observe the formation of new particles as a result of the decay of some particles or their collisions with other particles. The higher the energy of interaction of elementary particles, the more complicated physical phenomena are observed in the microworld, and the more complex theory is required for the theoretical analysis of these phenomena. Thus, at low energies, the process of scattering of elementary particles is elastic. This phenomenon can be described even within the framework of ordinary classical mechanics or classical field theory. To study more complicated atomic phenomena, it is necessary to apply the formalism of quantum mechanics, whereas the analysis of the processes of creation of particles requires the methods of quantum field theory. Note that the theory of interaction of elementary particles was historically developed in exactly this way, which was explained by the gradual increase in the energy of experiments. Hence, the basic postulates and concepts of classical mechanics and classical wave mechanics as well as the fundamental ideas of the quantum mechanics, were predominant in the construction of classical and quantum field theories. Clearly, it is worth noting that, despite the fact that the indicated heredity was decisive for the construction of quantum theory, it could also be the main cause of the appearance of numerous mathematical

2

Chapter 0

Introduction

Classical Mechanics: c D 1I „ D 0I N < 1 . Classical Field Theory: c < 1; „ ¤ 0; N D 1, u.x/ — (c-number)

& Quantum Mechanics: c < 1; „ ¤ 0; N < 1,  u.x/ — wave function (c-number) & . Quantum Field Theory: c < 1; „ ¤ 0; N D 1, u.x/ is a quantum field (q-number)

Figure 0.1. Relationship between theories.

problems in quantum field theory. This question can be a topic of subsequent investigation. The relationship between the theories mentioned above can be represented in the form of a block diagram (Fig. 1). In this diagram, we use the following notation: c is the velocity of information transfer (velocity of light in vacuum); „ D h=2, where h is the Planck constant; N is the number of degrees of freedom in the system; u.x/ is a field or the wave function of a particle. Note that the notion of field does not exist in classical mechanics, and u.x/ is an ordinary function (c-number) in classical field theory. In quantum mechanics, u.x/ is a wave function and also a c-number, whereas in quantum field theory, u.x/ is a q-number, i.e., it has the operator nature. Another historical aspect of construction of quantum field theory is connected with the notion of interaction between particles. Parallel with gravitational forces, there are three known types of interaction: weak, electromagnetic, and strong. Each of these types of interaction is characterized by its own coupling constant specifying the strength of interaction. Moreover, the forces of weak interaction act between all particles (except photons), the electromagnetic interaction exists between charged particles and photons, and the forces of strong interaction act between hadrons. The first attempts to describe the physical processes were connected with models dealing with a single specific interaction. Thus, the theories of quantum electrodynamics and the weak and strong interactions were developed by analyzing a single type of interactions. However, in numerous cases, physical phenomena are governed by both elec-

Chapter 0

Introduction

3

tromagnetic and strong interactions or even by all three types of interactions. Hence, the idea to develop a model combining all types of interactions is quite natural. The realization of this idea is closely connected with the analysis of the symmetries of elementary particles, which can be regarded as an extremely important part of the physics of elementary particles. The indicated properties of symmetry can be split into two classes: space–time symmetries and internal symmetries. From the mathematical viewpoint, the properties of symmetry are described by the corresponding groups of symmetry. The laws of interaction of elementary particles are invariant under certain symmetry groups. Moreover, the space–time symmetries are related to the geometry of the space, where the interaction of particles occurs, and are identical for all elementary particles. The invariance under a group of internal symmetry may be true only for a certain family of particles with a given specific form of interaction. Moreover, the fact of breaking of this symmetry may serve as an indication of the appearance of new (possibly unknown) particles in the system. Hence, quantum field theory is constructed on the basis of the laws of classical and quantum mechanics and the analysis of various symmetry groups connected with the physical laws of the microworld of elementary particles. In the history of the quantum theory of interacting fields, it is possible to distinguish three important periods of formation and development of close but, at the same time, fundamentally different concepts of introduction of interaction. Thus, within the framework of the standard approach, the interaction of fields is, in fact, postulated, although this postulate is based on the mechanism of introduction of electromagnetic interaction, whose substantiation is discussed in detail in Chapter 4 (Remark 4.3). Another approach is based on the principle of gauge invariance, which generalizes the principle of gradient invariance of electromagnetic fields. According to this principle, the Lagrangian of interaction is not postulated. It is derived from the corresponding principle of gauge invariance regarded as fundamental for the construction of the theory. This principle was proposed by Yang and Mills in 1954 [224]. However, this theory became indeed popular only at the end of the 1960s and the beginning of the 1970s after the works by L. D. Faddeev and V. N. Popov [47], B. S. De-Witt [34, 35, 36], G. ’t Hooft [198, 199, 201], A. A. Slavnov [186, 187], J. C. Taylor [196], etc. They marked the onset of a tremendous boom. In fact, the theory of gauge fields “saved” the standard approach to the theory of quarks (more exactly, to the theory of strong interactions) and led to the formation of an independent branch of the theory of elementary particles, namely, quantum chromodynamics. Actually, the beginning of the 1970s can also be regarded as the onset of the development of the theory of weak interactions and their merging into a single model of electroweak interaction. In addition, the development of nonstandard models gave no success in overcoming the mathematical problems encountered in the construction of the theory and even increased their number and complexity. This becomes especially clear in the attempts to include in the theory the gravitational interaction, which may significantly affect the physical processes running at superhigh energies.

4

Chapter 0

Introduction

A quite natural idea that the existing theory is not fundamental appeared in connection with the accumulation of numerous problems in the theory of interacting fields whose main purpose is, in fact, the construction of the unified theory of elementary particles. At the end of the last century, numerous researchers believed that the theory of relativistic string may play the role of the basis for the construction of the contemporary theory of interacting fields. However, the analysis of this theory lies beyond the scope of our book. Thus, we refer the reader, e.g., to the works [7, 79, 161, 232, 5]. It would be quite helpful to carefully read the historical surveys of the formation and development of the theory of quantized fields [212, 213] and the evolution of concepts and new ideas in the theory of the structure of elementary particles based on the theory of quarks [87, 105]. It would also be especially useful to read the historical essays devoted to the 50th anniversary of the Yang–Mills theory in the collection of papers edited by G. ’t Hooft [200]. In our book, we present a detailed analysis of the problems of construction of standard models and models of gauge fields.

Part I

Symmetry Groups of Elementary Particles

Part I Symmetry Groups of Elementary Particles

7

The symmetry properties are of primary importance for the investigation of various physical phenomena in the physics of interacting elementary particles. First of all, these relates to the classification of particles within the framework of the corresponding types of interactions, construction of the Lagrangians, determination of the integrals of motion (conserved quantities), etc. The properties of symmetry are described by the corresponding symmetry groups and an elementary particle is regarded as an object whose states form the basis of irreducible representation of a certain symmetry group. In Chapter 1 of this part, we briefly dwell upon the problems of relativistic invariance (covariance). This is a natural requirement to any physical theory because all physical laws should be independent of the reference system. The invariance of laws of the physics of elementary particles under the proper inhomogeneous Lorentz group is regarded as a fundamental postulate of the quantum field theory.

Chapter 1

Lorentz Group

In this chapter, we present some basic facts from the theory of the Lorentz group and its representations. For the more comprehensive analysis of these problems, the reader is referred to the works [204, 62, 134, 181, 185].

1.1

Euclidean and Minkowski Spaces. Relativistic Notation

The ordinary s-dimensional Euclidean space Es D Rs is defined as the Cartesian product .R1 /s (R1 is the real axis) in which the distance (metric) between two points x D .x 1 ; : : : ; x s / and y D .y 1 ; : : : ; y s / is given by the formula q E .x; y/ WD .x 1  y 1 /2 C    C .x s  y s /2 WD jx  yj: For s D 3; this space is denoted simply by E. The Minkowski space M D M4 (Md for d ¤ 4) is also defined as the Cartesian product R1  R3 but the real axis R1 D cT , where T D Œ1; C1, is the time axis and the metric structure of the Minkowski space can be defined as the structure of a 4-dimensional affine space with linear form M .x; y/2 D .x 0  y 0 /2  .x 1  y 1 /2  .x 2  y 2 /2  .x 3  y 3 /2 :

(1.1)

It is convenient to introduce a metric tensor G D .g /: g  D g ; g 00

D

g 11

g  D 0; D

g 22

if  ¤ ;

D g 33 D 1:

(1.2)

The points of the space x 2 M are called contravariant coordinate vectors and their coordinates are denoted by x  ,  D 0; 1; 2; 3. Under the transformations of coordinates, all vectors a transformed according to the same law as x  are called contravariant and their components are denoted by a : The vectors b transformed as @ / are called covariant and denoted by the 4-dimensional gradient @ WD .@ WD @x  b (with subscripts). The Greek sub- and superscripts .; ; ˛; ˇ; : : :/ are used to denote the space-time components varying from 0 to 3 (or from 0 to s in the d D .s C 1/-dimensional theory). The Roman letters .i; j; k; a; b; c; : : :/ are used only for the space components and vary from 1 to 3 (or from 1 to s). They are also used to indicate the components of the fields.

Section 1.1 Euclidean and Minkowski Spaces. Relativistic Notation

9

The transition from contravariant components to covariant components is realized by the tensor g : a D

3 X

g a WD g a

or

a D g  a :

D0

Here and in what follows, we assume that summation is carried out over the repeated indices. In some cases where summation is absent, this fact is specially indicated. Vectors from the Euclidean space E are marked in boldface. To avoid confusion in the notation, it is accepted that .a /3D0 D .a0 ; a1 ; a2 ; a3 / D .a0 ; a/; .a /3D0 D .a0 ; a1 ; a2 ; a3 / D .a0 ; a/; whereas for the 4-gradient, these rules are modified: .@ /3D0 D .@0 ; @1 ; @2 ; @3 / WD .@0 ; r /; .@ /3D0 D .@0 ; @1 ; @2 ; @3 / WD .@0 ; r /:

(1.3)

The Euclidean space Es turns into a complete Hilbert space if we introduce the scalar product of two vectors a and b by the formula ab D

s X

ai b i D ai b i :

iD1

Note that the scalar Euclidean product of two vectors a and b from E4 is defined as follows: (1.4) .a; b/E4 D a0 b 0 C a  b D a b  : The Minkowski space M is sometimes called a Hilbert space with indefinite or pseudo-Euclidean metric defined by formula (1.1) and scalar product a  b D .a; b/M D g a b  D .a; Gb/E4 D a0 b 0  a  b:

(1.5)

For this definition, the square of a vector v 2 M may take both positive and negative values. A vector v is called timelike vector if v 2 D v  v D .v; v/M > 0, spacelike vector if v 2 < 0, and isotropic or lightlike vector if v 2 D 0. Every isotropic vector v is associated with a one-dimensional isotropic subspace v WD ¹u 2 M j u D v,  2 Rº called a light ray. The union of all light rays (playing the role of generatrices) form the surface of a light cone C . In the ordinary 4-dimensional coordinate system, the equation of this cone takes the form C W s 2 WD M .0; v/2 D c 2 t 2  x 2  y 2  z 2 D 0;

v D .t; x; y; z/:

(1.6)

10

Chapter 1

Lorentz Group

x0 d

b

θ x

0

a

c

Figure 1.1. Light cone.

As a convenient representation of this surface, we can use its two-dimensional image (see Fig. 1.1) regarded as the projection of surface (1.6) onto the coordinate plane ¹x 0 ; x 1 º. From the physical point of view, this figure can be interpreted as the diagram of propagation of two signals with velocity c D cot  along the straight lines .ab/ and .cd / through the point 0, i.e., of an infinite number (continuum) of rays of this sort along surface (1.6). Surface (1.6) decomposes the space M into four domains. These domains are also called cones: V˙ WD ¹x 2 M j x 2 > 0; ˙x 0 > 0º; 2

or

0

V ˙ WD ¹x 2 M j x > 0; ˙x > 0º:

(1.7)

Moreover, VC .V C / is called an open (closed) upper cone or a future cone and V .V  / is called an open (closed) lower cone or a past cone. The cone N WD ¹x 2 M j x 2 < 0º

(1.8)

including all spacelike coordinate vectors is sometimes called a lateral cone. Remark 1.1. In what follows (with few exceptions), we use a customary system of units in which c D „ D 1. In this system of units, the energy, mass, inverse length, and inverse time have the same dimension. In order to restore the constants c and „ in the analyzed formulas, it is necessary to perform the change of variables X ! „˛X c ˇX X

(1.9)

in each quantity X appearing these formulas. The values of the exponents ˛X and ˇX can readily be found by analyzing the dimensions of the terms containing the

11

Section 1.2 Homogeneous Lorentz Group

quantities X . In addition, it is necessary to take into account the relationship between the relativistic quantities x 0 and p 0 and the nonrelativistic quantities t and E: x 0 D ct;

p0 D

E : c

Thus, for the momenta p  ; coordinates x  ; and mass m; we have ˛p D ˛m D 0, ˛x D 1, ˇp D 1, ˇx D 1, and ˇm D 2. This follows from the fact that the relativistic relationship between the energy and mass E 2 D c 2 p2 C m2 c 4 in the system of units with „ D c D 1 takes the form p 2 D m2 in the classical case and pO 2 D m2 , pO  D i @ in the quantum case.

1.2

Homogeneous Lorentz Group

e is defined as a group of real homogeDefinition 1.2. The general Lorentz group L neous linear transformations of coordinates of the Minkowski space M preserving the quadratic form (1.1) and the scalar product (1.5) of coordinate vectors x; y 2 M. The general form of a homogeneous linear transformation is given by the system of equations  (1.10) x 0 D ƒ x or, in the matrix form, x 0 D ƒx;

.ƒ/ WD ƒ ;

e: ƒ2L

(1.11)

Thus, it follows from Definition 1.1 and relations (1.4) and (1.5) that .x 0 ; y 0 /M D .x 0 ; Gy 0 /E4 D .ƒx; Gƒy/E4 D .x; ƒT Gƒy/E4 D .x; y/M D .x; Gy/E4 : This yields ƒT Gƒ D G

 or .ƒT /˛ g .ƒ/ˇ D ƒ ˛ g ƒˇ D g˛ˇ ;

(1.12)

where ƒT is the matrix obtained as a result of the transposition of ƒ. In view of the fact that det ƒT D det ƒ; relations (1.12) imply the following condition: .det ƒ/2 D 1

or det ƒ D ˙1:

(1.13)

Hence, the inverse transformation exists for any transformation ƒ: The product of two transformations ƒ1 and ƒ2 is defined as the transformation formed by the consecutive action of the transformations ƒ1 and ƒ2 . The transformation ƒ2 ƒ1 is

12

Chapter 1

Lorentz Group

defined by a matrix equal to the product of matrices of the transformations ƒ2 and ƒ1 . The equality .ƒ2 ƒ1 x; ƒ2 ƒ1 y/M D .x; ƒT1 ƒT2 Gƒ2 ƒ1 y/E4 D .x; ƒT1 Gƒ1 y/E4 D .x; Gy/E4 D .x; y/M implies that the product ƒ2 ƒ1 of two Lorentz transformations is also a Lorentz transformation. In other words, the collection of all Lorentz transformations forms a group. We now perform a brief analysis of all significant subgroups of the homogeneous Lorentz group playing independent roles in the investigation of symmetries in the theory of interacting particles. The Lorentz group contains a subgroup isomorphic to the group of three-dimensional orthogonal transformations O3 . In E4 ; this group is specified in the matrix form as follows:   1 0 ; ƒO3 D 0 R where R is a real 3  3 matrix such that RT R D RRT D 1: Further, relation (1.12) (with ˛ D ˇ D 0) implies that .ƒ00 /2 D 1 C

3 X iD1

.ƒi0 /2 D 1 C

3 X

.ƒ0i /2  1;

(1.14)

iD1

i.e., ƒ00  1 or ƒ00 1. The Lorentz transformations for which ƒ00  1 are called orthochronous Lorentz transformations. These transformations also form a group. The transformations of this semigroup possess the following important property: Proposition 1.3. The orthochronous Lorentz transformations preserve the cone of timelike positive vectors, i.e., transfer the vectors x for which x 0 > 0 and x  x D x 2 > 0 into the vectors x 0 for which x 00 > 0 and x 0  x 0 D x 02 > 0. Proof. For any timelike vector x D .x 0 ; x/ and any orthochronous transformation  ƒ D .ƒ /; we define a vector ƒ0 D .ƒ01 , ƒ02 ,ƒ03 / 2 R3 and write the Schwartz inequality for the scalar product of the vectors x and ƒ0 in R3 : j.ƒ0 ; x/R3 j jƒ0 j jxj: Further, by using (1.14), we represent (1.15) in the form q p jƒ01 x 1 C ƒ02 x 2 C ƒ03 x 3 j .ƒ00 /2  1 x2 :

(1.15)

(1.16)

13

Section 1.2 Homogeneous Lorentz Group

Then, for the timelike vectors .x2 < .x 0 /2 / such that x 0 > 0 and the Lorentz transformations with ƒ00 > 1; the right-hand side of (1.16) is smaller than ƒ00 x 0 . Hence, the transformed vector x 0 D ƒx possesses the component x 00 D ƒ00 x 0 C ƒ01 x 1 C ƒ02 x 2 C ƒ03 x 3 > ƒ00 x 0  jƒ01 x 1 C ƒ02 x 2 C ƒ03 x 3 j > 0: The fact that x 0 is a timelike vector follows from the definition of the Lorentz transformation .x 02 D x 0  x 0 D x  x D x 2 > 0/: Depending on the values of det ƒ and ƒ00 ; the Lorentz group can be split into four subsets (components): 1. a subgroup of proper orthochronous Lorentz transformations (or simply a proper Lorentz group) (sometimes it is also called a restricted homogeneous Lorentz " group) LC with det ƒ D C1 and ƒ00  1; #

2. proper nonorthochronous transformations .LC / with detƒ D C1 and ƒ00 1; 3. improper orthochronous transformations .L" / with det ƒ D 1 and ƒ00  1; 4. improper nonorthochronous transformations .L# / with det ƒ D 1 and ƒ00 1. Remark 1.4. The subgroup of proper orthochronous transformations is a 6-parameter Lie group. It is called a proper Lorentz group. The transformations from the other three components (they do not form subgroups of the Lorentz group) can be obtained from the proper orthochronous transformations as a result of the following three discrete transformations: 1. space reflection (or parity transformation):

2. time reversal:

b x D ƒis .x 0 ; x/ D .x 0 ; x/I P

(1.17)

bx D ƒi t .x 0 ; x/ D .x 0 ; x/I T

(1.18)

bT bx D ƒist .x 0 ; x/ D .x 0 ; x/: P

(1.19)

3. space-time reflection:

Thus, the general Lorentz group admits the following symbolic representation: e D L" CP bL " C T bL " C P bT bL " : L C C C C

14

Chapter 1

Lorentz Group

Remark 1.5. Note that the continuous transformations from this group are fundab, T b; and P bT b of the general mentally different from the discrete transformations P Lorentz group. This difference is connected with the fact that it is possible to pass from an inertial system to a different inertial system as a result of successive application of infinitesimally small transformations without changing the physical nature of the phenomena or objects, i.e., guaranteeing their invariance in various reference systems. At the same time, the space or time reflections may lead to the violation of the nature of some of these phenomena or objects. This follows from the data of experimental observations. e and can be repEvery Lorentz transformation belongs to one of the terms in L " b, T b; resented as the product of transformations from LC and the transformations P " " bT b if they do not belong to L . Hence, it suffices to study the group L and or P C C its possible representations. This group is a 6-parameter Lie group. The number of independent parameters specifying the group of proper orthochronous Lorentz transformations can be related to the rotation in six planes .x  x  / by angles ˛  . From the general theory of Lie groups, it is known that every Lorentz transformation can be specified by its generator. To find the generator, we consider infinitesimally small rotations by angles " . Thus, we get 1 ƒ D ƒ."/ D 1 C " l C O."2 /: 2 The generators satisfy the relation l D l and are linear operators (matrices) in the Minkowski space. Their matrix elements can readily be found (see, e.g., [181], Chapter 2, Section 2) and have form .l /˛ˇ D g˛ ıˇ C g˛ ıˇ : It is easy to see that they satisfy the following commutation relations: Œl ; l˛ˇ  D g˛ lˇ C gˇ l˛  gˇ l˛  g˛ lˇ : The linear span of the matrices l with real coefficients forms the Lie algebra of the Lorentz group.

1.3

Inhomogeneous Lorentz Group–Poincaré Group

It is easy to see that the pseudo-Euclidean form (1.1) between two points in the Minkowski space is invariant not only under the homogeneous transformations (1.10), (1.11) but also under the space-time translations and, hence, under more general linear transformations:   (1.20) x 0 D .Lx/ D ƒ x Ca :

15

Section 1.4 Complex Lorentz Transformations

These transformations are given by the matrix ƒ and a 4-vector a and denoted by L D .ƒ; a/. The set of transformations (1.20) forms a group if the product of two transformations L1 D .ƒ1 ; a1 / and L2 D .ƒ2 ; a2 / is defined as follows: L D L2 L1 D .ƒ2 ƒ1 ; a2 C ƒ2 a1 /: "

The family of inhomogeneous transformations (1.20) with ƒ 2 LC form a 10parameter continuous Lie group. The generators of shifts t along the axes x  are commuting and satisfy the following commutation relations with the generators of rotations (see [204, 62, 134, 181]): bl ; t˛ c D .g˛ t  g˛ t /: The operator L corresponding to the pure translation by a 4-vector a has the form L.1; a/ D e ia

1.4

 g  t 

D e ia

t 

:

Complex Lorentz Transformations

In analyzing the analytic properties of functions invariant under transformations from " the group LC , it is necessary to extend Definition 1.1 to the complex Minkowski space MC D MC 4 , where it is possible to define the following bilinear form for z1 D x1 C iy1 and z2 D x2 C iy2 with xi , yi 2 M, i D 1; 2: 

z1  z2 D .z1 ; z2 /MC D g z1 z2 :

(1.21)

Definition 1.6. The group of complex Lorentz transformations L .C/ is a group of transformations of the space MC preserving the quadratic form (1.21), i.e., .ƒz1 ; ƒz2 /MC D .z1 ; z2 /MC :

(1.22)

It is easy to see that requirement (1.22) leads to the same relation (1.12) for the complex matrices ƒ and to the same condition (1.13). Hence, L .C/ can be represented as the sum of two components L .C/ D LC .C/ [ L .C/;

(1.23)

where LC .C/ and L .C/ correspond to the transformations with det ƒ D C1 and det ƒ D 1; respectively. Unlike real transformations, LC .C/ is already a connected Lie semigroup of the group L .C/ (see, e.g., [23], Part V, Chapter 1.2). The generalization to the case of complex inhomogeneous transformations is trivial.

16

1.5

Chapter 1

Lorentz Group

Representations of the Lorentz and Poincaré Groups, Field Functions, and Physical States

In the quantum mechanics, every particle is associated with a wave function satisfying the Schrödinger equation and its physical meaning is connected with the probability density of finding the corresponding particle in a certain quantum state. In the relativistic theory, this role is played by a field function u (or a field) which is, generally speaking, multicomponent and depends not only on the space-time variables but also on the other (internal) variables, such as mass, spin, etc.). As a rule, this function satisfies a differential equation (or a system of equations in the case of interaction of many particles). The transition from one reference system to another reference system performed with the help of Lorentz transformations should not change the laws of motion, i.e., the indicated equation(s) must remain invariant. This means that the field u.x/ varies according to the following linear law: u.x/ ! u0 .x 0 / D S.L/u.x/ D S.ƒ; a/u.x/;

(1.24)

where the operator S.L/ is determined by the Lorentz transformation L D .ƒ; a/. Thus, every Lorentz transformation L D .ƒ; a/ corresponds to a linear operator S.L/: Moreover, it is clear that the unit element of the Lorentz group corresponds to the identity operator S.1; 0/ D 1 and the product of two elements of the Lorentz group corresponds to the product of two linear S.L/-transformations: S.L2 /S.L1 / D S.L2 L1 /. The system of operators S.L/ corresponding to these properties is called the linear representation of the Lorentz group. The form of the operator S.L/ depends on the space in which it acts, i.e., on the structure of the functions u. In the case where the number of components u.x/ is finite, the operator S.L/ is a finite-dimensional matrix and the corresponding representation is called a finite-dimensional representation of the Lorentz group. Sometimes, the space in which the representation of S.L/ acts can be split into subspaces invariant under the action of all transformations S.L/. In this case, the indicated representation is called reducible. If these subspaces do not exist, then the representation is called irreducible. Since all physical fields known at present are described by functions with finitely many components, for the analysis of the types of encountered fields and their classification, it is necessary to study all possible irreducible finite-dimensional representations of the homogeneous Lorentz group. The investigation of all representations of the Lorentz group can be reduced to the analysis of all irreducible representations of the proper Lorentz group. In what follows, we present some basic characteristics of finite-dimensional spinor representations of the proper Lorentz group. For the detailed description of the structure of these representations, we refer the reader to [62]. Note that each irreducible representation of the proper Lorentz group is determined by a certain integer or half-integer number j: This number is called the weight of the corresponding irreducible representation and corresponds to the maximum eigenvalue of

Section 1.5 Representations of Groups and Physical States

17

the generator of space rotations H3 D M12 : The basis whose vectors are eigenvectors of the operator H3 is called canonical. In order to give an elementary description of the procedure of construction of irreducible representations of the Lorentz group, we establish the relationship between this group and the group of complex unimodular second-order matrices SL.2; c/; i.e., 2  2 matrices A D ..aij //, .i; j D 1; 2/ satisfying the condition det A D a11 a22  a12 a21 D 1. We now consider a family of four matrices  , where 0 D 1; and the remaining three matrices are the well-known Pauli matrices:       0 1 0 i 1 0 ; 2 D ; 3 D : (1.25) 1 D 1 0 i 0 0 1 Every 4-vector of the Minkowski space x  is associated with an Hermitian matrix X by the rule  0  x C x 3 x 1  ix 2  : (1.26) X D x  D x 1 C ix 2 x 0  x 3 In view of the fact that Tr. 0 / D 2ı0 , we find 1 x  D Tr.X /; 2 and det X D .x 0 /2  .x 1 /2  .x 2 /2  .x 3 /2 D x 2 :

(1.27)

For any matrix A 2 SL.2; c/; we consider a transformation X ! X 0 : X 0 D AXAC ;

(1.28)

where AC is the Hermitian-conjugate matrix .AC D AT /. As a consequence of unimodularity of the matrix A (det A D det AC D 1), we have det X 0 D det X , which is equivalent (see (1.27)) to the condition x 02 D x 2 : Thus, according to Definition 1.1, transformation (1.28) describes a homogeneous Lorentz transformation of a 4-vector x  . To determine the relationship between the matrix A and the matrix of a Lorentz transformation ƒ, we substitute X 0 and X from (1.26) in relation (1.28):  C  X 0 D  x 0 D  ƒ  x D A A x :

Multiplying by  and performing the trace operation, we get 1 C ƒ  D Tr. A A /: 2

(1.29)

18

Chapter 1

Lorentz Group



It follows from relation (1.29) that every transformation ƒ can be specified by   two matrices ˙A, i.e., ƒ .A/ D ƒ .A/. We now consider the quantities called 2-component spinors (or spinors of rank 1),   1=2 : D 1=2 With the help of the matrix A; they are transformed according to the rule: ! 0 W ˛0 D A˛ˇ ˇ ;

˛; ˇ D ˙1=2

or

0 D A :

(1.30)

The spinor can be regarded as a vector in a linear complex two-dimensional space transformed under the Lorentz transformations (1.10) according to law (1.30). Since the correspondence ƒ ! ˙A is two-valued, the spinor is determined to within the sign (in any reference system). As a result of the Lorentz transformation, a couple of complex conjugate numbers  is transformed by using the matrix A D .A˛ˇ / as follows: ˙1=2 

0 ˛ D A˛ˇ ˇ : Instead of ˛ ; we can write ˛P and the corresponding transformation takes the form 0 ˛P D A˛P ˇP ˇP :

(1.31)

Transformation (1.30) realizes a two-dimensional representation of the proper Lorentz group called a spinor representation of rank 1, whereas transforma-tion (1.31) is called the conjugate spinor representation of rank 1. Similarly, we can consider the transformation of spinors of higher ranks: ‰˛1 ;:::;˛m I˛P 1 ;:::;˛P n D A˛1 ˇ1    A˛m ˇm A˛P

P

1 ˇ1

   A˛P

P

n ˇn

‰ˇ

P

P

1 ;:::;ˇm Iˇ1 ;:::;ˇn

;

(1.32)

which is simultaneously a spinor of rank m and a conjugate spinor of rank n. The set of these spinors forms a linear space of dimensionality 2mCn . Relation (1.32) gives a representation of the proper Lorentz group of rank .m; n/. The rank .m; n/ is connected with the weight .j; j 0 / by the formulas m D 2j;

n D 2j 0 :

In view of the fact that the spinors of rank .m; n/ are transformed as the product of m spinors of rank 1 and n conjugate spinors of rank 1, this representation is a product of irreducible representations with weight 1=2 and, in the general case, can be reducible. It turns out (see [62]) that every irreducible representation is realized in the space of symmetric spinors, i.e., spinors that do not change as a result of mutual permutations of indices both within the family ˛1 ; : : : ; ˛m and within the family ˛P 1 ; : : : ; ˛P n : It is

Section 1.5 Representations of Groups and Physical States

19

clear that the dimension of the corresponding linear space of action of the representa0 tion D .j;j / and, hence, the dimension of the matrix S.L/ D S.ƒ; 0/ in relation (1.24) are equal to (1.33) d D .2j C 1/.2j 0 C 1/: 0

It follows from (1.32) that the representation D .j;j / is single-valued if j C j 0 is 0 an integer and two-valued, if j C j 0 is a half integer. The matrix D .j;j / .A/ can 0 be written in the form of direct product of the matrices D .j;0/ .A/ and D .0;j / .A/ D 0 D .j ;0/ .AC /. We also note that the product of two representations D .j1 ;0/ and D .j2 ;0/ is reducible and can be decomposed into the sum D

.j1 ;0/

D

.j2 ;0/

D

jj1X Cj2 j

˚D .j;0/ :

(1.34)

j Djj1 j2 j

We now demonstrate several basic realizations of these representations.

1.5.1 Representation D .0;0/ It follows from (1.33) that the dimension d D 1 and, hence, S.ƒ; 0/ D Cƒ  1, Cƒ 2 C. The space in which this representation is realized is formed by onecomponent functions u.x/. The corresponding field is denoted by '.x/. The law of transformation (1.24) takes the form '.x/ ! ' 0 .x 0 / D Cƒ '.x/: Since the observables always contain invariant bilinear combinations of the form j'.x/j2 , we can set jCƒ j D 1. Moreover, if we consider the real fields, then Cƒ D ˙1. Hence, it is possible to realize two types of representations of rank 0. In the first case, any Lorentz transformation (1.20) corresponds to a transformation of the field '.x/ ! ' 0 .x 0 / D '.x/

(1.35)

equivalent to the transformation of the field function '.x/ ! ' 0 .x/ D '.ƒ1 .x  a//:

(1.36)

This field is called scalar and describes the so-called scalar mesons. The second type of representations is given by transformation (1.35), (1.36) for all continuous ƒ and the law of transformation of the field ' under the space reflection b (see (1.17)) has the form P '.x/ ! ' 0 .x 0 / D '.x/:

(1.37)

This field is called pseudoscalar. It describes pseudoscalar mesons. The particles transformed according to the representation D .0;0/ have spins equal to zero.

20

Chapter 1

Lorentz Group

1.5.2 Representations D . 2 ;0/ and D .0; 2 / 1

1

The dimension of this representation d D 2. The field functions transformed accord1 1 ing to the representations D . 2 ;0/ and D .0; 2 / are called 2-component spinors. The laws of transformation of these quantities are given by relations (1.30) and (1.31). 1 1 The basis vectors of the spaces in which the representations D . 2 ;0/ and D .0; 2 / are realized can be chosen to make the matrices realizing these representations mutually conjugated. The 2-component spinors describe the particles and antiparticles with half-integer spins. In the contemporary scientific literature, the indices ˛ and ˛P are often replaced by 1 the notation L and R used for the spinors on which the representations D . 2 ;0/ 1 and D .0; 2 / are, respectively, realized.

1.5.3 Representation D . 2 ; 2 / 1 1

In view of (1.30)–(1.32), the corresponding spinor ‰˛ˇP is transformed according to the law (1.38) ‰ 0 P .x 0 / D A˛ .ƒ/ANˇP ıP .ƒ/‰ ıP .x/: ˛ˇ

However, by virtue of relation (1.33), the dimension of the representation d D 4. Thus, it is possible to assume that the spinor ‰˛ˇP is transformed as a covariant 4-vector. To check this, by analogy with (1.26), we represent ‰˛ˇP in the form of decomposition: (1.39) ‰˛ˇP .x/ D V  .x/. /˛ˇP : Thus, by using (1.38) and (1.39), we get (see problem 4) V

0

 .x 0 / D ƒ  V .x/;

(1.40)

i.e., relation (1.39) establishes a one-to-one correspondence between the spinor ‰˛ˇP and a covariant 4-vector transformed according to formula (1.40). The other significant point is the invariance of physical states. In the next chapters, we present the definitions of classical and quantum states of a system of elementary particles. The requirement of relativistic invariance means that the transformations of the vectors of states must be realized according to the irreducible unitary representations of the Poincaré group. This means that the generators of representation M and P corresponding, respectively, to the generators of rotations l and the generators of shifts t are Hermitian matrices (or self-adjoint operators in the case of infinitedimensional space). The infinitesimal representation of the operator corresponding to transformation (1.20) has the following form: 1 U.L/ D U.ƒ; ıa/ D 1 C i " M C ıa P C O."2 ; .ıa/2 /: 2

(1.41)

Section 1.5 Representations of Groups and Physical States

21

The generators M and P satisfy the following commutation relations: ŒM ; M˛ˇ  D i.g˛ Mˇ C gˇ M˛  gˇ M˛  g˛ Mˇ /; bM ; P˛ c D i.g˛ P  g˛ P /

(1.42) (1.43)

and bP ; P c D 0:

(1.44)

These relations p differ from the corresponding relations for l and t solely by the factor i D 1. The problem of finding representations of the inhomogeneous Lorentz group is equivalent to the problem of determination of all representations of the commutation relations (1.42)–(1.44) by using the self-adjoint operators M and P . It turns out that the sole finite-dimensional unitary representation of the homogeneous Lorentz group is the trivial representation U.ƒ; 0/ D 1. The representation of the generators P corresponding to shifts along four coordinate axes is constructed in a very simple way because the set of all 4-dimensional shifts is a commutative subgroup of the Poincaré group [see (1.44)]. This means that all irreducible unitary representations of this subgroup are one-dimensional and the operator of shift by a 4-vector a takes the form  (1.45) U.1; a/ D e ia P : In the space of action of the corresponding generators , we can choose a basis in which the operator U.1; a/ simply multiplies a basis vector by the function exp .i a p /. Then all irreducible representations of the inhomogeneous Lorentz group can be classified according to the type of the vector p which can be spacelike, timelike, or isotropic. However, in view of the well-known relationship p 2 D .p 0 /2  p2 D m2 between the energy and momentum of a free relativistic particle, the representations are regarded as important for physical applications provided that p 2 D m2 or p 2 D 0 and p 0 > 0. Hence, the operator P 2 D P  P is an invariant of the group responsible for the spectrum of masses of relativistic particles. In other words, every value of the parameter m  0 characterizes the corresponding representation. The operator 1 W D M M  P˛ P ˛ C M˛ M ˛ P  P 2

(1.46)

is also an invariant of the group and commutes with all generators M and P . Thus, for any irreducible representation of the inhomogeneous Lorentz group, it is a multiple of the identity operator and its eigenvalues can be expressed in terms of the parameter specifying the analyzed representation. For m > 0; in the above-mentioned basis, we can choose a “rest system” in which W D m2 S2 , where S D .S1 ; S2 ; S3 /, and Si ; i D 1; 2; 3, are the generators of an irreducible representation of the 3-dimensional group of rotations. The eigenvalues of the operator S2 are numbers s.s C 1/, where

22

Chapter 1

Lorentz Group

s D 0; 1=2; 1; 3=2; : : : correspond to the values of spin of a particle with mass m. The dimension of the corresponding representation d D 2s C 1. If the mass is equal to zero, then it is impossible to choose the “rest system” because the corresponding relativistic particle moves with the velocity of light c in vacuum and p 0 D jpj. In this case, there are two types of representations. The first type corresponds to the case where both W D 0 and P 2 D 0: Thus, in order to characterize the representations, we choose the following eigenvalues of the operator of chirality: S.p/ D

†p Mp D ; jpj jpj

(1.47)

where M D .M23 ; M31 ; M12 /: The operator S.p/ is studied in Section 6.3.2 in more detail. The eigenvalues of the operator S.p/ correspond to the states of polarization of particle and are given by the formula p D s0 C n;

n D 0; ˙1; ˙2; : : : ;

where s0 D 0 and s0 D 1=2 for single- and two-valued representations, respectively. Hence, for s0 D n D 0; there exists a single state of the relativistic massless particle. At the same time, for p ¤ 0, there are only two independent states corresponding to the two states of polarization: in the direction of motion of the particle and in the opposite direction. Note that only two representations are realized in reality: with p D 1=2 corresponding to neutrinos (if neutrinos are indeed massless particles) and with p D 1 corresponding to photons. Finally, the second type of representations appears if W D ˛ 2 1, where ˛ is a real number. In this case, the corresponding infinite-dimensional representation is characterized by a continuous spin. Most likely, this type of representations does not correspond to any real particles. In conclusion, we note that all significant physical representations of the Poincaré group used in the relativistic theory of elementary particles are characterized by two parameters, .m; s/, where m is the mass of a particle taking arbitrary values m  0 and s D 0; 1=2; 1; 3=2; 2; : : : is the spin of this particle. Note that a more detailed description can be found in [181], Part II, [185]. See also [94].

Chapter 2

Groups of Internal Symmetries

It is known from the classical mechanics that the invariance of a theory with respect to certain symmetry groups (e.g., the groups of space and time translations or rotations) is connected with the existence of the laws of conservation (of energy, momentum, or angular momentum). Hence, the investigations aimed at finding new symmetry groups form one of the most important directions in the development of the theory of elementary particles. In the previous chapter, we have studied the groups connected with 4-dimensional space-time transformations. Parallel with these transformations, there exist transformations under which the space-time coordinates remain invariant but the field functions vary according to the law: x ! x0 D x u.x/ ! u0 .x 0 / D '.x; u/:

(2.1)

Transformations (2.1) are connected with the internal properties of the fields and the corresponding groups of transformations are called the groups of internal symmetries. It is clear that the comprehensive investigation of these groups can be a subject of a separate large monograph. Hence, in the present chapter, we briefly consider only the most frequently encountered class of groups, namely, the class of unitary groups generating the so-called unitary symmetry. The other groups encountered in the investigation of various theoretical aspects of the theory are studied in the context of the analysis of different specific fields and their interactions.

2.1

Abelian Unitary Group U.1/

The Abelian unitary group U.1/ is a one-parameter group generated by phase transformations of the form u.x/ ! u0 .x/ D e igq u.x/; (2.2)    u.x/ ! u0 .x/ D e igq u.x/; where g is the parameter of group transformation and the constant q is introduced for the sake of convenience and corresponds to the charge of particles described by the  fields u.x/. All physical quantities depend on the bilinear combinations u.x/u.x/ or their derivatives and are invariant under transformations (2.2). It is known from the general theory of Lie groups that all irreducible representations of the Abelian (commutative) group are one-dimensional. Hence, they are unitarily equivalent to

24

Chapter 2

Groups of Internal Symmetries

transformations (2.2). Transformations (2.2) are called gauge transformations of the first kind. Remark 2.1. The property of symmetry under transformations (2.2) is not necessarily connected with the law of conservation of the electric charge. This can be the law of conservation, e.g., of the baryon or lepton charge under the condition that the corresponding charge exists for a given type of interaction. If the group of transformations (2.2) is connected with the law of conservation of the electric charge, then the parameter q D e corresponds to the field of electrons, q D e to the field of protons or positrons, q D 0 to the field of neutrons, etc. If the group of transformations (2.2) is connected with the law of conservation of the baryon charge, then q D B D 1 for the proton and neutron fields and q D B D 0 for the electron-positron field. At the same time, if, e.g., transformation (2.2) is connected with the Lagrangian corresponding to the interaction for which the number of leptons is preserved, then q D L D 1 for the electron-positron, neutron, and muon fields, and q D L D 0 for all baryon and meson fields.

2.2

Charge Conjugation C

The operation of charge conjugation is a discrete transformation of high importance for the theory of strong interactions. It is defined in the space of wave functions as an operator that replaces the wave function of a particle by the wave function of the antiparticle: b ‰a D ‰aN : (2.3) C To describe the charge symmetry, we consider the particle and the antiparticle as two different states of the same particle that differ by the values of the charge quantum number q or some other quantum numbers in the case of neutral particles (if the particles and antiparticles are not identical). In more detail, we consider the form of the operator C in analyzing various specific fields (see Part 2).

2.3

Special Unitary Group SU.n/

First, we consider a group of unitary complex n  n matrices denoted by U.n/: With the help of these matrices, the n-component field u.x/ D ¹uj .x/ºiDn iD1 is transformed according to the law (2.4) u.x/ ! u0 .x/ D V u.x/: The property of unitarity V C V D I implies that jdet V j D 1, which enables us to represent V in the form V D e ig U;

U U C D 1 and

det U D 1:

(2.5)

Section 2.3 Special Unitary Group SU.n/

25

This means that we can separate a subgroup of unitary unimodular (det U D 1) matrices from the group U.n/: This subgroup is called a special unitary group S U.n/; and we have U.n/ D U.1/ ˝ S U.n/: The number of independent parameters for this group is equal to n2  1. Indeed, every n  n matrix has n2 complex or 2n2 real parameters. The condition of unitarity gives n2 equations and the condition of unimodularity, det U D 1, gives one more equation. Hence, the number of independent parameters is 2n2  n2  1 D n2  1. Thus, the corresponding Lie algebra of the group S U.n/ contains n2  1 generators. The detailed analysis of these groups and their representations can be found, e.g., in [143, 11, 148]. In what follows, we present only some most general facts required for the construction of the theory of elementary particles. In connection with the application of the groups of internal symmetries to fields of various types, including scalar, vector, spinor, and other fields, it is necessary to make the following technical remark: Remark 2.2. The component uj .x/ of the field u.x/ may itself have a complex multicomponent structure. Thus, if uj describes a scalar field, then it is a function. However, if uj corresponds to a vector field, then we get uj .x/ D ¹uj; º3D0 : In the case where uj describe fermion fields, they are 2- or 4-component spinors. This means that the matrices V and  (see Chapter 6) act upon different groups of indices.

2.3.1 SU.2/ Symmetry In analyzing the charge independence of nuclear forces, Heisenberg proposed to treat protons and neutrons as the same kind of particles (nucleons) which can be in two states (in the state of proton or in the state of neutron). The notion of isotopic spin was introduced to distinguish between these two states. The wave function of a nucleon is 2-component: ! ‰p : (2.6) ‰N D ‰n Any function of the form (2.6) can be represented as a superposition of the pure states of proton and neutron: ‰N .x/ D c1 ‰p .x/ C c2 ‰n .x/;     ‰p .x/ 0 ‰p .x/ D : ; ‰n .x/ D ‰n .x/ 0

26

Chapter 2

Groups of Internal Symmetries

Moreover, any two superpositions of this sort are connected with the help of a unitary matrix V with det V D 1 as follows: 0 .x/ D V ‰N .x/: ‰N

(2.7)

The set of matrices V can be treated as a representation of the group S U.2/ whose dimension coincides with the dimension of the group itself (i.e., as the fundamental representation). It is known that, in this case, the generators of the corresponding Lie group are given by the matrices 1=2k , where k , k D 1; 2; 3; are the Pauli matrices (1.25), i.e., 1 (2.8) V D V .g/ D e 2 igk k : The matrices k satisfy the relation Œj ; k  D 2i "j kl l ;

(2.9)

where "j kl are the structural constants of the group S U.2/ and "j kl is a completely antisymmetric tensor with "123 D 1. In the general case, the dimension of a representation can be determined via the value of the above-mentioned isotopic spin J : n D 2J C 1: The quantity J.J C 1/ is an eigenvalue of the squared isospin vector J D .J1 , J2 , J3 /, where J1 , J2 , and J3 are the generators of the corresponding representation and J takes integer or half-integer values by analogy with the vector of the ordinary spin operator. The projection of isospin J3 varies within the range J J3 J and takes n D 2J C 1 values, i.e., coincides with the dimension of the multiplet. The generators of the corresponding representation satisfy the same relations (2.9) and the corresponding transformation of wave functions takes the form 1

‰ 0 D e 2 igJ ‰: Hence, the doublet of a nucleon corresponds to J D 1=2. In addition to the doublet of a nucleon, the 2-dimensional representations of the group S U.2/ describe, e.g., the doublets of baryons „ D .„0 , „ / and mesons K D .K C , K 0 /: In order to distinguish between these doublets, Gell-Mann and Nishijima introduced the notion of hypercharge Y (Y D 1 for nucleons and Y D 1 for „-baryons). Moreover, the electric charge in the doublet is given by the formula 1 Q D J3 C Y: 2 Parallel with 2-dimensional representations of the group S U.2/, we can also mention the trivial 1-dimensional representation that describes isosinglets. This representation describes ƒ- and  -baryons and -mesons. In addition, we can also use

27

Section 2.3 Special Unitary Group SU.n/

representations of higher dimensions that describe, e.g., triplets (3-dimensional representations) of elementary particles, quartets, etc. As an example of triplets, we can mention the triplet of pions  C ,  0 , and   . The multiplets formed by antiparticles are transformed according to the conjugate representation of the group SU.2/. These representations are unitarily equivalent. If the matrices of representations are real, then the corresponding multiplets are selfconjugate. This means that the particles and antiparticles belong to the same multiplet. For more details, see [143].

2.3.2 SU.3/ Symmetry The existence of doublets with the same value of isospin but different values of some foreign (from the viewpoint of the groupSU.2/) quantum number Y (hypercharge) indicates that the SU.2/ symmetry is quite narrow and, thus, a wider internal symmetry (including the indicated doublets as components) must exist. Hence, the isospin group must be a subgroup of a new group of symmetries. The required symmetry was proposed by Gell-Mann and Ne’eman in 1961 [63, 137, 65]. It is described by the group of unitary unimodular 3  3 matrices denoted by S U.3/. Its eight generators satisfy the following commutation relations: Œj ; k  D 2ifj kl l ; j; k; l D 1; 8;

iD

p

1;

(2.10)

where fj kl are the structural constants antisymmetric with respect to the indices j , k, and l. They take the following values: f123 D 1; 1 f147 D f156 D f246 D f257 D f345 D f367 D ; 2 p 3 f458 D f678 D : 2 The structural constants corresponding to the other values of indices are equal to zero. For the sake of completeness, we also present the form of the matrices j : 0 0 0 1 1 1 0 1 0 0 i 0 1 0 0 1 D @1 0 0A ; 2 D @ i 0 0A ; 3 D @0 1 0A ; 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 i 0 0 0 4 D @0 0 0A ; 5 D @0 0 0 A ; 6 D @0 0 1A ; 1 0 0 i 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 7 D @0 0 i A ; 8 D p @0 1 0 A : 3 0 0 2 0 i 0

28

Chapter 2

Groups of Internal Symmetries

The generators of representation can be made proportional to the generators of the group itself, i.e., 1 (2.11) Tj D i ; ŒTj ; Tk  D ifj kl Tl : 2 They act in a space of 3-component spinors 1 0 ‰p ‰ D @ ‰n A  ¹‰i º3iD1 : ‰k As compared with (2.6), the spinor ‰ has three components, i.e., the S U.3/ triplet contains the isodoublet ‰1 , ‰2 and the isosinglet ‰3 . In the S U.3/-based theory, the generators of representation are connected not only with the components of isospin but also with the hypercharge Y . It turns out that the basis can always be chosen to guarantee that the first three generators Fj .j D 1; 2; 3; / coincide with the components of isospin Jj and the hypercharge Y is proportional to the generator F3 . The electric charge of the quarks   1 1 1 Q D J3 C Y D 3 C p 8 2 2 3 takes fractional values. To construct irreducible representations of higher dimensions, it is necessary to consider the spinor (tensor) functions of higher ranks: j ;:::;j

‰ D .‰k11 ;:::;kqp /:

(2.12)

In the general case, functions (2.12) describe a reducible representation of the group S U.3/. In order to exclude the representations of lower dimensions, we impose certain additional restrictions on functions (2.12). The irreducible representation according to which functions (2.12) are transformed is denoted by D.p; q/) and has the following dimension [i.e., the number of independent components of spinor (2.12)]: 1 n D n.p; q/ D .p C 1/.q C 1/.p C q C 2/: 2 This number characterizes the number of particles in the multiplet described by the representation D.p; q/. The simplest multiplets are n.0; 0/ D 1I

n.0; 1/ D n.1; 0/ D 3I

n.1; 1/ D 8I

n.3; 0/ D n.0; 3/ D 10;

n.2; 0/ D n.0; 2/ D 6I

etc. However, later, it became clear that, parallel with the singlet, the resonances are filled solely by octet 8 (mesons and baryons) and decuplets 10 and 10 (baryons). In connection with the special status of octets, it is said that the nature evolves in the

Section 2.4 Groups of Local Transformations. Gauge Group

29

eightfold way [65]. It is worth noting that the S U.3/ symmetry is, in fact, approximate, and the distribution of particles over multiplets may fail to describe the actual situation. The detailed presentation of the theory and applications of the S U.3/ symmetry can be found in [140]. The subsequent generalization is connected with higher S U.N / symmetries. The group S U.N / is described by .N 2 1/ independent parameters and has .N 2 1/ generators satisfying the same relations (2.10) solely with their own structural constants fj kl . The generators of representations of this group are M  M matrices (M > N ). They can be specified by relations (2.11). The S U.N / symmetries led to significant successes in the investigation and classification of particles and even made it possible to predict the existence of new particles. Moreover, the extension of the group of symmetries is connected with almost insurmountable difficulties caused by the necessity of combination of the symmetry properties with relativistic invariance (see, e.g., [144, 102]). A more detailed analysis of the symmetry groups and the discussion of the group properties of elementary particles can be found in [143, 87].

2.4

Groups of Local Transformations. Gauge Group

The transformations of internal symmetries (2.2)–(2.5) and (2.7) have the global character, i.e., they are identical at all points of the space-time M. However, from the elementary physical reasoning, it becomes clear that the indicated invariance cannot be preserved in the description of the major part of the actual physical phenomena. Indeed, the electromagnetic field appears if we place charged particles in a certain region of the space. This field is not homogeneous. Hence, it is possible to speak about the symmetry connected with the law of conservation of the electric charge only in a neighborhood of certain points of the space-time. In other words, the requirement of global invariance (2.2) should be replaced by the requirement of local invariance u.x/ ! u0 .x/ D e ig.x/q u.x/; 





u.x/ ! u0 .x/ D e ig.x/q u.x/

(2.13)

or, in the case of the non-Abelian transformations (2.4), (2.7), and (2.8), 1

u.x/ ! u0 .x/ D e 2 igk .x/k u.x/; 



1



u.x/ ! u0 .x/ D e  2 igk .x/k u.x/;

(2.14)

where k are the Pauli matrices (the constant 1=2 is introduced to express the generators via the Pauli matrices). Transformations (2.13) and (2.14) are called local gauge transformations of the first kind.

30

Chapter 2

Groups of Internal Symmetries

It is clear that transformations (2.13) and (2.14), like transformations (2.2), (2.7),  and (2.8), preserve the invariance of the combination of fields u u but break the symmetry of the entire Lagrangian because it necessarily contains combinations with  derivatives of the form @ u.x/ @ u.x/. In the same way, the invariance of the equations of motion (e.g., the Schrödinger equation for a nonrelativistic particle or relativistic equations studied in what follows) is also broken. To preserve the invariance of the theory under the local transformations (2.13) and (2.14), it is necessary to immediately consider the interaction of the field with a certain vector gauge field B .x/, i.e., with the electromagnetic field A .x/ in the case of transformations (2.13) or with the Yang–Mills fields Wk .x/ in the case of transformations (2.14). In this case, the fields B vary according to the gradient transformations x ! x 0 D x B .x/ ! B0 .x/ D B .x/  @ .x/;

(2.15)

where  is a certain differentiable function connected with transformations (2.13) and (2.14). The gradient transformations (2.15) are also called gauge transformations of the second kind. In more detail, these questions are considered in Part III. In conclusion, we briefly describe the structure of a gauge group fundamental for the construction of the theory of interaction of the gauge fields with the matter fields. The gauge group is defined as the direct product of groups Gx acting at every point x of the Minkowski space M: (2.16) G D ˝ Gx : x2M

Hence, every element S of the group G is a function on M and takes values in Gx . The principal role in the construction of the models of interaction of elementary particles is played by the interaction Lagrangians whose structure is based on the use of the above-mentioned group SU.N /, N D 1; 2; : : :, as the group Gx . From the practical point of view, the most extensively used representation of the group S U.N / is the fundamental representation. It can be realized in the space of fields with N components. These fields can be represented in the form of a column 1 0 1 : (2.17) D @ :: A : N

However, it is necessary to emphasize that these components i , i D 1; N , may also have the form of multicomponent functions and the representations of some other groups can be realized on these functions. In what follows, this situation is analyzed in the theory of interaction of the gauge fields with the matter fields representing the fields of quarks of various types. In this case, the group S U.N / and the Lorentz group act upon the superscripts and subscripts, respectively: ˛i , ˛ D 1; 2; 3; 4. By

31

Section 2.4 Groups of Local Transformations. Gauge Group

definition, the action of an element of the group S 2 Gx is specified by the formula .x/ ! .S /.x/;

S C S D S S C D 1;

det S D 1:

(2.18)

For any x 2 M; the N N matrix S D S! .x/ is described by .N 2 1/ independent parameters ! a .x/, a D 1; N 2  1, N > 2 and can be represented in the form S! .x/ D e i!

a .x/T a

;

(2.19)

where T a , a D 1; N 2  1, are called the generators of the group. For N D 1; the parameter ! a .x/ D !.x/ D q  g.x/ is a real function and T a  1. The generators T a are represented by numerical Hermitian matrices and satisfy the following relations: (2.20) ŒT a ; T b  D if abc T c ; Tr.T a / D 0;

Tr.T a T b / D 2ı ab :

(2.21)

The choice of the last trace is arbitrary and fixes the collection T a , a D 1; : : : ; N 2  1, as a basis in the corresponding Lie algebra. For this choice, the structural constants of the Lie algebra 1 f abc D Tr.ŒT a ; T b  T c / (2.22) 2i are real and antisymmetric with respect each couple of indices. The structural constants satisfy the relation (Problem 2.1): f abd f dce C f cad f dbe C f bcd f dae D 0;

(2.23)

obtained from relation (2.20) and the Jacobi identity ŒŒT a ; T b  ; T c  C ŒŒT c ; T a  ; T b  C ŒŒT b ; T c  ; T a  D 0:

(2.24)

Relation (2.23) coincides with (2.20) if the matrices T a are chosen in the form .T a /bc D if abc :

(2.25)

Hence, in this case, matrices (2.25) specify another representation of the group S U.N / whose dimension coincides with the number of generators, i.e., is equal to N 2  1. This representation is called adjoint and the matrices S! .x/ are real and orthogonal (Problem 2.2). Thus, the vectors subjected to the action of matrices of the group in the adjoint representation can be rewritten in the form of a real vector column B .x/ D ¹Ba .x/º, a D 1; : : :, N 2  1. In this case, the action of generators (2.25) upon the vector B .x/ is described by the formula .T a B /b .x/ D if abc Bc .x/:

(2.26)

32

Chapter 2

Groups of Internal Symmetries

The law of transformation of the vector B .x/ can be represented via the action of elements of the group S! .x/ upon the matrices A .x/ D Ba .x/T a :

(2.27)

Thus, relation (2.26) is equivalent to the action of the generator T a upon the matrix A .x/ according to the rule: T a ı A .x/ WD .T a B /b .x/T b : At the same time, the transformation A .x/ ! A! .x/ D S! .x/A .x/S!1 .x/

(2.28)

specifies the adjoint representation of the group Gx  S U.N / on the matrices (objects) A .x/ (Problem 2.3). In order to extend this formalism to the theory of interacting fields, we now consider the simplest possible case, namely, the group S U.1/  U.1/. In this case, the elements of the group are simple functions (factors) changing the phase of the function (see (2.13)). In the next part, in analyzing the interaction of electromagnetic fields with charged particles, we show that the quantum (classical) electrodynamics is invariant under the gradient transformations of the first kind (2.13) with q D e and (at the same time) under the transformations of the 4-vector potential A .x/: A .x/ ! A .x/ D A .x/  @ g.x/:

(2.29)

Relations (2.13) and (2.29) can be regarded as a realization of the representation of the group U.1/ on the set of fields ¹ .x/, A .x/º. Then the generalization of this situation to any gauge group is given by the following representations (see Problem 2.4): .x/ ! A .x/ !

!

.x/ D S! .x/ .x/;

A! .x/

D

S! .x/A .x/S!1 .x/

(2.30) 

i .@ S! .x//S!1 .x/;

(2.31)

where  is an arbitrary constant. In the case of QED, we set A .x/  A .x/, !.x/ D eg.x/; and  D 1=e. Relations (2.28), (2.30), and (2.31) should not necessarily be connected with the adjoint representation of the gauge group (2.16). Indeed, they can be regarded as a realization of an arbitrary representation on the objects ¹ .x/; A .x/º, where A .x/ are expressed via the gauge fields by relations (2.27) and has form of column (2.17) with the number of components corresponding to the dimension of representation (for the fundamental representation, this number is equal to N , i.e., coincides with the dimension of the group).

Section 2.4 Groups of Local Transformations. Gauge Group

33

By using (2.31) and the properties of generators (2.20) and (2.21), one can construct the representation of the gauge group directly on the set of fields ¹ .x/;Ba .x/º. To do this, we act by the generator T a upon both parts of Equation (2.31) and perform the trace operation Tr. Thus, in view of (2.19) and property (2.21), we arrive at the following law of transformation: Ba .x/ ! .B ! /a .x/ D ab .x/Bb .x/ C @ ! a .x/;

(2.32)

where

1 Tr.T a S! .x/T b S!1 .x//: (2.33) 2 The action of the trace operation upon the right-hand side of (2.33) is computed as follows (see, e.g., [127], Chapter 5, Section 5.9, problem 7):

ab .x/ D

e X Y e X D Y C

1 X ¹YX n º ; nŠ

(2.34)

nD1

where the operation ¹  º is defined on the Lie algebra by the commutators: ¹YX n º WD Œ: : : ŒY; X  ; : : : ; X  :

(2.35)

We now substitute X D i ! c .x/T c and Y D T a in relations (2.34) and (2.35). Further, we act by the generator T b upon the obtained equality from the right and perform the trace operation on both sides of the formula. Finally, in view of (2.21), we arrive at the series ab

.x/ D ı

ab

C i !Q

ab

1 n X i ac1 c1 c2 !Q !Q C    !Q cn1 b ; nŠ

(2.36)

nD2

where !Q ab WD if abc ! c .x/:

(2.37)

In view of the antisymmetry of the structural constants f abc ; the matrix !Q is always an .N 2  1  N 2  1/-dimensional antisymmetric matrix independently of the dimension of representation, i.e., of the dimension of the matrix S! .x/. Hence, !Q ab D !Q ba

and

e !Q D 0: Tr

(2.38)

Thus, for the matrix of rotation in the space of gauge fields, we get

.x/ D e i !Q

and

Det .x/ D 1:

The last equality is a consequence of the well-known relation e ln .x/ D expŒi Tr e !.x/ Det .x/ D expŒTr Q D e0 D 1

(2.39)

34

Chapter 2

Groups of Internal Symmetries

e means that the trace operation is performed and relation (2.38). Here, the symbol Tr over the indices a; b; c; : : : taking values from 1 to N 2  1. Thus, the final form of representation of the gauge group in the space of the fields ± ° .x/ D ¹ ˛i .x/º; B D ¹Ba º is given by relations (2.30), (2.32) with matrices (2.19)–(2.22) and (2.37)–(2.39). In conclusion, we note that Definition (2.37) and Relation (2.26) imply that

.x/  S! .x/ in the adjoint representation.

(2.40)

Chapter 3

Problems to Part I

Problem 1.1.

Prove that conditions 1) .x; y/M D 0 and 2) x 2 > 0 imply that y 2 < 0.

Problem 1.2.

Find a group of homogeneous Lorentz transformations for the twodimensional Minkowski space (i.e., ; ; ˛, ˇ 2 ¹0; 1º) corresponding to the physical situation in which the coordinate system ¹x 0 º moves along the x 1 -axis of an immobile system ¹xº, i.e., x 0 D x  for  D 2; 3. How many parameters has this group? Write the expressions for transformations via the relative velocity of two reference v systems v or in terms of the parameter ˇ D . c

Problem 1.3.

Find the generators of Lorentz rotations in six planes .x  x  /.

Problem 1.4.

Prove that the homogeneous Lorentz transformation x 0 ! ƒx corresponds to the following transformation of the covariant 4-vector V  .x/:  V  .x/ ! V 0 .x 0 / D ƒ  V .x/:

Problem 2.1.

Prove that the structural constants (2.22) satisfy relation (2.23).

Problem 2.2.

Show that the matrices S! .x/ whose generators act according to formula (2.26) are real and satisfy relation (2.18).

Problem 2.3.

Show that formula (2.31) gives a representation of the group Gx on the matrices A .x/.

Problem 2.4.

Show that transformations (2.30) and (2.31) form a group.

Part II

Classical Theory of the Free Fields

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory For the construction of the theory of classical fields, we use the Lagrange approach because the Lagrangian formalism proves to be the most convenient tool for the development of the relativistic theory in view of its obvious covariance. The Hamiltonian formalism (or the formalism of canonical variables) appears to be useful in finding the similarities between the classical and quantum theories. From the viewpoint of the general theory of relativity, this formalism is, at first sight, not invariant under Lorentz transformations because the time variable t D x 0 plays a specific role. However this feature of the formalism means, in fact, that we fix a reference system connected with the time variable. In this approach, the relativistic invariance is hidden and its verification is a separate (sometimes fairly nontrivial) problem. We also note that all calculations performed in this chapter remain true not only for the free fields but also for the classical interacting fields. The construction of the classical theory of evolution of the fields connected, according to the principle of wave-particle duality, with the corresponding elementary particles is based on the well-developed formalism of classical mechanics. For the sake of completeness of our analysis, we now briefly recall some basic concepts of classical mechanics used for the construction of the field theory.

4.1

Variational Principle and Canonical Formalism of Classical Mechanics

4.1.1 Lagrangian Equations If a mechanical system is formed by N material points moving and interacting in the space Rs ; then its state at time t is completely determined by the set of coordinates and velocities (4.1) ri .t /; rP i .t /; i D 1; N : However, for the major part of problems in the classical mechanics, it is convenient to pass from the Cartesian coordinates (4.1) to certain generalized coordinates qi ; qi 2 R1 and generalized velocities qP i .t / whose choice depends on the specific problem (e.g., to spherical or cylindrical coordinates). It is clear that qi D qi .r1 ; : : : ; rN ; t /; i D 1; n .n 6 sN /: The inequality n < sN holds in the case where some additional constraints are imposed on the system. These constraints can be specified by the

40

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

corresponding equations: 'l .r1 ; : : : ; rN ; t / D 0;

l D 1; m:

(4.2)

Then n D sN  m: Further, we consider the generalized coordinates qi ; i D 1; n; as independent variables in the case where the system has no constraints. In the presence of constraints, Equation (4.2) can be rewritten in terms of the coordinates qi : According to the Newton law, the process of motion of a system of N material points is described by the following system of sN equations: @U d .˛/ .mi ri / C D 0; .˛/ dt @ri

i D 1; N ; ˛ D 1; s;

where mi are the masses of particles and U is the potential energy of their interaction. As an advantage of the generalized coordinates qi ; we can mention the fact that, in most cases, constraints (4.2) are taken into account by the choice of these coordinates. This significantly simplifies the procedure of construction of the solution. The Lagrangian formalism of classical mechanics is more general. It enables one to perform a more comprehensive investigation of numerous important problems. Thus, for a finite mechanical system formed by N material point particles, the Lagrange function is defined as follows: 1X mi qP i  U.q1 ; : : : ; qn / L D L.q; q/ P D 2 iD1  X n  n X 1 2 mi qP i  Ui D Li ; D 2 n

iD1

(4.3)

iD1

where Ui and Li are the potential energy and the Lagrange function of the i th particle. According to the variational principle or the principle of the least Hamiltonian action, the character of motion of the Newton potential system of material points is determined by the following theorem: Theorem 4.1 ([4], Part II, Chapter 3, Section 13). The motion of a mechanical system in a time interval Œt0 ; t1 , t0 < t1 , coincides with the extremals of the action functional Zt1 AD

L.t /dt;

(4.4)

t0

where L.t / D T  U is the Lagrange function (T is the kinetic energy of the system and U is its potential energy).

Section 4.1 Variational Principle and Canonical Formalism of Classical Mechanics

41

Corollary 4.2. If L is expressed via the generalized coordinates qi and generalized velocities qP i ; then the principle of stationary action ıA D 0

(4.5)

yields the following system of differential equations (Lagrange equations): d @L @L  D 0; dt @qP i @qi

i D 1; n;

(4.6)

for the determination of the extremals qi .t / of action (4.4). Remark 4.3. Here and in what follows, we consider the systems whose Lagrangians do not depend on time explicitly.

4.1.2 Canonical Variables. Hamiltonian Equations Parallel with the generalized coordinates qi , the Hamiltonian formalism is based on the analysis of generalized momenta pi introduced by the formula pi D

@L : @qP i

(4.7)

The Hamiltonian of a classical system is the Legendre transform of the Lagrange function (with respect to the variables qP i .t /) regarded as a function of the generalized coordinates qi and the generalized velocities qP i (for a more detailed presentation, see [4], Part II, Chapter 3, Section 14): H.p; q/ D

n X

pi qP i  L.q; q/: P

(4.8)

iD1

The system of Lagrangian equations (4.6) is equivalent to the system of 2n Hamiltonian equations (Problem 4.1): @H ; pPi D  @q i

qP i D

@H : @pi

(4.9)

Equations (4.9) are called the canonical equations of motion and the set ¹qi ; pi ºniD1 is called the set of canonically conjugate variables. The transformations of coordinates under which the form of the Hamiltonian equations (4.9) is preserved for the new variables qi0 D qi0 .q1 ; : : : ; qn ; p1 ; : : : ; pn ; t /; pi0 D pi0 .q1 ; : : : ; qn ; p1 ; : : : ; pn ; t /; i D 1; n; are called canonical transformations.

(4.10)

42

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

4.1.3 Poisson Brackets. Integrals of Motion The Poisson brackets play an extremely important role in the classical mechanics. For a couple of dynamical characteristics of the system F D F .q1 ; : : : ; qn ; p1 ; : : : ; pn ; t / and G D G.q1 ; : : : ; qn ; p1 ; : : : ; pn ; t /; the Poisson brackets are given by the formula:  n  X @F @G @F @G ¹F; GºP  ¹F; Gº WD :  @qi @pi @pi @qi

(4.11)

iD1

It is easy to see that, in view of the canonical Hamiltonian equations (4.9), the total time derivative of the dynamic quantity F has the form @F dF D C ¹F; H º; dt @t

(4.12)

where H is the Hamiltonian of system (4.9). As follows from relation (4.12), in order that the dynamic characteristic F (which does not depend on time explicitly) be an integral of motion, it is sufficient that the Poisson brackets of this characteristic with the Hamiltonian be equal to zero. Equation (4.12) describes the evolution of the dynamic characteristic F: Thus, in the case where the role of the dynamic characteristic is played by the generalized coordinates and momenta, Eqs. (4.12) take the form dpi D pPi D ¹pi ; H º; dt

dqi D qP D ¹qi ; H º: dt

(4.13)

It is easy to see from Definition (4.11) that Eqs. (4.13) coincide with Eqs. (4.9). Moreover, the Poisson brackets have the following properties: ¹F; Gº D ¹G; F º;

whence

¹F; F º  0;

¹F; G1 C G2 º D ¹F; G1 º C ¹F; G2 º;

(4.14)

¹F; G1 G2 º D G1 ¹F; G2 º C ¹F; G1 ºG2 ; and, as a consequence, the following Jacobi identity is true (Problem 4.2): ¹G1 ; ¹G2 ; G3 ºº C ¹G2 ; ¹G3 ; G1 ºº C ¹G3 ; ¹G1 ; G2 ºº D 0:

(4.15)

The following fundamental Poisson brackets between the coordinates qi and momenta pj play a significant role in the construction of the quantum theory: ¹qi ; qj º D ¹pi ; pj º D 0; ¹qi ; pj º D ıij ; where ıij is the Kronecker symbol.

(4.16)

Section 4.1 Variational Principle and Canonical Formalism of Classical Mechanics

43

In Sections 4.1.1–4.1.3, we present the basic concepts of the classical mechanics of systems with n degrees of freedom. These concepts serve as the starting point for the primary quantization, i.e., for the construction of the canonical formalism of quantum mechanics in the case where the analyzed system has no additional constraints. However, for the construction of the quantum theory of gauge fields, this is insufficient because the indicated theory contains additional equations imposing, in fact, severe restrictions on the field functions. Hence, in the next section, we propose a brief presentation of the ideology of canonical formalism developed by P. Dirac for systems with constraints (see lecture 1 in [41]).

4.1.4 Canonical Formalism in the Presence of Constraints Consider a mechanical system in which 2n variables qi ; pi ; i D 1; n; are connected by m1 relations 'k .q1 ; : : : ; qn ; p1 ; : : : ; pn / D 0;

k D 1; m1 :

(4.17)

According to the terminology of Dirac [41], these relations are called primary constraints. Actually, the Hamiltonian of this system is not a single-valued function of the variables q and p: It is possible to define a more general Hamiltonian e DHC H

m1 X

uk 'k ;

(4.18)

kD1

where H is given by relation (4.8), uk are arbitrary coefficients, and 'k are functions of the generalized coordinates and generalized momenta appearing on the left-hand side of (4.17). In the general case, uk must be differentiable functions of qj and pj : It is worth noting that the idea of canonical formalism in the presence of constraints (4.17) is to assume that all qi and pi in Equation (4.18) and all calculations connected e are independent (i.e., 'k are not identically equal to zero). with the Hamiltonian H Conditions (4.17) are taken into account in the final formulas. Hence, for any dynamic characteristic G; the evolution equation (4.12) takes the form m1 X dG D ¹G; H º C uk ¹G; 'k º dt

(4.19)

kD1

because it follows from the previous remark that, despite the validity of conditions (4.17), it is possible that ¹G; uk º ¤ 0 but the terms ¹G; uk º'k D 0 in view of (4.17) even if uk are functions of qi and pj : As an important problem of this formalism, we can mention the problem of consistency of the equations of evolution and constraints (4.17). Since 'Pk must be identically

44

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

equal to zero, substituting the equality G D 'j in Equation (4.19), we arrive at the following consistency conditions: ¹'j ; H º C

m1 X

uk ¹'j ; 'k º D 0;

j D 1; m1 :

(4.20)

kD1

Together with the Lagrangian equations (4.6), these conditions impose certain restrictions on the choice of the Lagrange function L: As a rule, conditions (4.20) yield the following three types of relations: (1) identity 0 D 0; (2) new equations that do not contain arbitrary coefficients uk : j .q1 ; : : : ; qn I p1 ; : : : ; pn / D 0;

j D 1; m2 I

(4.21)

(3) equations containing the coefficients uk : Equations (4.21) are called secondary constraints. Further, it is necessary to perform the same calculations with conditions (4.21) as with relations (4.17). In other words, conditions (4.21) are regarded as primary, the consistency condition is deduced from the requirement P D 0; and finally, new conditions of the form (2) are established (if they exist) . The constraints obtained in this way (both primary and secondary) are denoted by 'j : 'j .q1 ; : : : ; qn I p1 ; : : : ; pn / D 0; j D 1; m;

(4.22)

m D m1 C m2 C    :

It remains to take into account Equations (4.20) containing the undetermined coefficients uk : These equations are regarded as equations for the unknown variables uk : In this case, it is always possible to find their solutions as functions of the variables qk ; pk ; k D 1; n; namely, uk D Uk D Uk .q1 ; : : : ; qn I p1 ; : : : ; pn /: The indicated solutions must always exist because, otherwise, the chosen Lagrangian is incompatible with the proposed formalism. Moreover, these solutions are not unique because it is always possible to add to the presented expressions an arbitrary solution of the homogeneous system mi X kD1

Vk ¹'j ; 'k º D 0:

(4.23)

Section 4.1 Variational Principle and Canonical Formalism of Classical Mechanics

45

Thus, we denote all independent solutions of system (4.23) by Vj k and get the following expression for the coefficients uk W uk D Uk C

mk X

cj Vj k ;

mk < m;

(4.24)

j D1

where cj are arbitrary coefficients. They can be functions of time. Hence, despite the requirement of consistency, it is impossible to remove arbitrariness in the evaluation of the dynamic characteristics. At first sight, the situation is quite strange. It is assumed that the state of the system is completely determined by the variables q and p: At the initial time t D 0; this state is uniquely determined by specifying the initial values q0 and p0 : At the same time, for any t > 0; the values of the generalized coordinates and momenta are determined ambiguously due to the presence of arbitrary coefficients cj : In fact, it is possible to get even a set of values of ¹q; pº corresponding to the same state because the state of the system is uniquely determined solely by the experimentally observed quantities (observables). The same situation is also observed in the classical electrodynamics where the role of generalized coordinates is played by the potentials of electric and magnetic fields. The potentials themselves are not observables and, hence, it is possible to change their values without changing the values of strength of the fields. In fact, just the values of strength of the electric and magnetic fields determine the state of the electromagnetic field. Hence, for constraints of a given type and a given Lagrangian which are not contradictory, we arrive at the evolution equation (4.19) in which the coefficients uk are given by relations (4.24). From the viewpoint of transition to quantum systems, all dynamic characteristics can be split into two types. According to the definition given by Dirac [41], we say that G is a dynamic characteristic of the first kind if its Poisson brackets with all 'k are equal to zero: ¹G; 'k º D 0;

k D 1; m;

and equalities (4.22) are satisfied. In other words, the Poisson brackets must be a linear homogeneous function of the constraints 'j ; j D 1; m: ¹'j ; 'k º D

m X

cj kl 'l :

(4.25)

lD1

Clearly, it is possible to assume that the right-hand side is a more complex function of the constraints 'l and vanish for 'l D 0: However, in this case, the situation becomes much more complicated and requires a more profound analysis with separate investigations for each special type of constraints (4.22). All other dynamic characteristics are regarded as variables of the second kind.

46

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

The same classification is used for the constraints 'k ; k D 1; m. Indeed, the constraints whose Poisson brackets with all other constraints are equal to zero are constraints of the first kind. At the same time, the Poisson brackets of constraints of the second kind with the other constraints are nonzero for at least one constraint. e is always a variable It is clear that the conditions of consistency (4.20) imply that H of the first kind, whereas the Hamiltonian H is a variable of the first kind only in the case where all constraints 'j are variables of the first kind. Hence, for constraints of the first kind, the equation for the dynamic characteristics of the first kind remains unchanged: dG e º: D ¹G; H º D ¹G; H dt In fact, this means that, in systems with constraints of the first kind, it is possible to remove “redundant” variables with the help of a certain canonical transformation and develop the Hamiltonian formalism with a smaller number of independent generalized coordinates and momenta. For constraints of the second kind, Dirac proposed to replace the Poisson brackets (4.11) with brackets of a different type ¹F; GºD WD ¹F; Gº 

S X

¹F; 's º.C 1 /ss 0 ¹'s 0 ; Gº;

(4.26)

s;s 0 D1

where C D ..Cij // is the matrix of constraints of the second kind. It can be constructed by using the following procedure: Assume that the first S constraints in the family of all constraints (4.22) are constraints of the second kind, whereas the remaining m  S constraints are constraints of the first kind. Then, by definition, Cij WD ¹'i ; 'j º: In view of the definition of constraints of the second kind, we have Cij ¤ 0 for i ¤ j: It turns out (for more details, see [41]) that the number of constraints of the second kind is always even and the matrix C always possesses the inverse matrix because det C ¤ 0: It is also possible to show (Problem 4.3) that the Dirac brackets (4.26) have the same properties (4.14) and (4.15) as the Poisson brackets. In addition, relation e is always a variable of the first kind imply that (4.26) and the fact that H eº  e ºD D ¹G; H ¹G; H

S X

e º D ¹G; H eº ¹G; 's º.C 1 /ss 0 ¹'s 0 ; H

s;s 0 D1

(4.27)

Section 4.1 Variational Principle and Canonical Formalism of Classical Mechanics

and ¹G; 'j ºD D ¹G; 'j º 

S X

¹G; 's º.C 1 /ss 0 Cs 0 j D 0

47

(4.28)

s;s 0 D1

for any dynamic characteristic G: This means that the equations of evolution of the dynamic variable G do not change their form if we replace the Poisson brackets in these equations by the Dirac brackets. In view of Equation (4.28), we get dG e ºD D ¹G; H ºD : D ¹G; H dt Later, Maskawa and Nakajima [129] demonstrated that if there are 2m constraints of the second kind in the system with ¹qj ; pj º; j D 1; n; then, with the help of canonical transformations, it is always possible to introduce new variables 0 0 ¹Q1 ; : : : ; Ql ; Q10 ; : : : ; Qm I P1 ; : : : ; Pl ; P10 ; : : : ; Pm º;

l C m D n;

(4.29)

for which the corresponding new constraints take the form Qj0 D Pj0 D 0;

j D 1; m:

In this system, the matrix of constraints can readily be constructed by using the relations ¹Qi0 ; Qj0 º D ¹Pi0 ; Pj0 º D 0; ¹Qi0 ; Pj0 º D ıij : Therefore,



   C11 C12 O 1 C D  ; 1 O C12 C22 where O and 1 are, respectively, the null and identity m  m matrices. Moreover, we have @G @G ¹G; Pj0 º D ; j D 1; m: ¹G; Qj0 º D  0; @Pj @Qj0 Then the Dirac brackets for F and G in the system of variables (4.29) take the form (in view of the fact that C 1 D C ) ¹F; GºD D ¹F; Gº C D

l  X j D1

m X   ¹F; Qs0 º¹Ps0 ; Gº  ¹F; Ps0 º¹Qs0 ; Gº sD1

@F @G @G @F  @Qj @Pj @Qj @PJ

 :

(4.30)

Hence, the right-hand side of (4.30) coincides with the Poisson brackets in the system of “truncated” variables: ¹Q1 ; : : : ; Ql ; P1 ; : : : ; Pl º:

48

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

The formalism presented above is used for the quantization of systems in the presence of additional constraints. For a more detailed presentation, see [41, 212].

4.2

From Classical to Quantum Mechanics. Primary Quantization

In the present section, one can find a brief description of the procedure of substantiation of the transition to a new formalism whose appearance at the beginning of the last century was explained by the impossibility of explaining various phenomena observed in the microworld from the viewpoint of classical mechanics. For a more detailed presentation, the reader is referred to the classical monographs on quantum mechanics (see, e.g., [40, 52, 115], etc.). Clearly, the transition to the new formalism was, first of all, the result of intuitive analyses based on the experimentally observed phenomena. Thus, it was shown, e.g., that some dynamic characteristics of the atomic systems take discrete values. From the mathematical point of view, this means that the indicated dynamical characteristics cannot be described by ordinary functions. In mathematics, the role of objects of this sort is played by operators acting in certain Hilbert spaces. The spectra of these operators can be discrete. Thus, it is clear that the eigenfunctions of these operators must describe the state of the system one of the characteristics of which is observed in the analyzed experiment. In connection with this idea, it is necessary to be able to answer the following natural question: What kind of operators corresponds to the dynamic characteristics and, first of all, to the canonical variables pj and qj ? It is clear that, for this purpose, it is necessary to consider the equations of motion (4.12) common for all dynamic characteristics, i.e., to construct a quantum analog of the indicated equations. In the classical approach, the evolution of each dynamic characteristic is determined by the Poisson brackets whose general algebraic properties are described by Equations (4.14). Thus, to construct the quantum equation, it is necessary to determine the quantum analog of the Poisson brackets on the basis of the most general properties (4.14). It turns out b and G b corresponding to the dynamic (see [40], Section IV) that, for the operators F characteristics F and G; the quantum Poisson brackets coincide (to within a constant) b and G; b i.e., with the commutator of the operators F

2

b ; G b  D a.F bG bG bF b /; ¹F; Gº D aŒF where a is an unknown constant.

(4.31)

Section 4.2 From Classical to Quantum Mechanics. Primary Quantization

49

Comparing equalities (4.31) and (4.16), we determine the commutation rules obeyed by the fundamental quantities, i.e., by the operators of canonical variables pOj and qOj : 1 ŒqO i ; qOj  D ŒpOi ; pOj  D 0; qO i pOj  pOj qO i D ıij : (4.32) a To determine the explicit form of the operators qO i and pOj ; it is possible, e.g., to use the hypothesis of wave-particle duality according to which the velocity of a material particle is equivalent to the group velocity of the wave packet. Then, for a free nonrelativistic particle, the wave ‰ can be represented in the form of superposition as follows: Z Q .p/e „i .prE t/ d p; ‰.r; t / D where the energy E and momentum p of the particle satisfy the well-known relation ED

p2 : 2m

(4.33)

By using this relation, we conclude that the function ‰ satisfies the following differential equation: „2 @ ‰.r; t /; (4.34) i „ ‰.r; t / D  @t 2m where  D r  r is the Laplace operator in Rs : This means that the classical relation (4.33) is transformed into the following operator equation: b D 1 pO 2 ‰: E‰ 2m b0  E b and the operator of momenThus, the operator of energy of the free particle H tum pO can be defined as follows: bDH b0 D i„ @ E @t

and

pO D i „r :

(4.35)

One can now readily show that the operators pOi and qOj satisfy the operator relation (4.32) if the operators qOj are defined as the operators of multiplication by the components of the vector r and the constant a is chosen in the form a D 1= i „:

(4.36)

Finally, it should be emphasized that these operators act in the Hilbert space of square-integrable functions L2 .Rs ; d r/; i.e., in the space of one-particle states for the major part of problems in the quantum mechanics. Equation (4.34) is called the Schrödinger equation for a free nonrelativistic particle. We also note that the operators corresponding to arbitrary dynamic characteristics F

50

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

can be found by using the mathematical theory of construction of functions of the operators pOj and qO i : However, since the operators pOj and qOj are not commuting, this procedure is nontrivial. The role of an analog of the classical equation of motion (4.12) for the operator corresponding to an arbitrary dynamic characteristic F is played by the Heisenberg equation (for all F that do not depend on time explicitly): i„

b dF b; H b  : D ŒF dt

(4.37)

The formalism described above is called the primary quantization of classical mechanical systems (the procedure of secondary quantization corresponds to the quantization of wave functions ‰). Finally, we describe the situation connected with the quantization of classical mechanical systems containing additional constraints (4.22) imposed on the variables pj and qj ; j D 1; n: First, we consider the case where all constraints (4.22) are of the first kind (see Section 4.1.4). In the quantum theory, Equation (4.22) should be replaced with the operator equations (4.38) 'Oj ‰ D 0; j D 1; m; where 'Oj are operators acting upon the wave function of state. Hence, (4.38) are equations for the function of state ‰: Equations (4.38) readily yield the following equations: Œ'Oj ; 'Ok  ‰ D 0; j; k D 1; m: For constraints of the first kind, this equation is consistent with (4.38). Indeed, in view of the consistency between relations (4.31) and rule (4.25), the commutator Œ'Oj ; 'Ok  is a linear homogeneous function of the quantities 'Ol ; l D 1; m and, according to (4.38), the state ‰ is nullified. The same type of consistency should be true for Equation (4.37). Indeed, for this purpose, it is necessary to show that i h b ‰ D 0; (4.39) 'Oj ; H 

b is either a homogeneous linear function i.e., the commutator of the operators 'Oj and H of the operators 'Ol ; l D 1; m; or a function nullifying the state ‰ obtained as a solution of (4.38). In this case, relation (4.39) is not a new equation for ‰: Hence, e can be replaced with an in this case (just as in the classical case), the Hamiltonian H b e containing the operators 'Oj : Actually, this is a consequence of invariance operator H of the theory under certain transformations allowing us to decrease the number of independent operator variables and return to the situation considered earlier. In the case of quantization of the systems with constraints of the second kind, the situation is absolutely different. To show this, we consider a simple example of the

51

Section 4.2 From Classical to Quantum Mechanics. Primary Quantization

following constraint (see [40]): q1 D 0;

p1 D 0:

The quantum version of this situation means that we consider the state of a system ‰ for which (4.40) qO 1 ‰ D 0 and pO1 ‰ D 0: However, these equations contradict relations (4.32) according to which, we have ŒqO 1 ; pO1  ‰ D i „‰

(4.41)

because, in view of relation (4.40), the left-hand side of Equation (4.41) may vanish even for ‰ ¤ 0; whereas its right-hand side vanishes only for ‰  0: However, in this situation, the solution is very simple. It is necessary to consider the states depending solely on the variables q2 ; : : : ; qn ; p2 ; : : : ; pn and use the same reasoning as above. For more complicated constraints [e.g., for p1 D 0;

q1 D f .q2 ; : : : ; qn I p2 ; : : : ; pn /;

Dirac proposed to replace (4.31) by a new rule according to which the commutator must be proportional to the new (Dirac) brackets (4.26):

2

h i b; G b D 1 ¹F; GºD : F  c

(4.42)

In this case, relation (4.30) implies, in fact, that the commutation relations computed by using the Dirac procedure [i.e., with the help of (4.42)] coincide with the commutation relations computed by the ordinary method (4.31) for a new reduced number of independent variables. The commutation relations between the variables remaining in the new family and the variables removed from the analyzed set can be obtained by using the equations of constraints. We consider the application of this method to the case of quantization of vector fields in Chapter 20, Sections 20.2.3 and 20.3, and Chapter 32. In conclusion, we make the following important remark: Remark 4.4. The situation with quantization becomes much more complicated if constraints (4.22) are given by nonlinear functions of the variables qj and pj : In this case, the procedure of finding the operators 'Oj is connected with a possibility of appearance of ambiguity caused by the order of action of the operators pO and qO in the corresponding expressions (these operators are not commuting). Hence, in each special case, it is necessary to check the consistency of all analyzed equations, i.e., the consistency of the equations of motion with all additionally imposed conditions.

52

Chapter 4

4.3

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

General Requirements to the Lagrangians of the Field Theory

For the continuous systems (e.g., for classical fields), the sum in relation (4.3) should be replaced by the integral Z 0 L D L.t / D L.x / WD d x L .x/; ƒ

where L is the density of the Lagrange function or the Lagrangian and ƒ R3 is the volume of the analyzed system. The corresponding expression for action in a time interval T D Œt0 ; t1  takes the following form: Z Z 0 0 A D dx L.x / D dx L .x/;  D ƒ  T M: (4.43) T



By analogy with the classical mechanics, it is assumed that L .x/ depends only on the fields ui .x/ (regarded as generalized coordinates) and their first derivatives (regarded as generalized “velocities”) and does not depend on the @ ui .x/ WD @u@xi .x/  coordinates x. Remark 4.5. Note that the quantities @ ui .x/ called generalized “velocities” are nothing but the covariant derivatives of the fields ui .x/: The actual generalized velocities corresponding to the notions of the canonical formalism are the quantities @0 ui .x/  @u@xi .x/ 0 : However, in order to construct a Lorentz-invariant theory from the very beginning, it is necessary to assume that the Lagrangian depends on @ ui .x/ with  D 0; 1; 2; 3: Moreover, we consider the theory in which the dependence of the Lagrangian on ui .x/ and @ ui .x/ is local, i.e., L .x/ does not depend on the fields and their derivatives concentrated at the other points. Hence, L .x/ D L .ui .x/; @ ui .x//:

(4.44)

In the general form, the nonlocal Lagrangian is given by the formula Z Lnonloc .x/ D dy F .ui .x/; ui .y/; @k ui .x/; @k ui .y//: The corresponding theory is called the nonlocal field theory. The most comprehensive presentation of this theory can be found in [44, 45]. As one of the most important postulated properties of the Lagrangian, we can mention its invariance under the complete inhomogeneous Lorentz group. This means

53

Section 4.4 Lagrange–Euler Equations

that, under transformation (1.20), we have L .x/ ! L 0 .x 0 / D F .u0i .x 0 /; @0 u0i .x 0 // D F .ui .x/; @ ui .x// D L .x/; i.e., the Lagrangian is transformed as a scalar. Finally, we return to the classical mechanics in which the Lagrangian is the sum of the kinetic and potential energies and assume that L is a real function (in the quantum theory, the Lagrangian is a self-adjoint operator). This means that the fields and their derivatives appear in the Lagrangian only in the form of bilinear combinations: 

ui .x/ui .x/I

4.4



@ ui .x/@ ui .x/I







  @ ui .x/ui .x/ C   ui .x/@ ui .x/:

Lagrange–Euler Equations

We assume that the principle of stationary action (4.4) exists both in the classical mechanics and in the field theory. One more condition is imposed on the field functions, namely, that the variations of the fields ıui vanish on the boundary of the 4-volume  2 M: (4.45) ıui .x/jx2@ D 0: Condition (4.45) means that the field on the boundary @ of the region  takes a certain constant value. It is natural to assume that this field simply vanishes on @ because the volume of  is always very large. In this case, we say that the field satisfies the Dirichlet boundary conditions on the boundary of the domain : Substituting the expression for the action (4.43) in relation (4.5), we find   Z Z @L @L ıui .x/ C ı.@ ui .x// ıA D ıL .x/dx D dx @ui .x/ @.@ ui .x//     Z @L @L @ D dx   (4.46) ıui .x/ D 0: @ui .x/ @x @.@ ui .x// 

In the second term in (4.46), we have changed the order of the operations ı and @ and performed the operation of integration by parts by using condition (4.45). The left-hand side of (4.46) must be equal to zero for any 4-volume : This means that @L @ @L   D 0; @ui .x/ @x @.@ ui .x//

(4.47)

where i D 1; 2; : : : ; N and N is the number of components of the field u.x/: This system of equations for the components of the field u.x/ is called the Lagrange–Euler equations. In view of the conditions for the Lagrangian formulated in Section 4.3, (4.47) is a system of differential equations of the first or second order.

54

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

Remark 4.6. Since the physical properties of the system are determined by the values of the action, the Lagrangian L specifying the indicated action by relation (4.43) is not single-valued. Indeed, if we choose e .x/ D L .x/ C @ F  .ui .x/; @ ui .x//; L e D 0 yields the then, in view of condition (4.45), the principle of stationary action ı A same equation (4.47) as the Lagrangian L :

4.5

Noether’s Theorem and Dynamic Invariants

As in the classical mechanics, it is necessary to know not only the equations of motion but also the integrals of motion of the system or the so-called dynamic invariants. The form of these invariants is established by the following theorem: Noether’s theorem. [142] Every s-parametric group of transformations of the field functions (and coordinates) for which the variation of action is equal to zero is associated with s dynamic invariants. Proof. We consider an arbitrary s-parametric Lie group and write infinitesimal transformations of this group: (4.48) x  ! x 0 D x  C ıx  ; ui .x/ ! u0i .x 0 / D ui .x/ C ıui .x/:

(4.49)

According to the general theory of Lie groups, these infinitesimal transformations can be represented via the parameters of the group ! n ; n D 1; s (i.e., via their infinitesimally small variations ı! n ): s X

ıx  D



X.n/ ı! n ;

nD1 s X

ıui .x/ D

‰i;.n/ ı! n ;

(4.50) (4.51)

nD1 

where X.n/ ; and ‰i;.n/ are structural constants depending on the analyzed group. Prior to writing the variation of action (4.43), we compute the variations of the field functions by taking into account the changes in their form at the same point: N i .x/ D u0 .x/  ui .x/ D u0 .x 0 /  ui .x/  u0 .x 0 / C u0 .x/ ıu i i i i  ıui .x/  @ ui .x/ıx  :

(4.52)

The last equality is written to within the infinitesimally small quantities of the first order in view of the relations u0i .x 0 /  u0i .x/  @ u0i .x/ıx  and u0i .x/ D ui .x/ C

55

Section 4.5 Noether’s Theorem and Dynamic Invariants

N i .x/: Thus, by using (4.50) and (4.51), we finally obtain ıu N i .x/ D ıu

s X



.‰i;.n/  @ ui .x/X.n/ /ı! n :

(4.53)

nD1

For the total variation of action, we find Z Z 0 0 0 ıA D L .x /dx  L .x/dx: 0

(4.54)



Under the transformation of coordinates (4.48), we get 0 ! : The Jacobian of this transformation specifies the transformed element of the volume dx 0 to within arbitrarily small quantities of the first order by the formula 1 0 00 01 02 03 3  X @.x ; x ; x ; x / @ıx A dx: (4.55) dx ' @1 C dx 0 D @.x 0 ; x 1 ; x 2 ; x 3 / @x  D0

N .x/; we Thus, by virtue of relations (4.55) and the similar relation (4.52) for ıL transform (4.54) as follows: Z Z N .x/ C @ .L .x/ıx  / ıA D dxŒıL .x/ C L .x/@ ıx   D dxŒıL 



Z

D

dx@



@L N ıui C L .x/ıx  : @.@ ui /



N via the In the last equality, we have used the representation of the variation ıL N ı-variations of the fields and their derivatives and the Lagrange–Euler equation of motion (4.47). Taking into account relations (4.50) and (4.53), we finally obtain ıA D 

s X

Z ı! n

nD1

where 

.n/ .x/ D 



@ .n/ .x/dx; 

@L   .‰i;.n/  @ ui .x/X.n/ /  L .x/X.n/ : @.@ ui /

(4.56)

In view of the arbitrariness of the 4-volume  and the parameters ı!n , we can write the equation of continuity in the form 

@ .n/ .x/ D 0:

(4.57)

56

Chapter 4

Lagrangian and Hamiltonian Formalisms of the Classical Field Theory

Integrating Equation (4.57) over the hyperplane x 0 D const and assuming that  the field (and, hence, .n/ ) vanish on the boundary of a spacelike surface (i.e., of the volume ƒ), we find Z 0 .x/ D const; n D 1; s: (4.58) J.n/ D d x .n/ Hence, every continuous s-parametric transformation of coordinates and the field functions (4.48), (4.49) corresponds to s integrals of motion J.n/ : 

Remark 4.7. The quantities .n/ are not uniquely defined. Thus, it is possible to add   to these quantities expressions of the form @ f.n/ satisfying the condition f.n/ D  f.n/ : In this case, the values of integrals (4.58) remain unchanged. Remark 4.8. If we consider a complex field ui .x/; then, in all relations containing the operation of summation over the components of the field (the sum over i ), this operation is carried out over all components of the field ui .x/ and all components of  the conjugate field ui .x/: We now consider some examples of fundamental groups corresponding to the most important integrals of motion.

4.6

Vector of Energy-Momentum

Consider a group of space-time translations along the axes x  : Then 

ıx  D ı!  ;

X.n/ D ın ;

‰i;.n/ D 0;

and relation (4.56) takes the form  .x/ WD T .x/ D

@L .x/ @ ui .x/  L .x/ı : @.@ ui .x//

(4.59)

@L .x/ @˛ ui .x/  g  L .x/: @.@ ui .x//

(4.60)

In the contravariant form, we get T  .x/ D g ˛ T˛ .x/ D g ˛

Z

The 4-vector P D

d x T 0 .x/

(4.61)

is a dynamic invariant. Comparing with the corresponding formulas of classical mechanics, we readily conclude that the zero component P 0 corresponds to the Hamiltonian function, i.e.,

57

Section 4.7 Tensors of Angular Momentum and Spin

represents the energy of the system, while the components P i ; i D 1; 2; 3; correspond to the vector of total momentum. Hence, in view of the covariance, the 4vector P  is called the 4-vector of energy-momentum and the tensor T  is called the energy-momentum tensor. Note that, in view of the ambiguity of its definition, the structure of the tensor T  may change. However, we are interested only the form of the uniquely defined vector P :

4.7

Tensors of Angular Momentum and Spin

Consider infinitesimal 4-rotations: 0

x  D x  C x ı!  :

(4.62)

In this case, the role of parameters !  is played by the angles of infinitely small rotations in the planes x  x  : In view of the antisymmetry !  D !  ; there are only six linearly independent parameters !  . < /; ; D 0; 1; 2; 3: Comparing (4.50) with (4.62) and taking into account the antisymmetry of the quantities !  ,   we determine the structural constants X.n/ D X appearing in the expression for   .n/ D  : ıx  D

X

 X ı!  D x ı!  D x ı ı! 

D xj‰0 .x/j2 D x0 : R1

It follows from this trivial example that the quantum state of a particle is connected with its classical state. We now describe the analogy between this trivial example and the quantization of classical fields. In the case of fields, the generalized coordinate is the field itself, i.e., the solution of the corresponding classical equation. Thus, it is clear that the quantum field must contain the classical solution as an internal component, and its mean value of the form (15.7) (over the physical vacuum) must coincide with the corresponding classical solution. In the same way, the energy of the ground state in the classic limit .„ ! 0/ must probably coincide with the minimum of the potential energy attained on the stationary solutions of the classical field equations. As for the scheme of canonical quantization of more complicated nonlinear models of the self-interacting fields which immediately takes into account the existence of the corresponding classical system of solitons, the reader can find it, e.g., in [160].

Chapter 16

Problems to Part III

Problem 11.1.

The gauge transformation (11.8) is called the Coulomb transformation if it leads to the potential B .x/ with B0 .x/ D 0 and r B.x/ D @k Bk D 0. Choose .x/ such that A00 .x/ D 0 and then 0 .x/ such that r  A00 .x/ D 0.

Problem 12.1.

Prove the validity of relation (12.7).

Problem 12.2.

Show that Lagrangian (12.12) is invariant under transformations (12.5), i.e., prove equality (12.14).

Problem 12.3.

b . Prove the Jacobi identity for the covariant derivatives D

Problem 13.1.

Show that transformations (13.29) and (13.24) turn Lagrangian (13.30) into (13.31).

Part IV

Second Quantization of Fields

Chapter 17

Axioms and General Principles of Quantization

Any theory is always based on certain fundamental principles (axioms) obtained as a result of observations, generalizations, or intuitive reasoning. Clearly, these principles depend on the level of comprehension of the natural phenomena in the given stage of development and may, in the course of time, turn out to be incomplete or even wrong. In this case, they are replaced with new principles and the old theory remains valid under certain assumptions or restrictions. Thus, this is true, e.g., for the classical mechanics which fails to explain the phenomena running in the microworld or the behavior of physical objects moving with very high velocities. Thus, the quantum mechanics and the special relativity theory were developed to explain these phenomena. They are based on the fundamentally different principles and axioms (as compared with the classical ideas) whose introduction made it possible to clarify various obscure phenomena. In the present part, we try to formulate some fundamental principles of the quantum field theory and give their physical (maybe, somewhat naive) substantiation.

17.1

Why Do We Need the Procedure of Second Quantization? Operator Nature of the Field Functions

The formalism of quantum mechanics enables us to describe systems in which the number of particles is the integral of motion and does not take into account the processes of their creation and annihilation running under the relativistic conditions. This circumstance explains the necessity of creation of a new mathematical technique capable of generalization of the available theory. Possible ways of this generalization follow from the concept of wave-particle duality according to which the Schrödinger description must be equivalent to the wave (field) description. In Chapter 1, we formulated (on the classical level) basic principles of this formalism. According to Noether’s theorem, the basic physical quantities (energy, momentum, spin, etc.) are determined in terms of the field functions, whereas according to the quantum ideas, they must be operators acting in the space of states. This means that the field functions must be “operators” 4 . Since the Schrödinger equation itself was obtained with the help of primary quantization, i.e., as a result of the replacement of the momenta p; coordinates b D i „@ t , respectively, q; and energy E by the operators pO D i „r , qO D q, and E 4

Here, we use the quotation marks because, as shown in what follows, these functions are the so-called operator-valued distributions but not operators.

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the procedure of new replacement of the field function of a particle by an operator is called the second quantization. The equivalence of the operator description (method of second quantization) of a system of N particles and the Schrödinger theory was established by P. Jordan and O. Klein [96] for particles with integer spin and by P. Jordan and E. Wigner [98] for particles with half-integer spin. In the most elegant form, this formalism was presented by V. Fock [60]. Thus, the formalism of second quantization appears independently of the problems of quantum field theory and can be regarded solely as a possible version of its construction. Here, we do not present the detailed analysis of this formalism. In the context of construction of the quantum field theory, this formalism appears in a natural way from the fundamental axioms of the theory. Prior to the analysis of these axioms, we recall three possible ways used in the quantum mechanics for the description of the physical phenomena or, in other words, three representations of the basic physical objects, states, and operators of physical quantities with the help of the so-called Schrödinger, Heisenberg, and interaction pictures.

17.2

Schrödinger, Heisenberg, and Interaction Pictures

In the quantum field theory, as in the quantum mechanics, the state of a physical system is completely characterized by the amplitude of state ˆ; i.e., an eigenvector of the operators of physical quantities which have specific values (i.e., can be measured in the state ˆ). The amplitude ˆ is a vector of a certain Hilbert space of states H . If ˆ corresponds to a pure physical state, then the norm of this vector is equal to 1: kˆk2H D .ˆ; ˆ/H D 1:

(17.1)

b does not have the exact value in the For a physical quantity B whose operator B b the mean value is is given by state ˆ (i.e., ˆ is not an eigenstate of the operator B), the formula b B D .ˆ; Bˆ/: (17.2) The physical system may evolve in time. This evolution can be described by using three equivalent representations (pictures) of the amplitudes ˆ and operators of the b dynamic quantities B. In the quantum mechanics, the Schrödinger picture is used most extensively. In the Schrödinger picture, the evolution of the system is described via the evolution of the amplitude of state ˆ D ˆ.t / with the help of the Schrödinger equation i

@ˆ.t / D Hˆ.t /; @t ˆ.0/ D ˆ0 ;

(17.3)

where H is the operator of total energy (Hamiltonian) of the system. We now formally assume that H is a self-adjoint .H  D H / operator in the space of states. It

Section 17.2 Schrödinger, Heisenberg, and Interaction Pictures

203

is independent of time if the system is closed. The operators of dynamic physical quantities are also independent of time, while their mean values (17.2) are functions of time via the amplitudes ˆ.t /. We can now formally 5 integrate Equation (17.3) and obtain its solution in the form ˆ.t / D e iH t ˆ0 D U.t /ˆ0 :

(17.4)

If the family of initial vectors ˆ0 for which relation (17.4) is meaningful is sufficiently large, then, instead of the evolution of states (17.4), we can consider the b Substituting (17.4) in the relation evolution of operators of the dynamic quantities B. for the mean value (17.2), we find 

b bH .t /ˆ0 /; B D B t D .ˆ0 ; U .t /BU.t /ˆ0 / D .ˆ0 ; B bH .t / D e iH t Be b iH t : B

(17.5)

Thus, we arrive at the Heisenberg picture in which the operators of dynamic quantities depend on time and the amplitudes of state are stationary quantities. Relation (17.5) yields the equation of motion in the Heisenberg picture: i

bH .t / @B bH .t /; H  : D ŒB @t

(17.6)

However, as a rule, the mathematical problem of finding the solutions of, say, the Schrödinger equation (17.3) for actual physical systems is quite complicated. In the quantum mechanics, this can be realized only in fairly simple cases or for fairly exotic exactly solvable models. In the quantum field theory, this problem is even more complicated. The most widespread method in the theoretical physics is the method of perturbation theory. According to this method, the total Hamiltonian of the system of interacting particles is represented in the form H D H0 C HI ; where H0 is the free Hamiltonian without interaction and HI is the interaction Hamiltonian which must be small in a certain sense as compared with H0 . Hence, it is sometimes convenient to somewhat simplify the Schrödinger evolution equation (17.3) by separating the process of free evolution. We seek the solution of Equation (17.3) in the form ˆ.t / D e iH0 t ‰.t /: Thus, Equation (17.3) yields the following equation for the vector ‰.t /: i 5

d ‰.t / D HI .t /‰.t /; dt

(17.7)

From the mathematical point of view, this is a fairly nontrivial problem (depending on the properties of the operator H ).

204 where

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HI .t / D e iH0 t HI e iH0 t :

The corresponding equation for the mean values (17.2) takes the form:     b iH0 t ‰.t / D ‰.t /B b t ‰.t / : B t D ‰.t /; e iH0 t Be

(17.8)

(17.9)

In the picture described by relations (17.7)–(17.9), the amplitudes and operators of the dynamic quantities vary as functions of time. This picture is called the interaction picture. In the literature, one can also find the name Dirac picture (see, e.g., [181], Chapter 11, Section 3).

17.3

Axioms of Quantization

AQ1. Existence of the Hilbert space of states H The physical state of a system is described by a function ˆ common for all fields and called the amplitude (vector) of state. This function belongs to a certain Hilbert space H . The relativistic law of transformation of states is determined by a continuous unitary representation of the inhomogeneous Lorentz group acting in H . Not all vectors from the Hilbert space H correspond to physical states. However, if ˆ describes a physical state, then it is a normalized vector in the space H , i.e., this vector satisfies Equation (17.1). On the contrary, the vector ˆ which is an eigenvector of the operator of a certain physical quantity may be not an element of H . Substantiation. This axiom is a consequence of the quantum-mechanical approach to the construction of the quantum field theory. The quantity jˆj2 can be interpreted as the probability density of finding the system in the state with the corresponding physical characteristics (momentum, spin, etc.) regarded as variables of the function ˆ. Remark 17.1. If we consider the phenomena specific for a certain group of particles (say, only for bosons or only for fermions), then it is possible to consider not the entire space H but only its certain subspace, e.g., HB or HF .

Section 17.3 Axioms of Quantization

205

AQ2. Basic axiom of quantization The operators of 4-momentum, the tensor of angular momentum, charge, etc., are infinitesimal operators of the corresponding transformations (i.e., representations in the space H ) of the inhomogeneous Lorentz group and can be expressed via the field operator functions by the same relations as in the classical theory.

Substantiation. According to AQ1, the representations of the Lorentz group, i.e., the operators Uƒ;a D UL corresponding to transformations (1.20), act in the space of states H : In the general theory of Lie groups for infinitesimally small Lorentz transformations, the operator UL can be represented in the form 1 Uƒ;a D 1 C iP  a C iM  ! ; 2

(17.10)

where P  and M  are called the infinitesimal operators (generators) and have the physical meaning of the vector of energy-momentum and the tensor of angular momentum, respectively. Similar relations can also be written for the groups of internal symmetries if we express the corresponding operators of representations in terms of the generators (charge, etc.). On the other hand, according to Noether’s theorem, P  , M  , charge Q, etc. are integrals of motion in the classical theory and can be expressed via the field functions ui .x/ by using relations (4.61), (4.67), and (4.73). Hence, AQ2 appears as a consequence of the principle of correspondence. AQ3. Operator-valued character of the fields The field ui .x/ is an operator-valued distribution, i.e., for a certain class of test functions E .M/ in the Minkowski space, the functional ui .f / D hui ; f i is an operator in H with dense domain of definition. If ui .x/ is a real field, then ui .f / is a self-adjoint operator in H . Substantiation. The fact that ui .x/ is an unusual function of the operator-valued character is actually a consequence of the fundamental axiom of quantization (AQ2), Noether’s theorem, and the quantum-mechanical approach. The fact that these quantities are operator-valued distributions but not ordinary operators becomes clear if we establish their commutation relations containing ı-functions or other distributions. The notation ui .f / D hui ; f i is often used in the form Z hui ; f i D ui .x/f .x/ dx;

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where the integral should be understood in a sense of the convolution of the distribution ui .x/ with a test function f .x/ 2 E . AQ4. Law of transformation of the field operators For the general inhomogeneous transformations of the Lorentz group x ! x 0 D Lx D ƒx C a;

(17.11)

the transformation properties of the operator field functions {transformed (in the classical theory) according to the law [see (1.24)] u0 .x/ D S.ƒ; a/u.ƒ1 .x  a//º

(17.12)

must satisfy the condition of consistency with the transformation properties of the amplitudes of states (transformed by using the operators UL ). This condition is expressed by the following operator relation: 

U L u.x/UL D S.ƒ; a/u.ƒ1 .x  a//

(17.13)

or, more strictly, in terms of the operators in the space H W 

U L u.fƒ;a /UL D S.ƒ; a/u.f /;

(17.14)

where fƒ;a .x/ D f .ƒ1 .x  a//. Substantiation. In the space of states H ; the representation of the Lorentz group is realized by the operators UL (see AQ2): ˆ ! ˆ0 D UL ˆ;

(17.15)

where UL is a unitary operator in the space H , i.e., 

UL1 D U



or U L UL D 1;

which follows from the condition of invariance of the norm of the vector ˆ .kˆ0 kH D kˆkH /. To clarify the law of transformation (17.13), (17.14), we consider a matrix element of the operator u.x/ between two states ˆ1 and ˆ2 : .ˆ1 ; u.x/ˆ2 / D u.x/: Q

(17.16)

207

Section 17.3 Axioms of Quantization

For the observer located in a coordinate system ¹x 0 º connected with the coordinate system ¹xº by the Lorentz transformation (17.12), the matrix element (17.16) should be computed by using the transformed amplitudes (17.15) at a new point x 0 :

0  ˆ1 ; u.x 0 /ˆ02 D uQ 0 .x 0 /: (17.160 ) It is clear that u.x/ Q and uQ 0 .x 0 / are the quantum-mechanical generalizations of the classical fields u.x/ and u0 .x 0 / connected by relation (1.24) [or (17.11)]. Hence, uQ 0 .x 0 / D S.ƒ; a/u.x/: Q

(17.17)

Thus, in view of (17.15) and (17.160 ), relation (17.17) implies the law of transformation (17.13). AQ5. Axiom of the spectral property The operator of energy-momentum P  is a self-adjoint operator in the space H : 2 Its spectrum lies in the upper light cone of the variable p D .p 0 ; p/; p 0  p2  0, p 0 > 0; and contains the point .0; 0; 0; 0/ corresponding to the vector ˆ0 ¤ 0 (P  ˆ0 D 0, ˆ0 2 H ) as an eigenvalue. The vector ˆ0 is called the vacuum vector or the ground state. The vector ˆ0 is invariant under the transformations UL : UL ˆ0 D ˆ0 :

(17.18)

AQ6. Axiom of local commutativity or the causality principle h

i  ui .x/; uj .y/ D 0 for ˙

.x  y/2 < 0:

(17.19)

Substantiation. This axiom is a consequence of the physical requirement called, for the local theories, the causality principle. This principle has the following meaning: Consider a physical quantity B.x/ at two points x, y 2 M of the Minkowski space separated by a spacelike interval, i.e., .x 0  y 0 /2  .x  y/2 < 0; (this means that the distance between these points in the space is larger than the distance passed by the signal emitted from the point x at time x 0 in the direction of the point y for the time period jy 0  x 0 j). Then, according to the quantum-mechanical ideas, the measurements of a physical quantity B.x/ at the points x and y and times x 0 and y 0 cannot affect each other. Hence, the operators of the measured physical quantity B at the points x and y and times x 0 and y 0 must commute.

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Thus, if the fields ui .x/ have a definite physical meaning or are linearly connected with the physical fields (as in the case of the electromagnetic field [see (8.4) and (8.5)], then their commutator must vanish for the spacelike intervals. However, if the fields do not have direct physical meaning and the physical quantities connected with these  fields depend on the bilinear combinations ui .x/ui .x/ [as in the case of the electronpositron field; see (6.115) or (6.119)], then the commutator of B.x/ and B.y/ may   turn into zero also in the case where the fields ui .x/, ui .x/ and ui .y/, ui .y/ are anticommuting for .x  y/2 < 0. Clearly, this does not mean that the outlined situation must be realized for all complex fields. However, by using this simple pure mathematical consideration, we can split all fields ui .x/ into two groups: The fields for which it is necessary to consider commutator in relation (17.19) (they are called boson fields) and the fields for which we must take the anticommutator in relation (17.19) (they are called fermion fields). As follows from the axioms presented in what follows, these statistics are connected either with particles with integer spins (Bose–Einstein fields) or with particles with half-integer spins (Fermi–Dirac fields). The corresponding particles are called bosons and fermions. AQ7. Axiom of canonical commutation relations Œui .x/; uj .y/˙;x 0 Dy 0 D Œi .x/; j .y/˙ D 0; Œuj .x/; k .y/˙;x 0 Dy 0 D i ı.x  y/ıj k ;

(17.20)

where k .x/ is a quantity canonically conjugated to the fields uk .x/. Substantiation. In fact, this axiom is a generalization of the quantum-mechanical axioms proposed by Bohr: ŒpOk ; qOj  D i „ıkj I

ŒpOk ; pOj  D ŒqO k ; qOj  D 0:

The role of generalized coordinate in the quantum field theory is played by the field itself. The canonically conjugated quantity (momentum) is computed according to Section 4.9 [see formulas (4.75)]. To guarantee the consistency with AQ6, it is necessary to choose the commutator for the Bose fields and the anticommutator for the Fermi fields.

209

Section 17.3 Axioms of Quantization

AQ8. Pauli theorem on the relationship between spin and statistics [152]6 The fields corresponding to particles with integer spins are quantized according to the Bose–Einstein statistics, whereas the fields corresponding to particles with half-integer spins are quantized according to the Fermi–Dirac statistics.

We present a detailed substantiation (“proof”) of this theorem in Section 21.5. Here, we only give some strong arguments in favor of its validity. Mathematically, this theorem can be formulated as follows: If the operators ui .x/ are Bose fields, i.e., the fields used for the description of particles with integer spins, then their commutator satisfies the relation: i h  (17.21) ui .x/; uj .y/ D B ij .x  y/1: 

If the operators ui .x/ are Fermi fields, i.e., the fields used to describe particles with half-integer spins, then their anticommutator satisfies the relation: i h  (17.22) ui .x/; uj .y/ D F ij .x  y/1: C

F In these relations B ij and ij are c-functions, i.e., the right-hand sides of (17.21) and (17.22) contain the operators equal to multiples of the identity operator 1. The F properties and form of the functions B ij and ij depend on the specific realization of the fields ui .x/. However, some basic properties of these fields can be studied even now. Thus, the AQ6 and AQ7 axioms imply that F B ij .x  y/ D ij .x  y/ D 0 at

.x  y/2 < 0;

@ B @ ij .x  y/jx 0 Dy 0 D 0 F .x  y/jx 0 Dy 0 D i ıij ı.x  y/: 0 @x @x ij Substantiation. We again consider the classical theory and the principle of correspondence. In the classical mechanics, an important role in the investigation of the evolution of the system is played by the Poisson brackets. Dirac proved that the commutator is the analog of the Poisson brackets for quantum systems (see Section 4.2 or [40]). The classical Poisson brackets of the field functions ¹ui .x/; uj .y/º D ij .x  y/ do not depend on the fields ui explicitly but depend on the difference of the arguments. By using, e.g., the expressions (5.20) and (5.21) or (6.47) and (6.48) for the Fouriertransforms of the free fields, we immediately conclude that, for the commutator of 6

The words “theorem” (“proof”) are used here, in fact, in a formal sense and do not have strict mathematical meaning.

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Axioms and General Principles of Quantization



the fields ui .x/ and ui .y/ to depend on the difference of the arguments x  y, it is necessary that the commutator or anticommutator of their Fourier-transforms ui .k/  and uj .p/ be proportional to the Dirac ı-function, i.e., i h  (17.23) ui .k/; uj .p/ ı.k  p/: ˙

To substantiate the choice of statistics (i.e., the commutation or anticommutation relations), we consider the formulas for the energy of free classical particles. Thus, in Chapters 5, 7, and 8 [see relations (5.60), (7.29), and(8.37)], for particles with integer spin (scalar and vector), it was shown that Z i h  C  .k/u .k/ C u .k/u .k/ : (17.24) P 0 D d k k 0 uC i i i i At the same time, for particles with half-integer spin (i.e., for the electron-positron field), we have [Chapter 6, relation (6.120)]: P0 D

2 Z X

    N a C .k/ N ; d k k 0 aC .k/a .k/  a . k/ s s s s

(17.25)

sD1

p

where k 0 D k2 C m2 is the free energy of particle with mass m and momentum kN and k 0 D jkj for massless particles. Moreover, the independent amplitudes as˙ in ˙ N relation (17.25) are connected with u˙ i (or ˛ .k/) by relation (6.76). ˙ ˙ In the quantum theory, the amplitudes ui .a / are operators. Therefore, in relations (17.24) and (17.25), we arrange them in the order in which the corresponding  functions ui and ui appear in the classical relations for P 0 . To substantiate the introduction of anticommutators for spinor fields, we consider the relation for the energy of the Dirac field in a finite volume ƒ. Then, by using the discrete representation for the fields (see, e.g., [26], Section 3.3), relation (17.25) can be rewritten in the form  p X   C  k 0 aC a  a a (17.26) k 0 D k2 C m2 : P0 D s;k s;k ; s;k s;k k;s

If we do not take into account the second term, then it becomes clear that the oper  ator Ns D aC s;k as;k is simply the operator of the number of particles with momentum k and spin s. According to the Pauli principle, which enables us to establish the agreement between the results of theoretical investigations and the experimental data, two or more electrons cannot occupy the same state. This means that the operator Ns must have the eigenvalues ns D 0; 1. We now show that this is realized in the case where the

211

Section 17.3 Axioms of Quantization 

˙ operators as;k and a˙ s;k satisfy the anticommutation relations of the form

h i i h ˙ ˙ ; as˙0 ;k0 D a˙ D 0; as;k s;k ; as 0 ;k C h i C h i  ˙ ; as0 ;k D a˙ D 0; as;k s;k ; as 0 ;k C C h i  a˙ s;k ; as 0 ;k D ıss 0 ıkk0 :

(17.27) (17.28) (17.29)

Indeed, relation (17.27) with s D s 0 and k D k0 implies that ˙ 2 2.as;k / D 0:

Then











C  C 2  2 C   C  Ns2 D aC s;k as;k as;k as;k D as;k as;k  .as;k / .as;k / D as;k as;k ;

i.e., Ns2 D Ns : This yields ns D 0; 1: As one more important argument in favor of the choice of commutation relations for the Bose fields and anticommutation relations for the Fermi fields, we can mention the requirement of positivity of the operators of energy for the free fields [relations (17.24)–(17.26)]. It is easy to see that the second term in the relations for the energy of Fermi fields (17.25) and (17.26) is negative. In order to make it positive C definite, it is necessary to permute the operators a s;k and as;k by taking into account the fact that the anticommutator of these operators gives (by assumption) the ı-function, i.e., i h  a˙ .k/; a .k/ D ıss 0 ı.k  k0 /: 0 s s C

Thus, operator (17.25) takes the form    XZ XZ C C  P0 D d k k 0 aC d kk 0 ı.0/: a C a a s;k s;k s;k s;k  s

s 

˙ In Section 17.5, it is shown that the operators as;k . a˙ s 0 ;k / can be interpreted as the operators of creation and annihilation of particles (antiparticles) with momentum k and spin s. The negative term in the new expression for the operator P 0 is an infinite constant. It should be omitted for the following reasons: If we define the vector of the vacuum state, i.e., the state 0 without particles in the space of states H ; then (17.30) Ns 0 D 0:

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Hence, in view of the fact that we consider the theory of the free field, the energy of this state is equal to zero, i.e., P 0 0 D 0:

(17.31)

In fact, this means that the order of operators of the fields in the relation for energy, as well as and in the formulas for the operators of the other dynamic quantities should be established from the very beginning. This yields the axiom of the so-called normal form for the operators of dynamic quantities. AQ9. Axiom of the normal form for the operators of dynamic quantities or “Wick regularization” In the procedure of quantization of the fields, i.e., in the representation of the dynamic quantities via the operators of fields, it is necessary to express the operators of free fields in terms of the operators of creation and annihilation and write them in the normal form (in the form of the normal product), i.e., the operators of creation should be placed to the left and the operators of annihilation to the right.

Note. In the coordinate space, it is better to denote the normal product by two colons placed to the left and to the right of the corresponding expression. Thus, we write C C C     W ui .x/uj .y/ W D uC i .x/uj .y/ C ui .x/uj .y/ C "uj .y/ui .x/ C ui .x/uj .y/; (17.32) where " D C1 for the Bose fields and " D 1 for the Fermi fields. For the first time, this product was introduced by G.-C. Wick [217]. Hence, it is also frequently called the Wick product.

Remark 17.2. The axiom AQ9 is formulated for the dynamic quantities which can be represented in terms of the operators of free fields. In the case of interacting fields, there is no partition into the operators of creation and annihilation. Hence, the process described above should be understood as a regularization of the ordinary product of Heisenberg fields. In order to illustrate this process, we write product (17.32) in the form (17.33) W ui .x/uj .y/ W D ui .x/uj .y/  . 0 ; ui .x/uj .y/ 0 /: For any number of fields, this relation represents the well-known Wick theorem considered in Section 23.4. If ui .x/ is the interacting Heisenberg field, then relation (17.33) should be replaced by a similar relation W ui .x/uj .y/ W D ui .x/uj .y/  .ˆ0 ; ui .x/uj .y/ˆ0 /;

(17.34)

Section 17.4 Relativistic Heisenberg Equation for Quantized Fields

213

where ˆ0 is the wave function of the physical vacuum, i.e., the ground state of the total Hamiltonian P 0  H D H0 C HI . In the case where x D y and i D j; relation (17.34) should be understood as the result of regularization of the product of fields: (17.35) W ui .x/2 W D lim W ui .x/ui .y/ W : y!x

The combinatorial structure of the normal product of several fields is determined by the Wick theorem (see Section 23.4).

17.4

Relativistic Heisenberg Equation for Quantized Fields

In Section 17.3, we have deduced the Heisenberg equation (17.6) for any quantumbH .t / by using transformation (17.5), i.e., its evolution. In the mechanical operator B relativistic case, we act in a similar way with regard for the general law of transformation (17.13) of any operator B.x/ under the general inhomogeneous Lorentz transformations (17.12). Assume that the operator B.x/ varies under the transformations of coordinates (17.12) in the space H with the representation UL D Uƒ;a according to the rule 

U L B.x/UL D SL B.ƒ1 .x  a//:

(17.36)

Here, the operator B.x/ is not necessarily a field. It can be the operator of generalized momentum or another operator depending on the coordinate variables and relativistically invariant in a sense of relation (17.13). We now choose a special case of Lorentz transformations, i.e., Lorentz translations .ƒ D 1/. Then, according to the main axiom of quantization AQ2 [see (1.45)], we get  (17.37) U.1; a/ WD UI;a D e iP a : In this special case, relation (17.36) can be rewritten in the form 

B.x C a/ D UI;a B.x/U I;a :

(17.38)

Assume that a D ıx  , i.e., we consider infinitely small Lorentz translations. Then we expand the operator function B.x C ıx/ on the left-hand side of (17.38) and the exponential function on the right-hand side of (17.37) in series. Omitting the terms of the second, third, and higher degrees in ıx  , we have B.x/ C

@B.x/  ıx D .1 C ig  P  ıx  /B.x/.1  ig  P  ıx  /: @x 

This yields the Heisenberg equation in the relativistic invariant form ig 

@B.x/ D ŒB.x/; P   : @x 

(17.39)

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Equation (17.39) is a quite general equation. For each specific field, it will take the corresponding form, by depending on the operator of energy-momentum P  . We now determine the form of these equations for a free real scalar field '.x/ (as an example of a Bose-field) and for a free electron-positron field .x/ (as an example of a Fermi-field). The initial condition for these equations are the equal-time commutation (anticommutation) relations (17.20).

17.4.1 Heisenberg Equation for a Free Scalar Field The classical free scalar field satisfies the Klein–Fock–Gordon equation (see Chapter 5). It is clear that a quantum field should also satisfy this equation. Therefore, Equation (17.39) written for the free scalar field '.x/ must be consistent with Equation (5.6). Consider firstly Equation (17.39) at  D 0 and B.x/ D '.x/: i

@'.x/ D Œ'.x/; P 0 : @x 0

(17.40)

The operator of energy P 0 calculated in Section 5.6 has the following form in the “x”-representation (coordinate representation): Z 1 d y.W @ '.y/@ '.y/ W Cm2 W '.y/2 W/: (17.41) P0 D 2 In order to calculate the commutator in formula (17.40), we use the elementary relation of commutation for operators a and b 2 : Œa; b 2  D Œa; b b C bŒa; b : Then we can take y 0 D x 0 in formula (17.41) for P0 ; by taking the energy conservation law into account. In view of the equal-time commutation relations (17.20) (with regard for (5.63) for a real field), we have  Z @'.x 0 ; y/ @'.x/ @'.y/ D dy W W '.x/; i @x 0 @x 0 @y 0 ;x 0 Dy 0 Z @'.x 0 ; y/ @ C dy W Œ'.x/; '.y/;x 0 Dy 0 W @y k @y k Z C m2 d y W '.x 0 ; y/Œ'.x/; '.y/;x 0 Dy 0 W Z @'.x/ D d y@0 '.x 0 ; y/i ı.x  y/ D i : @x 0 Hence, Equation (17.39) for '.x/ is the identity. It is easy to verify that the analogous result is valid for  D k D 1; 2; 3:

215

Section 17.4 Relativistic Heisenberg Equation for Quantized Fields

Consider now Equation (17.39) for the generalized momentum of a scalar field .x/ D @0 '.x/. Then, as in the previous case, we obtain  Z @'.x 0 ; y/ @'.x/ @'.y/ @2 '.x/ D dy W ; W i .@x 0 /2 @x 0 @x 0 @y 0 ;x 0 Dy 0  Z @'.x 0 ; y/ @ @'.x/ C dy W ; '.y/ W @y k @y k @x 0 ;x 0 Dy 0  Z @'.x/ C m2 d y W '.x 0 ; y/ ; '.y/ W @x 0 ;x 0 Dy 0 Z Z @2 '.x 0 ; y/ 2 D i dy ı.x  y/  m i d y'.x 0 ; y/ı.x  y/: (17.42) @y k @y k In view of Definition (5.5), this yields .  m2 /'.x/ D 0: For the second term of (17.42), we used the formula of integration by parts and assumed that the field vanishes at infinity. Hence, the Heisenberg equation for @0 '.x/ coincides with the Klein–Fock–Gordon equation for '.x/.

17.4.2 Heisenberg Equation for a Free Electron-Positron Field Consider now Equation (17.39) at  D 0 and B.x/ D i

@

˛ .x/ DŒ @x 0

˛ .x/; P

0

˛ .x/:

 ;

(17.43)

where ˛ .x/ is the operator of Fermi-field which satisfies the equal-time anticommutation relations ˛ .x/;

Œ Œ



ˇ .y/C;x 0 Dy 0

˛ .x/ ˇ .y/C;x 0 Dy 0

D ı˛ˇ ı.x  y/; DŒ



˛ .x/;



ˇ .y/C;x 0 Dy 0

(17.44) D0

which are a consequence of axiom AQ7 (formulas (17.20)) and relations (6.129) for the canonical generalized momenta of fermion fields. We now write formula (6.118) for P 0 in the form Z   i 0 P D d y. .y/@0 .y/  @0 .y/ .y//: (17.45) 2 Here, let us take y 0 D x 0 (since, by the energy conservation law, P 0 is independent of time). It is easy to verify, by substituting (17.45) in (17.43) and calculating

216

Chapter 17

Axioms and General Principles of Quantization

the commutator (with regard for the anticommutation relations (17.44)), that Equation (17.43) becomes the identity. In the same way, the Heisenberg equation for the 



momentum .x/ D i .x/ leads to the analogous identity for the field .y/. In order to deduce the equation for the field .x/, we assume that the Dirac equations (6.7) and (6.16) are valid not only for the classical fields .x/ and N .x/; but also for quantum fields. Then relations (6.7) and (6.14) yield i @0 .x/ D i  0  k @k .x/ C m 0 .x/; 

(17.46)





i @0 .x/ D i @k .x/ 0  k  m .x/ 0 : Hence, relation (17.45) for P 0 takes the form Z    1 d yΠ.y/ 0 i  k @k .y/ C m .y/  P0 D 2 



.i @k .x/ k  m .y// 0 .y/:

(17.47)

Let us substitute (17.4.2) in (17.43) and use relations (17.44). Then we obtain @

˛ .x/ D @xZ0 1 d y¹Œ D 2

i

Œ

˛ .x/;

˛ .x/; .i @k





 0 k .y/ ˇ ˇ i  @k

k ˇ .y/ˇ

 mıˇ



 .y/

0 ˇ .y//

C mı

  .y/

;x 0 Dy 0

 .y/;x 0 Dy 0 º:

By opening the commutators and grouping the terms to obtain the anticommutators, we rewrite the last formula in the matrix form, Z   1 @ d y¹ı.x  y/ 0 i  k @k .y/ C m .y/  .i @k ı.x  y/ k  i 0 D @x 2 m 0 .y//ºx 0 Dy 0 D  0 .i  k @k C m/ .x/; (17.48) where we used again the formula of integration by parts and the commutation relations of -matrices (6.4) in the first term in the second brackets. Multiplying (17.4.2) by  0 , we obtain the equation for .x/: .i   @  m/ .x/ D 0: It coincides with the Dirac equation (6.7). Remark 17.3. At the first glance, the results following from formulas (17.45)– (17.4.2) are tautological. But they mean: by starting from the assumption that the operators .x/ and N .x/ satisfy the Dirac equation (6.7) and (6.17), we made conclusion that this assumption is consistent with the assertion about the correspondence of the Heisenberg equation (17.43) to the Dirac equation under the condition that the fields

.x/ and



.y/ satisfy the equal-time anticommutation relations.

217

Section 17.5 Physical Content of Positive- and Negative-Frequency Solutions

17.5

Physical Content of Positive- and Negative-Frequency Solutions of Equations for Free-Field Operators

It is seen from the previous sections (Sections 5.3 and 6.3.1) that the free fields as solutions of the corresponding equations have the structure  ui .x/ D uC i .x/ C ui .x/;

where u˙ i .x/ D

1 .2/3=2

Z

e ˙ikx ı.k 2  m2 /uQ ˙ i .k/d k:

In quantum theory, the quantity u˙ i .x/ are relativistic invariant (in the sense of (17.13)) operators and satisfy the Heisenberg equation (17.39). Let us write this equation for the Fourier-transforms uQ ˙ i .k/:   C Q i .k/; ŒuQ C i .k/; P  D k u  ŒuQ  i .k/; P 

D

k  uQ  i .k/:

(17.49) (17.50)

Let now ˆp be the state amplitude with a certain value of the 4-vector of energymomentum. In other words, (17.51) P  ˆp D p  ˆp :  Consider the new states uQ C Q i .k/ˆp and u i .k/ˆp . Let us act by the operator P on these states and use relations (17.49) and (17.50) and condition (17.51):   C P  uQ C QC Q i .k/ˆp D .p  C k  /uQ C i .k/ˆp D u i .k/P ˆp C k u i .k/ˆp ;    P  uQ  Q Q i .k/ˆp D .p   k  /uQ  i .k/ˆp D u i .k/P ˆp  k u i .k/ˆp :  with the Hence, the vectors uQ ˙ i .k/ˆp are again the eigenstates of the operator P eigenvalues p  ˙ k  and k 2 D m2 . This can be interpreted as if the operator uQ C i .k/ has “created” a free particle with momentum k, by acting the state amplitude ˆp , and the operator uQ  i .k/ has annihilated a free particle with momentum k. Such terminology is commonly accepted, and creation and annihilation of a particle the operators u˙ i .k/ are called the operators of p 0 with momentum k and with the energy k D k2 C m2 .

Remark 17.4. The calculations performed in the present chapter should be considered on the pure formal level. The rigorous mathematical substantiation of quantized fields and the quantities connected with these fields by some mathematical relations is presented in what follows.

Chapter 18

Quantization of the Free Scalar Field

We now establish the form of the quantum fields for the specific values of spin and mass of the particles. First, we consider this problem for a very simple field, namely, for the real scalar field corresponding to particles with spin 0, mass m; and charge 0.

18.1

Commutation Relations. Commutator Functions

In the substantiation of the axiom AQ8, we made a conclusion that the commutator (or anticommutator) of the Fourier transforms of the field operators must be proportional to the Dirac ı-function [see formula (17.23)] in order to guarantee the invariance of the commutator function of the fields ui .x/ under the group of translations. We now find the exact expression for this commutator for the real scalar field '.x/ by using the axiom AQ7 of equal-time commutation relations (17.20). In view of relation (5.62) (for the real field), this yields Œ'.x/; @0 '.y/;x 0 Dy 0 D i ı.x  y/:

(18.1)

Then we write the formula for the field '.x/ D ' C .x/ C '  .x/ in the form of a Fourier integral (see (5.20)–(5.22)) Z ikx Z ikx 1 e e 1 C '.x/ D ' .k/d k C '  .k/d k; (18.2) p p 3=2 0 0 .2/3=2 .2/ 2k 2k whence we obtain Z ikx e 1 @0 '.x/ D .i k 0 /' C .k/d k p 3=2 .2/ 2k 0 Z ikx e 1 .i k 0 /'  .k/d k: (18.3) C p 3=2 .2/ 2k 0 Let us multiply (18.2) by p 0 e ipx and then by p 0 e ipx . In addition, we multiply (18.3) by i e ipx and by i e ipx ; respectively, and integrate with respect to the Lebesgue measure d x. Taking formula (5.58) into account and summing the obtained equations, we have Z 1 e ipx 0 Œp '.x/  i @0 '.x/; d x ' C .p/ D p .2/3=2 2p 0 Z e ipx 1 '  .p/ D d x Œp 0 '.x/ C i @0 '.x/: p .2/3=2 2p 0

Section 18.1 Commutation Relations. Commutator Functions

219

In view of (18.1) and (5.58), this yields Z e ipxiky 1  C Œ' .p/; ' .k/ D d xd y i.p 0 C k 0 /Œ@0 '.y/; '.x/;x 0 Dy 0 p 0 0 .2/3 2 p k D

e ip

0 x 0 ik 0 y 0

p 2 p0k0

jx 0 Dy 0 .p 0 C k 0 /ı.p  k/:

p p Hence, with regard for the relations p 0 D p C m2 , k 0 D k2 C m2 ; and p D k, we have Œ'  .p/; ' C .k/ D ı.p  k/: Analogously, we obtain the commutators e ip x ix y .p 0  k 0 /ı.p C k/ D 0: Œ' .p/; ' .k/ D ˙ p 2 p0k0 0 0

˙

0

0

˙

Hence, the Fourier-transforms of the fields ' ˙ .x/ satisfy the following commutation relations: Œ' ˙ .p/; ' ˙ .k/ D 0; Œ'  .p/; ' C .k/ D ı.p  k/:

(18.4)

Relations (18.4) allow us to calculate the commutator of the fields '.x/ and '.y/: Œ'.x/; '.y/ D Œ' C .x/; '  .y/ C Œ'  .x/; ' C .y/ ; Z d p ip.xy/ 1 1 e WD D C .x  y/; Œ' C .x/; '  .y/ D  3 0 .2/ 2p i Z d p ip.xy/ 1 1 Œ'  .x/; ' C .y/ D e WD D  .x  y/: .2/3 2p 0 i

(18.5) (18.6)

The functions D ˙ .x  y/ which are determined by integrals (18.5) and (18.6) can be rewritten in the covariant form with regard for formula (5.19) as follows: Z i dp .p 0 /e ip.xy/ ı.p 2  m2 /; D C .x  y/ D  .2/3 (18.7) Z i D  .x  y/ D dp .p 0 /e ip.xy/ ı.p 2  m2 /: .2/3 Finally, the complete Pauli–Jordan commutator function takes the form D.x  y/ D D C .x  y/ C D  .x  y/ Z i e ip.xy/ ı.p 2  m2 /".p 0 /dp; D .2/3

(18.8)

220

Chapter 18

Quantization of the Free Scalar Field

where ".p 0 / D .p 0 /  .p 0 / D

p0 ; jp 0 j

(18.9)

and the complete commutator of the fields '.x/ and '.y/ is 1 Œ'.x/; '.y/ D D.x  y/: i Remark 18.1. In the literature, one can find (see, e.g., [181, 16]) a somewhat different division of the Pauli–Jordan function .x  y/ ..x  y/ D D.x  y// into positive- and negative-frequency parts: ˙ .x  y/ D  .x  y/: But the commutator of the fields '.x/ and '.y/ remains the same: 1 Œ'.x/; '.y/ D D.x  y/ D i .x  y/: i The Pauli–Jordan function, i.e., integrals (18.5) and (18.6), can be calculated explicitly. Such calculation was executed in [26], Section 15 and gave the function D.x/ D

p m 1 ".x 0 /ı./  p ./".x 0 /J1 .m /;  D x 2 ; 2 4 

(18.10)

where J1 .z/ is the Bessel function of the first kind. It is seen from the explicit form of the function D.x/ that it is invariant under the group of Lorentz orthochronous transformations and becomes zero at  < 0. Hence, the commutator 1 Œ'.x/; '.y/ D D.x  y/  0 at i

.x  y/2 < 0;

which agrees with axiom AQ6 of local commutativity.

18.2

Complex Scalar Field

Unlike a real scalar field, a complex scalar field corresponds to charged scalar bosons with charges of both signs ˙e and is characterized by two independent components  Re '.x/ and Im '.x/ or (equivalent ones) '.x/ and '.x/. Then, according to axiom AQ7 (see (17.20)) and relations (5.62) and (5.63), the corresponding equal-time commutation relations take the form i h  D i ı.x  y/; (18.11) '.x/; @0 '.y/ ;x 0 Dy 0 i h '.x/; @0 '.y/ D i ı.x  y/: (18.12) 0 0 ;x Dy

221

Section 18.3 Operator Relations for Dynamic Invariants

It is easy to see that relation (18.12) is complex conjugate to (18.11). If we repeat the process in Section 18.1, the commutation relations (18.4) are replaced by i h i h  (18.13) '  .p/; ' C .k/ D '  .p/; ' C .k/ D ı.p  k/: 



All other commutators are zero. In the coordinate representation, we have, respectively, h

i 1  '  .x/; ' C .y/ D D  .x  y/I  i i h 1  ' C .x/; '  .y/ D D C .x  y/I  i h i 1  '.x/; '.y/ D D.x  y/:  i

18.3

(18.14)

(18.15)

Operator Relations for Dynamic Invariants

In Section 5.7, we obtained the relations for dynamic invariants in the classical theory of a scalar field, namely, for energy-momentum (5.60) and charge (5.61). By virtue of the axioms of quantization AQ8 and AQ9 and the arguments substantiating these axioms, we need to permute the operator of creation ' C .k/ and the operator of anni hilation '  .k/ in the mentioned relations or to introduce the sign of a normal product (see formula (17.32)) in the formulas for P  , j  ; and Q in the coordinate space (formulas (5.60) and (5.61)). Finally, these relations takes the form Z p    (18.16) P D d k k  Œ' C .k/'  .k/ C ' C .k/'  .k/; k 0 D k2 C m2 Z

and QDq





d kŒ' C .k/'  .k/  ' C .k/'  .k/:

(18.17)

Chapter 19

Quantization of the Free Spinor Field

19.1

Commutator Functions of Fermi Fields

As in the previous chapter, we start from the equal-time anticommutation relations (17.20). In view of (6.129), they take the form (; D 1; 4/   D ı ı.x  y/;  .x/;  .y/ 

C;x 0 Dy 0

  .x/;  .y/ C;x 0 Dy 0 D





 .x/;  .y/

We now write the expressions for the operators and (6.79), (6.80)]:  .x/ D

1 .2/3=2

Z d k e ikx Z

(19.1)



2 X

C;x 0 Dy 0

 .x/ and



D 0:

 .y/ [see (6.47)–(6.49)

0

s ;C v .k/asC0 .k/

s 0 D1

2 X 1 ikx s 0 ; C dk e v .k/as0 .k/; 3=2 .2/ 0 s D1 Z 2 X  1 C d p e ipy v s;C  .y/ D  .p/as .p/ 3=2 .2/ sD1 Z 2 X 1   ipy C d p e v s;  .p/as .p/: .2/3=2 sD1 

(19.2)

(19.3) 0

ipx from the left and (19.3) by v s ; .k/e iky Let us multiply (19.2) by v s;  .p/e  from the right. Integrating and summing with respect to x and  (respectively, with respect to y and ) and taking relations (5.58), (6.81), and (6.83) into account, we 

obtain the operators a˙ and a˙ in terms of the operators fields , Z 1  as˙ .p/ D d x e ipx v s; .p/ .x/; .2/3=2 Z  1 0  iky d y e a˙ .k/ D .y/v s ; .k/: s0 .2/3=2



:

223

Section 19.1 Commutator Functions of Fermi Fields

Then relations (19.1), (6.81), and (6.83) yield 

C Œa s .k/; as 0 .p/C D ıss 0 ı.k  p/;  Œas .k/; aC s 0 .p/C

Dı

ss 0

(19.4)

ı.k  p/: 



C 0  0 All other relations of anticommutation between asC .k/, a t .p/, as 0 .k /, and a t 0 .p / 0 0 .s; t; s ; t D 1; 2/ become zero. It is easy now to calculate the anticommutator of the  fields .x/ and N .y/ . N .y/ D .y/ 0 / which determines the commutator function of Fermi-fields:

1 S˛ˇ .x  y/ WD Œ i D

˛ .x/;

1 .2/3

N ˇ .y/C D Œ

Z d k e ik.xy/

C ˛ .x/; 2 X

N  .y/C Œ ˇ

;

v˛;C .k/vN ˇ

 ˛ .x/;

N C .y/C ˇ

.k/

D1

Z 2 X 1 ;C ik.xy/ v˛; .k/vN ˇ .k/ C dk e .2/3 D1 Z O .k  m/˛ˇ 1 D dk e ik.xy/ 3 .2/ 2 0 Z O 1 ik.xy/ .k C m/˛ˇ C e d k; .2/3 2k 0

(19.5)

where formulas (6.84) and (6.85) are used. With regard for formulas (5.19) and (18.9), we can easily rewrite the commutator function S.x  y/ in the covariant form Z i (19.6) e ik.xy/ ı.k 2  m2 /".k 0 /.kO  m/ d k: S.x  y/ D .2/3 Comparing it with relation (18.8), we see that S.x/ D .i   @ C m/D.x/:

(19.7)

It follows from relation (19.7) that this function satisfies the Dirac equation .i   @  m/S.x/ D .i   @  m/.i @ C m/D.x/ D .  m2 /D.x/ D 0: Since the function D.x  y/=0 at .x  y/2 < 0, the anticommutator of the fields .x/ and N .y/ becomes zero inside the light cone (i.e., at points separated by a spacelike interval), which agrees with axiom AQ6. As in the case of the commutator function of a scalar field (see (18.8)), the function S.x/ can be divided into the positive- and negative-frequency parts: S.x/ D S C .x/ C S  .x/; S ˙ .x/ D .i   @ C m/D ˙ .x/

(19.8)

224

Chapter 19

Quantization of the Free Spinor Field

or Z O i ikx .k  m/ e d k; S .x/ D .2/3 2k 0 p k 0 D k2 C m2 : ˙

(19.9)

In conclusion, we present also the commutation relations for charge-conjugate fields which correspond to antiparticles. Using relations (19.5) and the definition of charge-conjugate fields (6.100), we have c;˙

.x/ D C N ˙ .x/> ;

N c;˙ .x/ D



c;˙

.x/ 0 :

(19.10)

With regard for properties (6.95), we obtain i N c;C .y/ D ˇ C i h c; c;C N D ˛ .x/; ˇ .y/ h

c; ˛ .x/;

C

19.2

1  S .x  y/; i ˛ˇ 1 C S .x  y/: i ˛ˇ

(19.11)

Dynamic Invariants of a Free Spinor Field

In Section 6.4, we established the form of basic dynamic invariants, the 4-vector of energy-momentum and the charge for classical spinor fields (see formulas (6.120), (6.128)). But, according to the axioms of quantization AQ8 and AQ9, the operator formulas corresponding to these quantities should be rewritten in the normal form. In other words, we should permute the operators of creation aC .k/ and annihilation 

a  .k/ and take into account that they satisfy anticommutation relations. Then the corresponding formulas take the form Z P D

d k k Z

Q D e

2   X    C aC .k/a .k/ C a .k/ a .k/ ; s s s s sD1 2  X

    C aC s .k/as .k/  as .k/as .k/ :

dk

(19.12)

(19.13)

sD1

For the corresponding relations in the “x” space, we can preserve the formulas (6.119) and (6.127) by introducing the sign of normal product on both sides.

Chapter 20

Quantization of the Vector and Electromagnetic Fields. Specific Features of the Quantization of Gauge Fields 20.1

Quantization of the Complex Vector Field

As a specific feature of the quantization of vector fields, as compared with the quantization of scalar fields, we can mention the fact that, in addition to the determination of the commutation relations for the components of the vector field V  .x/, this process should be made consistent with the additional Lorentz condition (7.2) guaranteeing the positivity of the operator of energy P 0 . In the classical case, the operator P 0 is expressed via three linearly independent amplitudes of the corresponding linear po larizations aj˙ .k/ and aj˙ .k/ [see (7.23)]: Z 0

P D

3   p X   2 2 dk k C m ajC .k/aj .k/ C aj .k/ajC .k/ :

(20.1)

j D1

It follows from the form of P 0 that the conditions of axioms AQ5 and AQ9 are satisfied if the indicated three independent amplitudes aj .k/ are quantized according to the Bose–Einstein statistic, i.e., they must satisfy the following commutation relations:  .k0 / D ıj k ı.k  k0 / (20.2) Œaj .k/; aC k and the rule of adjointness is identical to the corresponding rule in the classical theory, 

.aj˙ .k// D aj .k/;

(20.3)

but in a sense of the operator adjointness in the Hilbert space of states H . Remark 20.1. Postulating the commutation relations (20.2) only for the spatial components aj .k/, j D 1; 2; 3 which are coefficients of the expansion of potentials V ,  D 0; 1; 2; 3; in the longitudinal and transverse polarizations (see Section 7.3) in .x/ , the classical theory, we choose, in fact, the quantities Vj .x/ and  j .x/ D @.@@L 0 Vj .x// j D 1; 2; 3; as the canonical variables. At the end of this section, we show that such choice is consistent with the theory of quantization of systems in the presence of constraints which was briefly described in Section 2.3.2).

226

Chapter 20 Quantization of Gauge Fields

Hence, the quantization of a vector field leads to the formulas for the vector of energy-momentum, the charge, and the spin projection on the motion direction which can be written, according to Section 7.4, in terms of the amplitudes of circular polarizations: Z P



D

dk k

dk





.bjC .k/bj .k/ C bjC .k/bj .k//;

(20.4)

3 X





.bjC .k/bj .k/  bjC .k/bj .k//;

(20.5)

j D1

Z S3 D

3 X

j D1

Z QDq



dk

2 X





.1/j 1 .bjC .k/bj .k/  bjC .k/bj .k//:

(20.6)

j D1

We recall that the operators P  and Q in terms of the amplitudes of linear polar ization aj˙ .k/, aj˙ .k/ are the same since transformations (7.28) are canonical. We now calculate the commutation relations for the fields V .x/. First of all, we note that all relations for classical fields and the polarization amplitudes (7.19)–(7.23) also hold for the operators of quantized fields. In view of relation (20.2), the expansions of the operator amplitudes V˙ .k/ in the polarization amplitudes a˙ .k/ (7.23), and properties of the polarization vectors (7.22), we have     kj kk C 0 0  0 .k / D Œ V .k/; V .k / D ı.k  k / ı C : (20.7) ŒVj .k/; V C   jk j k k m2 Analogously, we calculate the commutation relations between the time components 

V0 .k/, V 0 .k0 / and between the time component V0 .k/ and the space components 

V j .k0 / .j D 1; 2; 3/ with regard for their connection (7.20):   k2 C 0 C 0  ŒV  ı.k  k/ 0 .k/; V0 .k / D ŒV0 .k/; V 0 .k / D m2   k0 k0 D 1 C ı.k  k0 /; m2   k0 kj C 0 C 0  ŒV  ı.k  k0 /: 0 .k/; Vj .k / D ŒV0 .k/; V j .k / D m2

Equations (20.7) and (20.8) can be presented as a single Equation     k k  C 0  C 0 ı.k  k0 /: ŒV  .k/; V .k / D ŒV .k/; V  .k / D g C m2

(20.8)

(20.9)

227

Section 20.1 Quantization of the Complex Vector Field

Finally, using the formulas of the Fourier transformation (7.15) and relations (18.7) and (18.8) for the commutator functions D  .x  y/ and D.x  y/, we have 



 C ŒV .x/; V C  .y/ D ŒV  .x/; V .y/   1 @2 D i g C 2   D  .x  y/; m @x @x 

ŒV  .x/; V .y/ WD iD .x  y/;   @2 1 D .x  y/ D g C 2   D.x  y/: m @x @x

(20.10) (20.11) (20.12)

It is easy to verify that relations (20.10)–(20.12) are consistent with the equations of motion (7.1) and with the Lorentz condition (7.2). Both assertions are valid due to the equation .x  m2 /D.x  y/ D 0: 2 To P verify the first and second assertions, we act by the operators x  m and  @=@x on both sides of Equation (20.11), respectively,    @ 1 @2 g C 2   D.x  y/ Œ@ V  .x/; V .y/ D i @x m @x @x   @  D i  1  2 D.x  y/ D 0: @x m

Remark 20.2. Equations (20.7)–(20.12) establish the commutation relations for the vector field V on the basis of relations (20.2) and expansions (7.20) and (7.23) which assume, in fact, the additional constraint (7.2) (or (7.19)). This allowed us to conserve the classical pattern and to obtain relation (20.4) for the operator of energymomentum which ensures the positivity of the energy. In this case, conditions (7.2) and (7.19) were not broken, and, at the transition to the operator-valued expressions for V .x/; we used Lagrangian (7.3). In addition, it is easy to see that the canonical commutation relations take the form Œ k .x/; Vj .y/;x 0 Dy 0 ¤ i ıj k ı.x  y/;

j; k D 1; 2; 3;

(20.13)

according to the requirements of axiom AQ7, if the generalized momentum is calculated with the use of Lagrangian (7.3), i.e., by Equation (7.11). In the same way, it is easy to find (Problem 20.1), by using Equations (20.11), (20.12), (18.5), (18.6), and (5.58), that the equality in relation (20.13) can be ensured if the generalized momentum   .x/ corresponding to the generalized coordinate V .x/ is defined by the formula 

  .x/ D g  F 0 .x/;

F .x/ D @ V .x/  @ V .x/:

(20.14)

228

Chapter 20 Quantization of Gauge Fields

In order to obtain relation (20.14) for the generalized momentum   .x/, it is necessary to consider a Lagrangian of the free vector field in the form   1 L .x/ D  F  .x/F  .x/ C m2 V  .x/V  .x/: 2

(20.15)

For a real vector field, the right-hand side of (20.15) should be multiplied by 1=2. It is worth noting that Lagrangians (7.3) and (20.15) are equivalent if the condition (7.2) is satisfied. Starting from Lagrangian (20.15), we can apply the method of canonical quantization in the presence of constraints in the system. Setting  D 0 in (20.14), we obtain a primary constraint (20.16) '1 .x/ WD  0 .x/ D 0 due to the antisymmetry of the tensor F . For simplicity, let us consider a real vector field, whose Lagrangian can be obtained from (20.15), by multiplying it by 1=2. Then the Lagrange–Euler equation (4.47) takes the form   (20.17) @ F  .x/ C m2 V  .x/ D 0: The condition of consistency of constraint (20.16) and Equations (20.17) follows from (20.17) at  D 0 W 

'2 .x/ WD @k k .x/  m2 V 0 .x/ D 0: This equation is a secondary constraint. It is consistent with the equations of motion (20.17), Therefore, no other constraints are present. The constraints '1 .x/ and '2 .y/ determined at the same time x 0 D y 0 belong to constraints of the second kind (see Section 4.1.4), since their Poisson brackets ¹'1 .x/; '2 .y/ºx 0 Dy 0 D m2 ı.x  y/ ¤ 0: The corresponding matrix of constraints Cij .x; y/ D ¹'i .x/; 'j .y/ºx 0 Dy 0 ;

i; j D 1; 2;

takes the form C11 D C22 D 0;

C12 D C21 D m2 ı.x  y/;

and the inverse matrix is as follows: .C 1 /11 D .C 1 /22 D 0;

.C 1 /12 D .C 1 /21 D 

1 ı.x  y/: m2

Clearly, we note that the formalism presented in Section 4.1.4 was developed for a finite number of canonical variables.

229

Section 20.2 Quantization of an Electromagnetic Field

In the case where the field depends on a continuous variable, C is an operator matrix, i.e., the matrix elements of C are integral operators, whose kernels are the matrix elements Cij .x; y/, and the formula C C 1 D C 1 C D 1 should be considered as the equality Z d z C.x; z/C 1 .z; y/ D ı.x  y/1;

(20.18)

where 1 is the identity 2  2-matrix. Then, according to the Dirac proposition (4.42) and formulas (4.27) and (4.28), b .x/ and G.y/ b the commutator of the operators F which correspond to the classical quantities F .x/ and G.y/ is defined by the formula

5 7

b .x/; G.y/ b (20.19) ŒF  D i „¹F .x/; G.y/ºP Z 2 X d x0 d y0 ¹F .x/; 's .x0 /ºP .C 1 /ss 0 .x0 ; y0 /¹'s 0 .y0 /; G.y/ºP :  i„

8

s;s 0 D1

We recall that the symbol b above Poisson brackets means that we should firstly calculate the classical Poisson brackets, and then to “announce” the final expression as an operator. In order to obtain the commutation relations for the canonical variables V j .x/ and k .x/, we must take into account that, by the definition of Poisson brackets, ¹V j .x/; k .y/ºPD ı.x  y/ıj k ; ¹V j .x/; V k .y/ºPD ¹j .x/; k .y/ºP D 0:

(20.20)

Substituting V  and  in formula (20.19) instead of F and G and the left-hand sides of (20.16) and (20.17) instead of '1 and '2 , we obtain the commutation relations (Problem 20.2) i„ @i ı.x  y/; m2 ŒV i .x/; j .y/D  i „ı.x  y/: 

ŒVi .x/; V 0 .y/D 

(20.21)

All other commutation relations are equal to zero. We note that V  .x/ WD V  .0; x/ and  .x/WD .0; x/ in Equations (8.20) and (8.21), and the equal-time commutation relations between V  .x/ and  .y/ at x 0 D y 0 will have the same form.

20.2

Quantization of an Electromagnetic Field

20.2.1 Specific Features and Difficulties of the Quantization of an Electromagnetic Field In addition to the Lorentz condition (8.8), the classical electromagnetic field also satisfies the condition of transversality which is related to the fact that the rest mass of a

230

Chapter 20 Quantization of Gauge Fields

photon is equal to zero (see Section 8.3). This implies that, as compared with a vector field, the components of the potential of the electromagnetic field are more dependent and have only two linearly independent components. From the physical viewpoint, this means that real photons, as quanta of the electromagnetic field, can be only in two states (transverse components). Therefore, the specific feature of the quantization of the electromagnetic field is, first of all, in that the operators of the electromagnetic potential must satisfy conditions (8.8) and (8.14), and the whole scheme of quantization must be relativistic invariant. In addition, since the rest mass of photons is equal to zero, we cannot use the scheme of quantization which was used in the case of massive vector fields. Therefore, we will not consider now conditions (8.8) and (8.14) which ensured the positivity of energy in the classical theory. As in the scalar case (see Section 18.1), we use axiom AQ7 concerning the form of the canonical (equaltime) commutation relations. In the case of the electromagnetic field, they take the form Œ  .x/; A .y/;x 0 Dy 0 D i ı ı.x  y/;

(20.22)

Œ .x/;  .y/;x 0 Dy 0 D ŒA .x/; A .y/;x 0 Dy 0 D 0; where  .x/ D g   D @0 A .x/:

(20.23)

As for scalar and spinor fields (see Sections 17.4.1 and 17.4.2), the commutation relations (20.22) are consistent with the Heisenberg equation (17.39) at B.x/ D A .x/: For B.x/ D  .x/; Equation (17.39) coincides with the wave equation (8.9) (Problem 20.3). Taking formula (8.15) into account and executing calculations analogous to those in Section 18.1, we obtain ˙ ŒA˙  .k/; A .p/ D 0; C  ŒA  .k/; A .p/ D g ı.k  p/:

(20.24)

Hence, the operator-valued potentials take the form ŒA .x/; A .y/ D ig  D0 .x  y/; Z 1 e ik.xy/ ı.k 2 /".k 0 /d 4 k: D0 .x  y/ D .2/3 i

(20.25)

The function D0 .x/ can be calculated in the explicit form [97]: Z 1 D0 .x/ D e ikx sin k 0 x 0 ı.k 2 /".k 0 /d 4 k .2/3 Z Z1 d k ikx 1 1 0 e sin.jkjx / D d k sin kjxj sin kx 0 D .2/3 jkj 2 2 jxj 0

Section 20.2 Quantization of an Electromagnetic Field

D

231

1 1 Œı.x 0  jxj/  ı.x 0 C jxj/ D ".x 0 /ı.x 2 /; 4jxj 2

which agrees with relation (18.10) at m D 0. We recall that, due to the equation of motion (8.9), four linearly independent solutions A .x/ have form of the superposition of plane waves (8.13) or (8.15). In the same way as in Section 8.5, A˙  .k/ can be presented in the form of an expansion in  (see relations (8.32)). linear polarizations e Taking relations (8.31) into account, we obtain the commutation relations for the polarization amplitudes:  .k/; aC .p/ D g ı.k  p/: Œa

(20.26)

C / and annihilation .a / of These operators are called the operators of creation .a  temporal . D 0/, longitudinal . D 3/; and transverse . D 1; 2/ photons. In view of these relations, axiom AQ9, and the fact that the electromagnetic field is quantized by Bose–Einstein statistic, we rewrite the formula for P  (the first equality in (8.37)) as an operator in the space of states in the form (see Section 8.5, formula (8.33)) Z Z  .k/A .k// D d k k  .g  aC .k/a .k//: (20.27) P  D d k k  .g  AC  

It is easy to see from Equation (20.27) that the operator P 0 is not positive definite. In the classical case, we used the Lorentz conditions and that of transversality and represented the formula for the energy in the form (8.37) (the second equality), which ensures the positivity of the energy. It turns out that, in the quantum case, the commutation relations (20.25) are not consistent with the Lorentz condition. Indeed, relation (20.25) yields Œ@ A .x/; A .y/ D ig  @ D0 .x  y/ ¤ 0; and, hence, condition (8.8) is not satisfied, which does not allow us to represent the operator P  (20.27) in the form (8.37). In addition, the commutation relations (20.26) are not consistent with the physical C .k/ and a .k/ are the operators interpretation, according to which the operators a  of creation and annihilation of photons with corresponding polarization. Indeed, for  D k D 1; 2; 3 we obtain the ordinary commutation relations Œak .k/; alC .p/ D ıkl ı.k  p/; and the above-mentioned interpretation is valid. But the component a0 .k/ satisfies the relations (20.28) Œa0 .k/; a0C .p/ D ı.k  p/:

232

Chapter 20 Quantization of Gauge Fields

But this relation is not consistent with the requirement that the electromagnetic field 

be real (see (8.16)). Indeed, let us multiply (20.28) by f .k/f .p/, integrate (in the meaning of distributions) with respect to d k and d p; and calculate the vacuum average of free photons. Then the left-hand side takes the form 

. 0 ; a0 .f /a0C .f / 0 / D .a0C .f / 0 ; a0C .f / 0 / D ka0C .f / 0 k2 > 0

(20.29)

and is positive, whereas the right-hand side is Z  jf .k/j2 d k < 0; which gives the contradiction. In Equation (20.29), we used the notation Z ˙ a0 .f / D a0˙ .k/f .k/d k and the condition that the operator a0 .f / is the operator of annihilation adjoint to the operator a0C .f /: a0 .f / 0 D 0: At the first glance, this inconsistency can be removed by assuming the following: for temporal photons, the operator a0 .k/ is the operator of creation, and a0C .k/ is the operator of annihilation. Such interpretation eliminates this inconsistency, but leads to other difficulties related to that the operator of energy P 0 will not be bounded below. Indeed, for example, the state containing a single temporal photon is an eigenstate of the operator of energy with a negative value of the energy: P 0 a0 .k/ 0 D ŒP 0 ; a0 .k/ 0 D k 0 a0 .k/ 0 ;

k 0 D jkj:

To eliminate the indicated difficulties, S. Gupta [83] and K. Bleuler [18] proposed independently a method that allows one to avoid, in a certain sense, the contradiction and to construct a relativistic invariant theory. This approach is briefly described in the following subsection.

20.2.2 Gupta–Bleuler Formalism S. Gupta and K. Bleuler started from the fact that, really, “temporal photons” are not exist in the nature. They appear only at the relativistic description of an electromagnetic field, when one passes from the observable electric E and magnetic H fields to the unobservable 4-component potential A . This allows one to reject generally the condition of self-adjointness7 of the component A0 .x/ (or a0 .k/) and to consider it 7

Here and in what follows, clearly, it is necessary to understand that we consider the self-adjointness of the operators A0 .f / (or a0 .f /) for f belonging to some class of test functions.

233

Section 20.2 Quantization of an Electromagnetic Field

antiself-adjoint. In other words, 

 .a0˙ .k// D a 0 .k/ D a0 .k/;

and, according to (8.31) and (8.32), we have 

   .A˙ 0 .x// D A0 .x/ D A0 .x/: ˙ .k/ can be rewrite in the following way: Then the conditions of adjointness for a 

˙  .k// D a .a  .k/ D g a .k/;

(20.30)

and the commutation relations take the form 

C  C Œa  .k/; a .k/ D g˛ Œa˛ .k/; a .k/ D g˛ g˛ ı.k  k/ D ı ı.k  k/:

For the operators A .x/; we obtain  1 ŒA .x/; A .y/ D ı D0 .x  y/: i

We note that the self-adjointness of the operator of energy-momentum is not broken by such replacement, since the formula for P  includes the bilinear combination aC .k/a .k/. But it turns out that we can also keep, in some sense, the self-adjointness of the operators a .k/. To this end, it is necessary to introduce the so-called indefinite metric in the space of states. This means: if some self-adjoint operator  such that 2 D 1;

a .k/ D g a .k/ and  0 D 0

(20.31)

is defined in the Hilbert space of states H with the scalar product .ˆ; ‰/H , ˆ, ‰ 2 H ; then, in the space H ; we can introduce a new “scalar product” hˆ; ‰i WD .ˆ; ‰/H :

(20.32)

Relative to this new scalar product, the operators a .k/ are self-adjoint. Indeed, hˆ; a .k/‰i D .ˆ; a .k/‰/H D g .ˆ; a .k/‰/H 

D g .a .k/ˆ; ‰/H D g g˛ .a˛ .k/ˆ; ‰/ D ha .k/ˆ; ‰i: In the next subsection, we show how to construct the operator  and the corresponding form (20.32) in the Fock space. The bilinear form (20.32) corresponds to all requirements for the scalar product except for the positivity, i.e., hˆ; ˆi ¤ kˆk2H :

234

Chapter 20 Quantization of Gauge Fields

The quantity hˆ; ˆi can be negative or equal to zero. But the main thought by S. Gupta and K. Bleuler was in the replacement of the Lorentz condition @A .x/ˆ D 0, ˆ 2 H , which led to contradictions (they were discussed in the previous subsection), by some weaker condition which would be consistent with physical ideas and would give the positive definiteness of the energy operator P 0 . Gupta–Bleuler hypothesis. Every physical state of the electromagnetic field must satisfy the condition (20.33) @ A  .x/‰ D 0:

We show that relation (20.33) is the Lorentz condition in a weak form, i.e., in the sense of averages. Indeed, relation (20.33) yields: if ‰1 and ‰2 are the physical states, in which equality (20.33) is satisfied, then  h@ A .x/‰1 ; ‰2 i D h@ AC  .x/‰1 ; ‰2 i D .‰1 ; @ A .x/‰2 /H

D .‰1 ; @ A  .x/‰2 /H D 0;

(20.34)

where we used relations (20.30), (20.31), and (20.33). We denote the set of such physical states by P H . Hence, condition (20.33) is satisfied for every ‰ 2 P. We now show that condition (20.33) ensures the positivity of the energy for all physically admissible states. We rewrite condition (20.33) in the momentum representation (i.e., by passing to the Fourier representation): L .k/‰ WD k  A  .k/‰ D 0;

k 0 D jkj:

(20.35)

Condition (20.35) can be somewhat simplified with regard for the transversality of electromagnetic waves (see Section 8.5). Therefore, as in the classical case where relations (8.35) and (8.36) were obtained from the Lorentz condition (8.34) for the polarized amplitudes a˙ .k/, we obtain an analogous relation for quantized amplitudes  .k/ which are realized on all vectors from the set P. The set P is defined by the a formula (20.36) P WD ¹ˆ 2 H j.L .k/ˆ D 0º : Then, taking (8.31) and (8.32) into account, we rewrite the condition L .k/ˆ D 0 as follows: jkj.a0 .k/  a3 .k//ˆ D 0 or

a0 .k/ˆ D a3 .k/ˆ:

(20.37)

235

Section 20.2 Quantization of an Electromagnetic Field

Using relations (20.31), (20.30), and (20.37) for ˆ 2 P, we obtain hˆ;Œa0C .k/a0 .k/  a3C .k/a3 .k/ˆi D .ˆ; Œa0C .k/a0 .k/  a3C .k/a3 .k/ˆ/H D .ˆ; Œa0C .k/  a3C .k/a0 .k/ˆ/H

(20.38)

D .Œa0 .k/  a3 .k/ˆ; a0 .k/ˆ/H D 0: In view of (20.27) and (20.38), we have Z  hˆ; P ˆi D d k k  hˆ; Œa1C .k/a1 .k/ C a2C .k/a2 .k/ˆi:

(20.39)

This result follows from that the contributions of “temporal” and “longitudinal” photons in any physically admissible state ˆ 2 P to the energy-momentum have the same value and opposite signs and, hence, compensate each other. In Section 22.4, we construct a Hilbert space of states, in which the operators a and P  act, and the set P: Moreover, we show that, for any ˆ 2 P; the following conditions are satisfied: (20.40) ˆ D ˆ and hˆ; ˆi D .ˆ; ˆ/F D .ˆ; ˆ/F  0: Then, relation (20.39) at  D 0 yields hˆ; P 0 ˆi D .ˆ; P 0 ˆ/H  0: This fact gives possibility of a physical interpretation of the Gupta–Bleuler quantization. Remark 20.3. A significant consequence of the above-presented construction is the arbitrariness in the choice of the vectors of states from the set P. In more details, this point will be described in Chapter 22 (see also [181], Chapter 9, Section 2). Now, we demonstrate it by the example of the choice of the vacuum state. In view of the commutation relations (20.24), it is easy to verify that the operators L .k/ commute 0 with the operators LC .p/ D p  AC  .p/, since p D jpj, i.e., p 2 D 0: e 0 of the form This means that any vector e0 D

1 Z X

0 d p1    d pN !N .p1 ; : : : ; pN /

N D0

N Y LC .pj /

0 ; p 2 j D1

where 0 is a state without any particles which is defined by formulas (17.30) and 0 satisfies the condition (17.31), and the sequence of functions !N 1 Z X N D0

d p1    d pN

N Y j D1

0 jpj j2 j!N .p1 ; : : : ; pN /j2 D 1;

236

Chapter 20 Quantization of Gauge Fields

belongs to the set P and can play the role of the vacuum state, since e 0 D 0; L .k/

e 0 D 0; ai .k/

i D 1; 2;

and the mean value of the energy-momentum e 0 / D 0: e0; P  . Such arbitrariness of the choice of the vacuum state follows from the obvious fact of the independence of the physical quantities of gauge transformations (see Section 22.4 for more details).

20.2.3 Canonical Method of Quantization First, though the Gupta–Bleuler method described in the previous subsection in detail ensures the covariant formulation of the process of quantization, it arouses the sensation of some “violence” over the space of states. As a result, the quality of physical interpretation is lost. Second, the introduction of an interaction based in the modern theory on the principle of gauge invariance (see Chapter 4) will require more transparent approaches which would be consistent with this principle. In this subsection, we analyze very briefly the canonical method of quantization on the basis of the gauge-invariant Lagrangian (8.18). In fact, E. Fermi [51] quantized the electromagnetic field using exactly the canonical method. But, due to the obvious noncovariance, this process has got no popularity. Unfortunately, even starting from (8.18), the Dirac method cannot be applied to the systems with constraints, though it was applied to a vector field, since the primary constraint  0 .x/ D 0

(20.41)

@i  i .x/ D 0

(20.42)

and the secondary one do not involve the canonically conjugate variable A0 .x/. Therefore, these constraints are of the first kind. This means that, in addition to the variable A0 .x/, we must exclude one more variable from the system of independent variables. Possibly, only a single means remains: to eliminate one of the variables by using the gauge invariance (i.e., to fix a gauge (see Section 8.8)). Clearly, the explicit form of Lorentz-covariance is lost in this case. But this concerns only the variables A .x/ but not the theory in general. The most widespread gauge is the Coulomb one, where one can simply take A0 .x/  0 (see formula (8.56) with D ' D 0) after the elimination of the variable A0 .x/ in the case of a free field. The last three variables satisfy the condition of transversality (see (8.12)–(8.14)): '1 .x/  @i Ai .x/:

(20.43)

237

Section 20.2 Quantization of an Electromagnetic Field

As in the quantization of a vector field, we take x 0 D 0 within the canonical formalism. Hereafter, we use the notation ui .x/ D ui .0; x/. The secondary condition coincides with (20.42): (20.44) '2 .x/  @i  i .x/ D 0: Unlike constraints (20.41) and (20.42), constraint (20.43) and (20.44) are of the second kind (see Section 4.1.4), and the matrix operator of constraints takes the form (Problem 20.4) C11 .x; y/ D C22 .x; y/ D 0; C12 .x; y/ D C21 .x; y/ D ı.x  y/:

(20.45)

For the inverse matrix, its matrix elements are as follows: .C 1 /11 .x; y/ D .C 1 /22 .x; y/ D 0; .C 1 /12 .x; y/ D .C 1 /21 .x; y/ D 

1 : 4jx  yj

(20.46)

The invertibility of the matrix-operator C should be understood in the sense of (20.18) (see Remark 20.2), by using the above-presented formula x

1 D 4ı.x  y/; jx  yj

x; y 2 R3 :

It is also easy to calculate nonzero Poisson brackets of the canonical variables Ai .x/, i .x/, i D 1; 2; 3; with connections 'j .x/, j D 1; 2: ¹Ai .x/; '2 .y/ºPD 

@ ı.x  y/; @x i

(20.47) @ ¹i .x/; '1 .y/ºPD  i ı.x  y/: @x Then, according to formula (4.42) or (20.19), we obtain the following equal-time commutation relations .x 0 D y 0 D 0/:   1 @2 j ŒA .x/; k .y/ D i ıj k ı.x  y/ C i j k ; 4jx  yj @x @x (20.48) j k ŒA .x/; A .y/ D Œj .x/; k .y/ D 0: We note that the proposed scheme of quantization is invariant under spatial rotations and translations (see [17] for more details). In conclusion, we emphasize: for a free field A0 .x/  0 due to (8.56) (since .x/  0 in the absence of charges), the form of the canonical momentum variable k coincides with that of (20.23) at  D k, and the commutation relations (20.48) are satisfied by the fields Aj .x/ which are given by formulas (8.15), where ! j kk k ı.k  p/: ŒAj .k/; AC .p/ D ıj k  2 k k

238

20.3

Chapter 20 Quantization of Gauge Fields

On the Quantization of Gauge Fields

In the previous subsection, we described some difficulties arising in the quantization of the free electromagnetic field which is gauge-invariant. The gauge invariance means that the theory involves unphysical variables, on which the observable quantities must not depend. This helps us to fix a gauge (i.e., this unphysical variable) to make the calculations required for the solution of a specific problem most convenient. In addition, for a free (noninteracting) field, the gauge invariance even helps to explicitly consider the gauge-noninvariant Lagrangian (8.17) and to use the scheme of covariant quantization which eliminates all difficulties in the Gupta–Bleuler formalism. We consider the situation arising in the quantum theory of interacting fields in what follows and now try to apply the formalism of canonical quantization to the Yang–Mills theory. The canonical variable corresponding to the generalized momentum canonically adjoint to the field Wa is a D

@LYM a  D F0 @.@0 Wa /

(20.49)



(we recall that Wa D W a; ). Then, according to the axioms of canonical quantization, we have ŒWa .x/; b .y/;x 0 Dy 0 D i ıab g ı.x  y/:

(20.50)

Taking  D D 0 and, for example, a D b D 1 in formula (20.50), we obtain ŒW01 .x/; 01 .y/;x 0 Dy 0 D i ı.x  y/: a (see (12.11)) that It is clear from the form of relation (20.49) and F0

0a .y/  0: Hence, the gauge-invariant and canonical quantization turn out inconsistent. As in the case of an electromagnetic field, this difficulty can be eliminated, if we consider the explicitly nongauge-invariant Lagrangian e Y:M: D  1 F a F   1 .@ W a /2 ; L  4  a 2˛ a has the form (12.11). where the parameter ˛ fixes a type of gauge, and F Then we obtain 1 a  g0 .@ Wa /: a D F0 ˛ In the case of an electromagnetic field, the gauge invariance holds in physically admissible states, where the Lorentz condition (see (20.34)) is satisfied in the sense of the mean value due to the Gupta–Bleuler hypothesis (20.33). This scheme is possible

Section 20.3 On the Quantization of Gauge Fields

239

only because the fields A .x/ are free and their Lagrangian is a quadratic function of the fields A .x/. This enables one to explicitly construct the physically admissible states and the vacuum state (see Remark 20.3). Unlike the free electromagnetic field, the free Yang–Mills field is not free in a sense accepted above. In addition to the terms quadratic in the field, Lagrangian (12.12) also contains cubic terms and terms of the fourth degree, i.e., a kind of self-interaction of e 0 ) is no longer the ground state of the “free” Hamilthe field. The vacuum state (or tonian H0;Y.-M. . Hence, the quantization of the Yang–Mills fields (construction of the quantum fields) should be considered in the context of construction of the interacting quantum fields. This problem is studied in Chapter 8.

Chapter 21

CPT . Spin and Statistics

The property of invariance of the theory under proper Lorentz transformations has a universal character, i.e., must be true for all interactions without exception. However, this property can be violated for some discrete transformations. Recall that, e.g., in the reactions of decay of elementary particles governed by the weak interactions (ˇdecay), the space parity is broken. This experimental fact must be taken into account in order to get the correct expression for the Lagrangian of the system [see (14.9) and (14.10)]. In the present chapter, we give the general rules of transformation of the quantized fields under the transformations of charge conjugation, space reflection, and time reversal and briefly describe the properties of invariance of the theory under these transformations. b be one of the discrete transformations C b, P b ; or T b introduced by (1.17), Let X b in the real Minkowski space is deter(1.18), and (2.3). The action of the operator C mined by the identity operator b x D x D .x 0 ; x/: C The transformations of classical fields were defined in Part II (Chapters 5–8): X b u.x//; u.x/ ! ub .x/ D .X

b were defined, according to the type of fields u.x/; by where the operator-matrices X formulas (5.36)–(5.38), (6.100), (6.105), (6.108), and (8.50)–(8.52). In the case of quantum theory, a representation of the group Lorentz is realized , we denote the operators corresponding to these in the space of states H . By Ub X are unitary. Moreover, under transformations. According to AQ1, the operators Ub X b the amplitudes of states are transformed by the law the transformations X; X ˆ ! ˆb D Ub ˆ: X

(21.1)

Then the field operators must be transformed by the operator law, and, according to the axiom of consistency AQ4, we obtain 

b u.x/; Ub u.x/Ub DX X X

(21.2)

b is the same operator, as in classical theory. This operator acts on the operator where X column u.x/, for which the number of components is determined by the relevant field.

Section 21.1 The Transformation of Charge Conjugation

21.1

241

The Transformation of Charge Conjugation

For a complex scalar field, the law of charge conjugation takes the form  C C '.x/ ! 'b .x/ D U b '.x/Ub D b ' '.x/: C C

(21.3)

Since the twofold transformation (21.3) is the identity one, we have C 2 jb 'j D1

and



U 2 D U 2 D 1: b b C C

In other words, the unitary operator UC is simultaneously self-adjoint. If the field C is also real, and we have '.x/ is real, then b '

C b ' D ˙1:

(21.4)

C are simultaneously eigenvalues of the operator U ; In this case, the numbers b ' b C whose eigenvector corresponds to the one-particle state of a free particle. Indeed, if 0 is the wave function of the free vacuum, then the amplitude of a one-particle state is p ˆ1 .x/ D ' C .x/ 0 D '.x/ 0 :

Then, using (21.3), we have  p b C C p ˆ .x/ D Ub '.x/U b UC 0 D b Ub ' '.x/ 0 D ' ˆ1 ; C 1 C C b

where we used the invariance of the bare vacuum and condition (21.4). The quantum C is called charge parity of neutral particles which are described by the field number b u

u.x/. For a complex field, we obtain 1  p C a ˆ .x/ D  '.x/ 0 D b Ub ' ˆ1 .x/; C 1 b C ' C p UC ˆa1 .x/ D b ' ˆ1 .x/;

where ˆa1 .x/ is the amplitude of a one-particle state for antiparticles. Hence, the vector of state can be an eigenvector of the operator UC only if the charge of the system is equal to zero. Analogous relations are satisfied for spinor, vector, and electromagnetic fields. Respectively, using (6.100), (7.34), and (8.50), we have  b C .x/ D U

C .x/Ub D b C N T .x/; b C C  C T N .x/ ! N C .x/ D U N .x/U D b .x/C 1 b b C C

.x/ !

(21.5) (21.6)

242

Chapter 21 CP T . Spin and Statistics

(here, T stands for the operation of transposition of a spinor N ),   b C C V .x/ ! Vb .x/ D U b V .x/U D  V   .x/; V b C C

(21.7)

C A .x/Ub D A .x/: A .x/ ! Ab  .x/ D U b C  C

(21.8)



Relations (21.3), (21.5)–(21.8) together with the condition of invariance of the vacuum state ˆ D ˆ0 (21.9) Ub C 0 (see, e.g., [16], Part 15). This can be considered as the equation for the operator Ub C allows also to find whether a theory is invariant under the operation of charge conjugation. If such invariance holds, the equality 

Ub L .x/Ub D L .x/; C C

(21.10)

where L .x/ is the total Lagrangian of a system of interacting fields, holds. We also note that the operator of total charge of the system anticommutes with UC : 

Ub QUb D Q: C C

21.2

The Transformation of Space Reflection

Under the transformation of amplitudes (21.1) which corresponds to the space reflection, we will act as in the case of the operation of charge conjugation. Comparing (21.2) with the form of the corresponding transformations for a classical field (formulas (5.37), (6.105), (7.33), and (8.51)), we obtain the following rules of transformation of fields (scalar, spinor, vector, and electromagnetic ones):  P P .x/ D U b '.x/Ub D b '.x/ ! 'b ' '.ƒis x/; P P 

P 0 .x/Ub D b  .ƒis x/; b P P  P P N N .x/ ! N b N .x/U D b .ƒis x/ 0 ; .x/ D U b b P P  P P  .x/ D U b V .x/Ub D b V .x/ ! Vb V g V .ƒis x/; P  P  P A .x/Ub D g  A .ƒis x/: A .x/ ! Ab  .x/ D U b P  P

.x/ !

b P .x/ D U

(21.11) (21.12) (21.13) (21.14) (21.15)

The action of ƒis is defined by relation (1.17). Unlike the operation of charge conjugation, the twofold space reflection (the identity operator) for scalar particles yields P 2 .b ' / D 1;

P b ' D ˙1:

243

Section 21.3 The Transformation of Time Reversal

P is always real. Its value is called the internal space Hence, the phase factor b ' parity (or simply the parity) of a particle. The same is true for vector fields. As for spinor fields, we should take into account their two-valued property under rotations by an angle of 2. Indeed, the twofold space reflection can be written sometimes as P 2 / can take both values: C1 or 1: In other words, the full rotation. Therefore, .b

P 2 ¹1; C1; i; Ci º. It is easy to verify (Problem 21.1) that the charge-conjugate C  c satisfies the relation spinor b  P 0 b b C Ub Ub D b  C .ƒis x/: P P

(21.16)

P P D b , then the state transformation Formula (21.16) implies that if we choose b laws for a particle and an antiparticle are characterized by opposite signs. C and b P depends on a particle which We note that the choice of the phase factors b u

u

is described by the field u.x/. This means that, even for fields of the same type, C P (or b ) can be different for different say, for a spinor field , the phase factor b Fermi-particles.

21.3

The Transformation of Time Reversal

In quantum theory, the transformations related to the time reflection are essentially different from analogous transformations corresponding to the charge conjugation and the space reflection. First of all, we note that, in classical theory, the field u.x/ does not correspond to the quantum-mechanical wave function of a particle but is connected  with it by certain relations: '  .x/ or '  .x/ in the case of scalar bosons or (6.90) and (6.101) for the electron-positron field. Therefore, the law of transformation of fields in quantum mechanics is given, respectively, by formulas (5.38), (6.108), (7.35), and (8.52) to be consistent with the reversibility of time. In quantum theory, the one-particle wave functions which correspond to specific values of quantum numbers (momentum, spin, etc.), are defined as projections of the vector u.x/ 0 on the one-particle state with the same quantum numbers (see, e.g., [17, Section 100]). In other words, ‰.p;:::/ .x/ D N‰ .u.x/ 0 ; ˆ1 .p; : : ://H ; where

(21.17)

ˆ1 .p; : : :/ D uC .p/ 0 ;

the operator uC .p/ is the operator of creation of a particle with quantum numbers .p; s; : : :/, and N is the normalization factor. Hence, transformation (21.2) corresponding to the time reversal must satisfy two main conditions in the quantum case. The first condition is the invariance of the comT .x/, and the second is the conservation of mutation relations for transformed fields ub

244

Chapter 21 CP T . Spin and Statistics

the quantum-mechanical pattern, i.e., the transformation of the wave function (21.17) by the rule  T T .x/ ! ‰b .x/ D b .ƒ x/: (21.18) ‰ .p;:::/

.p;:::/

‰

.p;:::/

it

is linear It turns out that these both conditions cannot be satisfied, if the operator Ub T unitary operator in the space H . We demonstrate this assertion by the example of a free complex scalar field. Let rules (21.2) take the form  T # U TO '.x/UT D b ' ' .ƒi t x/;   T #  Ub '.x/Ub D b ' .' .ƒi t x/ ; T T 

where ' # coincides with the field ' itself or with the complex conjugate field '. 

and Ub on the commutation relations (18.15) from Let us act by the operators U b T T is a linear unitary operator, then we obtain the the left and right, respectively. If Ub T relation (it can be understood in the sense of the action on vectors from the space H )    1 D.x  y/ D ŒU b '.x/Ub ; Ub '.y/Ub  T T T T  i T 2 # #  D jb ' j Œ' .ƒi t x/; .' .ƒi t y//  8 9 < 1i D.ƒi t .x  y//; if ' # D ' = T 2 D jb 'j :  1 D.ƒ .y  x//; if ' # D ' ; it i T 21 D jb D.x  y/; 'j i

(21.19)

where we considered the property of the commutator function (see (18.8) or (18.10)) under the change x 0 ! x 0 . Hence, we obtain the contradiction, since (21.19) can be satisfied only at D.x  y/  0. To remove this contradiction, we need to interpret as “antiunitary” operator which is defined in the space H by the the operator Ub T relation ˆ ; Ub ˆ / D .ˆ1 ; ˆ2 / D .ˆ2 ; ˆ1 /; ˆ1 ; ˆ2 2 H : .Ub T 1 T 2

(21.20)

on a From this relation, we can easily obtain the rule of action of the operator Ub T linear combination of the vectors ˆ1 and ˆ2 (Problem 21.2): .˛1 ˆ1 C ˛2 ˆ2 / D ˛N 1 Ub ˆ C ˛N 2 Ub ˆ : Ub T T 1 T 2

(21.21)

In other words, the operator defined by equality (21.20) is antilinear. This allows us to remove the contradiction, by defining the law of transformation (21.2) as b T T 0 D b U 1 '.x/Ub ' '.ƒi t x/ D ' '.x ; x/: T b T

(21.22)

245

Section 21.3 The Transformation of Time Reversal

Then, in order to verify the invariance of the commutation relations (18.15) under transformations (21.22), we should take relation (21.21) into account, which ensures, in the given case, the equality 1 1 D  D.x  y/ U 1 D.x  y/Ub T b T i i and the invariance of the commutation relations (18.15). The definition of the operator by equality (21.20) yields Ub T kUb ˆkH D kˆkH ; T

ˆ2H:

In other words, the operator Ub conserves the norm of vectors of the Hilbert space T and, therefore, (together with its antilinearity (21.21)) is called antiunitary. But this term is not commonly accepted in the mathematical literature. In addition, we avoid the identification of the operator U 1 with the operator U  , since the notion of adjoint b b T T operator for nonlinear operators is not single-valued (see, e.g., Footnote 1 in [107], Chapter 1, Section 2.2). It is also easy to verify condition (21.18). Indeed, definition (21.17) at u D ', a representation of the field operator in the form (18.2), and the commutation relations (18.13) imply obviously that 1 e ip x Cipx ; ‰.p;:::/ .x/ D N‰  p .2/3=2 2p 0 0 0

p0 D

p p2 C m2 :

Then, by using relations (21.22) and (21.20), the condition of invariance of vacuum,

D 0 ; Ub T 0 T /2 D 1, we obtain the relation and the condition .b ' T T ‰b .p;:::/ .x/ D N‰ .' .x/ 0 ; ˆ1 .p; : : ://H 

D N1 . 0 ; '  .x/Ub ' C .p/U 1 0 T b T N‰  D . 0 ; ' .x/' C .x/' C .p/ 0 / b T  '

T D b ' Nˆ

 1 0 0 T e ip x ipx D b ' ‰ .p;:::/ .ƒi t x/ 3=2 .2/

which coincide with (21.18).  We note that the choice of the field ' instead of the field ' on the right-hand side of formula (21.22) results in that the vector of state of a particle ‰1 .p; : : :/ D

246

Chapter 21 CP T . Spin and Statistics

p

ˆ1 .p; : : :/ D ' C .p/ 0 would transit to the state of an antiparticle with the help of b: the transformation T 

Ub ˆ .p; : : :/ C Ub ' C .p/U 1 0  D ' C .p/ 0 D ˆa1 .p; : : :/ : T 1 T b T p

Concluding this section, we present the form of transformation relations for the basic operator fields (scalar, spinor, vector, and electromagnetic ones): T T .x/ D U 1 '.x/Ub D b '.x/ ! 'b ' '.ƒi t x/; T b T T 1 3 T .x/ ! b .x/ D U 1 .x/Ub D b   .ƒi t x/; T b T T T N .x/ ! b N .ƒi t x/ 3  1 ; .x/ D U 1 N .x/Ub D b T b T

D TV g  V .ƒi t x/; V .x/ ! VT .x/ D U 1 V .x/Ub T b T A .x/ ! AT .x/ D U 1 A .x/Ub D g  A .ƒi t x/: T b T

21.4

(21.23) (21.24) (21.25) (21.26) (21.27)

CPT -Theorem and the Connection of Spin and Statistics

It is easy to verify that almost all Lagrangians of free fields are invariant under each b, P b ; and T b separately. The exclusion is the Lagrangian of a free of the operations C C .x/ does not satisfy neither the neutrino field (6.153). The point is in that the field b additional condition (6.148), nor (6.150). In order to satisfy these conditions (under the condition that .x/ satisfies condition (6.148)), it is necessary else to perform the b -transformation. In other words, the field P 





Pb C

b .x/ D U b

.x/Ub D Ub U C .x/Ub UP Pb C Pb C P b C b

satisfies condition (6.148) (Problem 21.3). In the same way, we can verify that Lab and C b , and is grangian (6.153) is not separately invariant under the transformations P b b invariant under P C (Problem 21.4). But it turns out that any theory (in the frame of the reasoning and with the above-considered conception of construction) is invariant under the product of all three transformations: bDP bC bT bDP bT bC bDC bP bT bDC bT bP bDT bP bC bDT bC bP b: ‚

(21.28)

This result can be formulated in the form of the following proposition. CPT -theorem. Let the theory of interacting fields be described by the Lagrangian L .x/ which satisfies the conditions (1) L .x/ is local (see Section 4.2)W L .x/ D F .ui .x/; @k ui .x//I

247

Section 21.4 CP T -Theorem and the Connection of Spin and Statistics

(2) L .x/ is invariant under proper inhomogeneous Lorentz transformationsW 

U .ƒ;a/ L .x/U.ƒ;a/ D L .ƒ1 .x  a//;

"

ƒ 2 LC I

(3) the operator L .x/ is a symmetric (Hermitian)8 operator in the space H I (4) the operator L .x/ is given by a normal (Wick) product (see AQ9) of fields which enter into its definition; (5) the operators of quantized fields (which enter the definition of a Lagrangian) with integer spin satisfy the commutation relations (i.e., they commute under the symbol of normal product), and the operators of fields with half-integer spin satisfy anticommutation relations (i.e., they anticommute under the symbol of normal product). b bDP bC bT b/, i.e., Then the theory is ‚-invariant .‚ D L .x/: U 1 L .x/Ub ‚ b ‚ This theorem was first proposed by G. Lüders and B. Zumino [125, 126] in the following version: if a relativistic invariant theory (i.e., it is invariant under the transfor" b , it must be also invariant mations from LC / is invariant under the space reflection P bT b. Earlier, W. Pauli [152] proved, in fact, under a product of the transformations C bC bT b-transformation under condithe invariance of the theory of free fields under P tion of its invariance under the restricted Lorentz group at the proof of the theorem of the connection of spin and statistics for free quantized fields (we consider this proof in what follows). The more detailed version of the proof can be found in the work by W. Pauli [153]. Within the axiomatic approach to the construction of quantum field theory (see Chapter 8), the CP T -theorem was proved by R. Jost [99] (see also [101, 22, 23]). In conclusion, we show that the CP T -theorem is valid for the following types of interaction: the interaction of a spinor field with a real scalar field (LYuk D W N .x/ .x/'.x/ W) and the 4-fermion interaction describing the ˇ-decay n ! p C N e  C : Lˇ .x/ D g W Np .x/  .1   5 / n .x/ N e .x/ .1   5 /  .x/ W C g W N n .x/.1 C  5 /  p .x/ N  .1 C  5 / e .x/ W

(21.29)

D Lpne .x/ C Lnpe .x/: To this end, we now write the transformation of the fields , N ; and ' which b Using formulas (21.5)–(21.8), (21.11)–(21.14), corresponds to the transformation ‚. 8

The Hermite property of L .x/ should be understood in the sense of the definition of L .x/ as an operator-valued distribution.

248

Chapter 21 CP T . Spin and Statistics

and (21.23)–(21.27), we can calculate (Problem 21.5) that U 1 '.x/Ub D U 1 U 1 U 1 '.x/Ub UP Ub ‚ C b T b b b b ‚ T P C C b T D b P b '.ƒ x/ D '.x/; i ' ' '  b C b P b T 1 2 3

D       N T .x/; U 1 .x/Ub ‚ b ‚ C b T T U 1 N .x/Ub D  b P b .x/ 3  2  1 : ‚ b ‚

(21.30) (21.31) (21.32)

It is easy to verify with regard for (21.30)–(21.32) that U 1 W N .x/ .x/'.x/ W U‚ b ‚ C 2 b T 2 D jb j jP j2 jb j W T .x/ N T .x/'.x/ W D W N .x/ .x/'.x/ W : It is worth noting that we used condition 5 of the CP T -theorem in the last equality, i.e., we involved the connection of spin and statistics. We now verify the invariance of Lagrangian (21.29). In view of (21.30)–(21.32), the first factor in the first term (21.29) takes the form U 1 g W Np .x/  .1   5 / n .x/ W Ub ‚ b ‚ ı2  5 D .1/ g W N n .x/.1 C  / 

p .x/

W:

We note that the factor .1/ı2 appears due to theaction of the antilinear operator   on the matrix elements   ˛ˇ D .1/ı2 ˛ˇ . Ub T An analogous result holds for the second factor of the first term. By multiplying, we obtain

 U‚1 Lpne .x/U‚ D Lnpe .x/ D Lnpe .x/; which yields the invariance of the total interaction Lagrangian.

21.5

Proof of the Pauli Theorem

In Chapter 17 (axiom AQ8), we formulated the Pauli theorem which gives the rule of the quantization of fields depending on their spin. In this section, we substantiate briefly this theorem by the example of free fields. First of all, we note that the boson and fermion fields must satisfy the Heisenberg equation (17.6) @ui .x/ D Œui .x/; P 0  : i @x 0 With regard for the representations of these fields in the form of their Fourier-transforms (see (18.2), (19.2), and (19.3)), we rewrite this equation for the operator amplitudes uQ ˙ i .k/ in the following form: 0 0 ˙ ŒuQ ˙ Q i .k/: i .k/; P  D k u

(21.33)

249

Section 21.5 Proof of the Pauli Theorem

˙ ˙ Here, uQ ˙ i takes one of the values ' .k/ for a scalar field, as .k/, s D 1; 2; for a ˙ spinor field, and b .k/,  D 0; 1; 2; 3; for a vector field. The corresponding formulas for the operator P 0 are given by formulas (17.24) and (17.25), where the rules of commutation of operators are not yet taken into account. It is easy to verify that Equation (21.33) is valid only if the commutators or anticommutators of these fields satisfy the relations 







Qj˙ .k/˙ D ŒuQ ˙ Qj˙ .k/˙ D ŒuQ ˙ Qj˙ .k/˙ D 0; ŒuQ ˙ i .p/; u i .p/; u i .p/; u 



ŒuQ  QjC .k/˙ D ŒuQ  QjC .k/˙ D ıij ı.p  k/: i .p/; u i .p/; u These relations are equivalent, since the commutators or anticommutators of the fields  ui .x/ and ui .y/ are c-functions, i.e., they are multiple to the identity operator. In other words,  (21.34) Œui .x/; ui .y/˙ D ˙ i i .x  y/: We now show that such relation is true for fields with integer spin only in the case of a commutator. Let ui .x/ D '.x/ be a complex scalar field, and let relations (21.34) be satisfied in the case where we take anticommutators. It is easy to calculate (as in the case with commutators) that 1  Œ'.x/; '.y/C D D.x  y/: i The function D.x  y/ is antisymmetric in the time variable and symmetric in the space variables. In view of this fact, we obtain 



Œ'.x/; '.y/C C Œ'.y/; '.x/C D 0: But, in this case at x ! y, each of four terms is strictly positive, and, therefore, we obtain the contradiction. An analogous result is valid for vector fields, since the corresponding function C i i .x  y/ is given by the action of the operator including an even number of derivatives (see (20.12)) on the function 1i D.xy/ and thus conserves the same properties of nonparity in the time variable and parity in the space one. Hence, the presented arguments imply that the fields corresponding to particles with integer spin must satisfy the commutation relations. For the fields with half-integer spin, we cannot use similar arguments. However, the arguments related to the positive-definiteness of the operator of energy P 0 for fields with half-integer spin and presented in the substantiation of axiom AQ8 complete the proof of the theorem.

Chapter 22

Representations of Commutation and Anticommutation Relations In the previous chapters, we have studied the free quantized fields as operator-valued distributions satisfying certain commutation relations. These commutation relations can be regarded as an abstract algebra. The problem of finding the representations of this algebra is an important independent problem. In addition, the principal dynamic quantities, such as energy-momentum, current, charge, etc., are represented via these abstract field operators. Hence, the construction of representations for the algebraic relations (18.4), (19.4), and (20.2) is equivalent to the construction of the space of physical states and amplitudes with the help of which it is possible to find the basic physical quantities of the theory. Unlike the quantum mechanics in which the number of particles is an integral of motion, the quantum field theory describes physical processes in which the number of particles may vary and take arbitrary values. Thus, the vectors of physical states must be functions of numerous variables. The Hilbert space of states satisfying this requirement was first constructed by Fock [60] (the history of this problem can be found, e.g., in [181], Part 6) and, hence, is called the Fock space. The Fock space is a simple obvious generalization of the space of states from the quantum mechanics. Therefore, the method of second quantization in the Fock space becomes most popular, although the representations of commutation relations in other spaces are also studied in the scientific literature. In the present chapter, we give a detailed description of representations in the Fock space.

22.1

General Structure of the Fock Space

First, we define the space of states in the most general situation without analyzing the type and structure of particles and their dynamic states (i.e., whether they are free or .j / interacting). Let H1 be the Hilbert (complex) space of states for a single relativistic particle of the j th type. Thus, for free particles, examples of spaces of this sort can be found in Chapter 2. We now construct the following spaces: s F0 D C; .j1 /

FN D H 1

.jN /

˝    ˝ H1

;

where the symbol ˝ denotes the tensor product (see, e.g., [170], Section II.4).

(22.1)

251

Section 22.1 General Structure of the Fock Space

In fact, formula (22.1) does not involve the case where the system of N particles contains the same (identical) particles. In addition, these particles can be bosons or fermions. According to the general principle of indistinguishability of identical particles in quantum theory (see, e.g., [115], Part IX), the corresponding functions must be symmetric with respect to “boson variables” and antisymmetric with respect to “fermion variables”. Let a system of N particles be composed of K types of bosons and L types of fermions, and let the numbers of particles of each type be mk , k D 1; K, and ml , l D K C 1; : : : ; K C L, respectively. Then the structure of the space FN corresponding to this situation is as follows:   ˝s m1 ˝s mK .1/ .K/ ˝    ˝ H1 FN D H1   ˝a mKC1 ˝a mKCL .KC1/ ˝ H1 ˝    ˝ H1KCL : (22.2) .1/

Here, ˝s mk means the symmetric tensor product of the space H1 by itself mk times, and ˝a ml is, respectively, the antisymmetric tensor product by nl times. We can consider N as a vector N D .m1 ; : : : ; mK , mKC1 ; : : :, mKCL / with jN j D m1 C    C mK C mKC1 C    C mKCL . The Fock space is now a completed direct sum F D

1 M

FN :

(22.3)

N D0

The elements of the space F are sequences (columns) 1 F0 BF1 C C F DB @F2 A : :: : 0

(22.4)

Moreover, F 2 F , if kF k D

1 X

!1=2 kFN k2FN

< 1;

(22.5)

N D0

where the norm and the scalar product in the space FN are completely determined .j / by the corresponding quantities in the one-particle Hilbert spaces H1 and by the rules of construction of these quantities in a tensor product of Hilbert spaces (see, e.g., [170], Section II.4). In what follows, we present specific examples of such spaces.

252

22.2

Chapter 22

Representations of Commutation and Anticommutation Relations

Representations of Commutation Relations for a Free Real Scalar Field

22.2.1 The Fock Space of Free Scalar Bosons The one-particle Hilbert space corresponding to a scalar particle (scalar boson) can be chosen in the same way as in classical theory (see Section 5.4). In the momentum space the corresponding N -particle space of states is the space of symmetric functions FN .p1 ; : : : ; pN /, for which Z 2 .d p/N jFN .p1 ; : : : ; pN /j2 < 1; (22.6) kFN kFN WD R3N

i.e., HN D L2 .R3N / and the Fock space F D FB is constructed by Equations (22.3)– (22.5).

22.2.2 Operators of Creation and Annihilation in the Fock Space. Momentum Representation First, we construct a representation of commutation relations for a real scalar field: Œ'  .k/; ' C .p/ D ı.k  p/  1; Œ' ˙ .k/; ' ˙ .p/ D 0:

(22.7)

It is seen from the first relation in (22.7) that the result of the action of a commutator on any function FN must be a function taking the form ˆN D ı.k  p/FN .p1 ; : : : ; pN /:

(22.8)

Clearly, it does not belong to the Fock space and is a distribution. This follows from the fact that the field '.k/ is an operator-valued distribution. If we choose two functions f and g from the space of basic functions (e.g., D [206]), then the quantities Z (22.9) '  .f / D h'  ; f i D '  .k/f .k/d k and C

C

' .g/ D h' ; gi D

Z

' C .p/g.p/d p

(22.10)

(here, the integrals must be considered in the sense of the pairing operation) are operators in the Fock space. According to (22.7), they must satisfy the commutation relations Œ'  .f /; ' C .g/ D .f; g/H1  1; Œ' ˙ .f /; ' ˙ .g/ D 0; where 1 is the identity operator in the space FB .

(22.11)

Section 22.2 Representations of Commutation Relations for a Free Real Scalar Field

253

First, we define heuristically the action of the “operators” ' C .k/ and '  .p/ on a Fock column F D ¹FN º1 N D0 as follows. We assume that the action of the “operators” ' C .k/ and '  .p/ on F creates new columns, whose components can be calculated by the Equations .' C .p/F /N .p1 ; : : : ; pN / N 1 X ı.p  pj /FN 1 .pi ; : : : ; pj 1 ; pj C1 ; : : : ; pN /; Dp N j D1 p .'  .k/F /N .p1 ; : : : ; pN / D N C 1 FN C1 .k; p1 ; : : : ; pN /:

(22.12) (22.120 )

Then it is easy to find that .'  .k/' C .p/F /N .p1 ; : : : ; pN / p D N C 1.' C .p/F /N C1 .k; p1 ; : : : ; pN / p N C1 Dp ı.p  k/FN .p1 ; : : : ; pN / N C1 p N N C1 X Cp ı.p  pj /FN .k; p1 ; : : : ; pj 1 ; pj C1 ; : : : ; pN /; N C 1 j D1

(22.13)

and .' C .p/'  .k/F /N .p1 ; : : : ; pN / N 1 X ı.p  pj /.'  .k/F /N 1 .p1 ; : : : ; pj 1 ; pj C1 ; : : : ; pN / Dp N j D1 p N N X Dp ı.p  pj /FN .k; p1 ; : : : ; pj 1 ; pj C1 ; : : : ; pN /: (22.14) N j D1

Subtracting, we obtain (22.8). The integration of relations (22.12)–(22.14) with the functions f; g 2 D gives the first relation in (22.11) on a vector F 2 FB . But, in order to consider that the representations of commutation relations are constructed in FB , we should prove that the operators ' ˙ .f / for f 2 D can be given in FB on some everywhere dense set of vectors. Such dense set in FB is the set D0  FB of finite columns. The vector F 2 D0 , if there exists a number N0 such that for FN  0 N > N0 : Every vector F 2 D0 has the own number N0 , i.e., N0 D N0 .F /. The property of the set D0 to be everywhere dense follows from the definition of this property (see, e.g., [169], Section I.2) and the absolute convergence of series (22.5) for any F 2 FB . Indeed, for any F 2 FB such that F … D0 , we construct

254

Chapter 22

Representations of Commutation and Anticommutation Relations

a sequence of vectors F .n/ 2 D0 in the following way: .n/

FN D FN

for N n

and

Then kF

.n/

2

 Fk D

1 X

.n/

FN  0

for N > n:

kFN k2FN :

(22.15)

N Dn

The convergence of series (22.5) implies that the right-hand side (22.15) tends to zero as n ! 1. Hence, we can approximate any vector from FB by the sequence F .n/ 2 D0 ; which means that the set D0 is dense in FB . We now show that, for any vector F 2 D0 ; the vectors ' ˙ .f /F 2 D0 . The finiteness of the sequence ' ˙ .f /F follows from the formulas determining their action. Indeed, relations (22.12), (22.9), and (22.10) yield at once 1 X f .pj /FN 1 .p1 ; : : : ; pj 1 ; pj C1 ; : : : ; pN /; .' C .f /F /N .p1 ; : : : ; pN / D p N j D1 Z p .'  .f /F /N .p1 ; : : : ; pN / D N C 1 d p f .p/FN C1 .p; p1 ; : : : ; pN /: (22.16) N

Hence, the operator ' C .f / increases the number of components of the vector F by transforming the vector

 F D F0 ; F1 .p1 /; : : : ; FN0 .p1 ; : : : ; pN0 /; 0; : : : (22.17) in the vector NX 0 C1  1 f .pj /FN0 .p1 ; : : : ; ' C .f /F D 0; f .p1 /F0 ; : : : ; p N0 C 1 j D1  pj 1 ; pj C1 ; : : : ; pN0 C1 /; 0; : : : ; (22.18)

whereas the operator '  .f / decreases the number of components by one: Z p Z  ' .f /F D d p f .p/F1 .p/; 2 d pf .p/F2 .p; p1 /;  p Z : : : ; N0 d pf .p/FN0 .p; p1 ; : : : ; pN0 1 /; 0; : : : :

(22.19)

Section 22.2 Representations of Commutation Relations for a Free Real Scalar Field

255

Hence, it remains to prove that .' ˙ .f /F /N .p1 ; : : : ; pN / 2 FN . According to definition (22.6) of the norm in FN ; we have k.' C .f /F /N k2

ˇ ˇ2 ˇ ˇ N X ˇ ˇ 1 N ˇ D .d p/ ˇ p f .pj /FN 1 .p1 ; : : : ; pj 1 ; pj C1 ; : : : ; pN /ˇˇ ˇ N j D1 ˇ R3N Z Z N d pjf .pj /j2 .d p/N 1 jFN 1 .p1 ; : : : ; pN 1 /j2 Z

R3

D N kf

R3.N 1/ 2 kH1 kFN 1 k2FN 1 :

(22.20)

In calculations, we used the Cauchy–Buniakowski–Schwarz inequality: 11=2 0 11=2 0 N N N X X X aj bj @ jaj j2 A @ jbj j2 A : j D1

j D1

j D1

With regard for the definition of norm (22.5), the finiteness of the vector F (22.17), and estimate (22.20), we obtain the inequality p (22.21) k' C .f /F kF N0 C 1 kf kH1 kF kF : Quite analogously for the operator '  .f /; we have p k'  .f /F kF N0 kf kH1 kF kF :

(22.22)

Since kf kH1 < 1 and kF kF < 1 for f 2 D and F 2 D0 , the vectors ' ˙ .f /F 2 D0 . But it is seen from equalities (22.21) and (22.22) that the operators ' ˙ .f / are unbounded in FB . This is easily verified if we write, by analogy with (22.18) and (22.19), the vector columns ' C .f /F and '  .f /F for F which is not finite. Then we can p always choose a vector F 2 FB so that k' ˙ .f /F kFB D C1 due to the factors N arising in the N -th component of the vectors ' ˙ .f /F though kF kFB < 1. Hence, the operators satisfying the commutation relations (22.11) are unbounded operators with the dense domain of definition D.' C .f // D D.'  .f // D D0 : The field ' is a real field. Hence, for real f 2 D; the operator '.f / D ' C .f / C / must be a self-adjoint operator in the space FB . First, we show that it is a symmetric (see the definition, e.g., in [169], Section VIII.2) operator in FB . To this end, it is sufficient to show that, for any G and F 2 D0 ; the following equality holds: '  .f

.F; ' C .f /G/ D .'  .f /F; G/:

(22.23)

256

Chapter 22

Representations of Commutation and Anticommutation Relations

Indeed, .F; ' C .f /G/ D

Z 1 X N D1

D

1 X N D1

GN 1 .p1 ; : : : ; pj 1 ; pj C1 ; pN /

Z .d p/N R3N

D

R3N

N 1 X .d p/N FN .p1 ; : : : ; pN / p f .pj / N j D1

p N f .pN /FN .pN ; p1 ; : : : ; pN 1 /GN 1 .p1 ; : : : ; pN 1 / D

Z 1 X N D0

R3N

.d p/N

p

Z N C1

d pf .p/FN C1 .p; p1 ; : : : ; pN /

R3

GN .p1 ; : : : ; pN / D .'  .f /F; G/:

In the first row, the sum begins from N D 1, since .' C .f /G/0 D 0 (see (22.18)). In the proof, we used the symmetry of the functions FN .p1 ; : : : ; pN / and GN .p1 ; : : : ; pN /.

22.2.3 Vacuum State of Free Particle. Cyclicity of Vacuum. Set of Exponential Vectors For a system of free noninteracting particles, vacuum is, obviously, the state without particles. The vector of such state must be an eigenvector of the operator of energy and the operator of the number of particles with the eigenvalue equal to zero. The operator of energy is presented in terms of the operators of creation and annihilation by the formula Z p 0 (22.24) P D d k k2 C m2 ' C .k/'  .k/: Using formulas (22.12) and (22.120 ), it is easy to calculate the components of the vector P 0 F for F 2 FB : 1 0 N q X p2 C m2 A FN .p1 ; : : : ; pN /: (22.25) .P 0 F /N .p1 ; : : : ; pN / D @ j

j D1

It becomes clear from formulas (22.24) and (22.25) that the operator of the number of particles takes the form Z b N D d k ' C .k/'  .k/ and acts on the vector F 2 FB in the following way: b F /N .p1 ; : : : ; pN / D NFN .p1 ; : : : ; pN /: .N

(22.26)

Section 22.2 Representations of Commutation Relations for a Free Real Scalar Field

257

b , as the operators By using formula (22.26), it is easy to verify that the operator N /, is defined on the everywhere dense set of finite vectors D0 and remains this b D0 D D0 . The operator of energy P 0 is defined not on all vectors set invariant, i.e., N from D0 , but only on such ones, for which the right-hand side of (22.25) belongs to the space FN , i.e., it is square integrable function. This holds if, for FN 2 FN ; Z .dp/N .pj2 C m2 / jFN .p1 ; : : : ; pN /j2 < 1 for j D 1; : : : ; N:

' ˙ .f

R3N

Whence, it is seen that P 0 can be defined on all vectors from D0 which have the components FN .p1 ; : : : ; pN / with a bounded support in the space R3 in each of the variables pj , i.e., on the set D0loc WD ¹F 2 D0 j FN .p1 ; : : : ; pN / D 0; if jpj j  Bj ; j D 1; N ; N D 1; N0 º:

(22.27)

Each vector F is characterized by the own constants Bj and N0 . The set D0loc D0 is also everywhere dense in FB . Using formulas (22.12), it is easy to verify (Problem 22.1) that, for F; G 2 D0loc ; .P 0 F; G/ D .F; P 0 G/

(22.28)

.P 0 F; F /  0:

(22.29)

and i.e., P 0 is a symmetric densely given nonnegative operator in the space F . Formulas (22.28) and (22.29) yield trivially the essential self-adjointness of the operator P 0 (see, e.g., [169], Section VIII.2). It is easily seen from formulas (22.25) and (22.26) that vectors of the form F n D ¹FN º1 N D0 ; .n/

.n/

FN D ıN n Fn .p1 ; : : : ; pN /;

b: where ıN n is the Kronecker delta, are eigenvectors of the operators P 0 and N 1 0 n q X pj2 C m2 A F .n/ ; P 0 F .n/ D @ j D1

b F .n/ D nF .n/ : N It is clear that n D 0 corresponds to the vector F .0/ . By normalizing this vector to 1, we obtain the so-called free (“bare”) vacuum 0 1 1 .0/ B C 0 F B C: D

0 D (22.30) @ 0A kF .0/ kFB :: :

258

Chapter 22

Representations of Commutation and Anticommutation Relations

We now show that the vector 0 is a cyclic vector of the family of field operators ¹'.f /; f 2 D F1 º. By definition (see, e.g., [169], Section VII.2), this means that the linear span of vectors ¹'.f1 /    '.fN / 0 º;

fj 2 D; j D 1; N ;

(22.31)

is everywhere dense in F . DN0 Indeed, using formula (22.12), it is easy to verify that any vector F D ¹FN ºN N D0 2 D0 can be written as follows: F D

Z N0 X 1 .d p/n Fn .p1 ; : : : ; pn /' C .p1 /    ' C .pn / 0 : p nŠ nD0

(22.32)

R3n

On the other hand, in view of the relations '.fj / D ' C .fj / C '  .fj / and

'  .fj / 0 D 0;

(22.33)

each vector of set (22.31) can be rewritten in the form (22.32) with Fn .p1 ; : : : ; pn / f1 .p1 /    fn .pn /; and vice versa. This means that an arbitrary vector F 2 D0 can be approximated by linear combinations of such vectors. In the conclusion of this subsection, we present one more set of vectors called the set of exponential vectors which enters the set of the so-called analytic vectors ( [169], Section VIII.5). The set of exponential vectors is defined by the formula μ ´ 1 X 1 C ' C .f /n 0 ; f 2 H1 : De WD F 2 FB j F D e f WD e ' .f / 0 D nŠ nD0 (22.34) It is easy to verify directly from estimate (22.21) that the series on the right-hand side of (22.34) is convergent in the norm in the Fock space. In addition, it is very easy to calculate the norm or the scalar product of two vectors of the form (22.34) as  C    C  C .e f ; e g / D e ' .f / 0 ; e ' .g/ 0 D 0 ; e ' .f / e ' .g/ 0 D e .f;g/ ; (22.35) where we used the Baker–Hausdorff formula (see, e.g., [127], Section 5.10]) 1

1

e a e b D e aCbC 2 Œa;b  C 4 Œa;Œa;b   C :

(22.36)

This formula, the commutation relations (22.11), and condition (22.33) yield relation (22.35).

Section 22.2 Representations of Commutation Relations for a Free Real Scalar Field

259

As the set of finite vectors D0 , the set of exponential vectors De , i.e., its linear span, is an everywhere dense set in FB (see [93]). P This follows from the fact that, for the vector e g with g D jnD1 si fj ; its strong (in the sense of the norm) derivative with respect to s1 ; : : : ; sn at s1 D    D sn D 0 coincides with the vector ' C .f1 /    ' C .fn / 0 : The linear span of these vectors is everywhere dense in FB . On the basis of other reasonings, it is true that each strong derivative is the limit of a linear combination of exponential vectors as well.

22.2.4 Construction of Representations of Commutation Relations for a Complex Scalar Field The complex scalar field describes charged mesons. Hence, particles and antiparticles have different quantum characteristics. This means that two types of particles are present in the system, and the corresponding vectors of states depend on two collections of variables. It is worth noting that a system of free charged particles is clearly some mathematical idealization, since we neglect the long-range Coulomb interaction, by considering that the particles are located at significant distances from one another. In this case, a vector of the Fock space (22.4) has the form 1 0 F0 C B F10 .k1 / C B C B F01 .p1 / C B C: : (22.37) F DB :: C B C B BFmn .k1 ; : : : ; km I p1 ; : : : ; pn /C A @ :: : The functions Fmn .k1 ; : : : ; km I p1 ; : : : ; pn / are symmetric in the variables k1 ; : : : ; km and separately in the variables p1 ; : : : ; pn . The scalar product of two elements F and G is given by the formula .F; G/ D

1 Z X

Z m

.d k/

.d p/n Fmn .k1 ; : : : ; km I p1 ; : : : ; pn /

n;mD0

 Gmn .k1 ; : : : ; km I p1 ; : : : ; pn /; and the norm takes the form kF k D .F; F /1=2 : The vector F 2 F , if kF k < 1. The operators of creation and annihilation of particles and antiparticles satisfy re lations (18.13). We now write these relations for the operators ' ˙ .f / and ' ˙ .g/,

260

Chapter 22

Representations of Commutation and Anticommutation Relations

where f and g are real functions from the space of basic functions D.R3 /: 

Œ'  .f /; ' C .g/ D .f; g/H1  1; 

Œ' .f /; ' C .g/ D .f; g/H1  1:

(22.38) (22.39)

Relations (22.38, 22.39) are satisfied on the set of finite vectors D0 , if the operators ' .g/ and '  .f / act on column (22.37) by rules (22.16) for the first group of vari ables k1 ; : : :, km , and the operators ' C .f / and '  .g/ act by the same rules (22.16) for the second group of variables p1 ; : : : ; pn . The eigenvectors of the operators of b are vectors of the form energy P 0 and charge Q  1 .M;N / F .M;N / D Fmn ; C

m;nD0

.M;N / Fmn

D ıM m ıN n Fmn .k1 ; : : : ; km I p1 ; : : : ; pn /;

where ıM m and ıN n are the Kronecker deltas, and 1 0 M N X X P 0 F M;N D @ kj0 C pk0 A F M;N ; j D1

kj0 D

q

kD1

kj2 C m2 ; pk0 D

q

p2k C m2 ;

b .M;N / D .qM  qN /F .M;N / : QF This means that the system contains M particles with charge .Cq/ and N particles with charge .q/.

22.2.5 Construction of Representations of Commutation Relations in the Configuration Space. Relativistic Invariance of a Free Field For simplicity, we restrict ourselves by the consideration of real scalar field. The commutator of the operator-valued distributions ' C .x/ and '  .y/ was calculated in Section 18.1: 1 Œ'  .x/; ' C .y/ D D  .x  y/: i In Section 5.6, we constructed the one-particle and N -particle Hilbert spaces corresponding to the classical solutions of the Klein–Fock–Gordon equation. In the coordinate representation, they were constructed with the help of the unitary Fouriertransformation (5.47), if the scalar product is given by formula (5.48). For a quantum field, the construction of H1 and HN  FN is the same, and the Fock space FB is constructed according to the procedure described in Section 22.1. Using Equations (22.12), formulas presenting ' ˙ .x/ in terms of ' ˙ .k/ (18.2), and the formulas

Section 22.2 Representations of Commutation Relations for a Free Real Scalar Field

261

for a Fourier transformation (5.47) of the functions FN .p1 ; : : : ; pN /, we have .' C .x/F /N .x1 ; : : : ; xN / D N 1 X1  D .xj  x/FN 1 .x1 ; : : : ; xj 1 ; xj C1 ; : : : ; xN /; Dp N j D1 i p .'  .x/F /N .x1 ; : : : ; xN / D N C 1FN C1 .x; x1 ; : : : ; xN /:

(22.40a) (22.40b)

Equations (22.40) allow us to verify axiom AQ4 of the relativistic invariance of a free scalar field. We make it for the inhomogeneous orthochronous Lorentz transformations: (22.41) x 0 D ƒx C a; ƒ00 > 0; det ƒ D C1: In this case, axiom AQ4 (Equation (17.13)) can be rewritten in the form 

Uƒ;a '.x/U ƒ;a D '.ƒx C a/:

(22.42)



The action of the operators Uƒ;a and U ƒ;a is defined by the formulas

 .Uƒ;a F /N .x1 ; : : : ; xN / D FN ƒ1 .x1  a/; : : : ; ƒ1 .xN  a/ and



.U ƒ;a F /N .x1 ; : : : ; xN / D FN .ƒx1 C a; : : : ; ƒxN C a/: Then 

.Uƒ;a ' C .x/U ƒ;a F /N .x1 ; : : : ; xN / 

D .' C .x/U ƒ;a F /N .ƒ1 .x1  a/; : : : ; ƒ1 .xN  a// N  1 X 1  1 D .ƒ .xj  a/  x/.U ƒ;a F /N 1 .ƒ1 .x1  a/; : : : ; Dp N j D1 i

ƒ1 .xj 1  a/; ƒ1 .xj C1  a/; : : : ; ƒ1 .xN  a// N 1 X1  D .xj  .ƒx C a//FN 1 .x1 ; : : : ; xj 1 ; xj C1 ; : : : ; xN / Dp N j D1 i

D .' C .ƒx C a/F /N .x1 ; : : : ; xN /; where the Lorentz-invariance of the functions D  .x/ (see (18.10)) is used. Analogously, we can verify relation (22.42) for the operator '  .x/. To obtain the law of transformation of fields for arbitrary Lorentz transformations, we should supplement (22.41) by the formulas of the transformation of fields under and Ub (see (21.11) and (21.22)). the space and time reflections Ub P T

262

Chapter 22

Representations of Commutation and Anticommutation Relations

The validity of formulas (21.11) and (22.42) can be easily seen, if we give Ub by P the formula P N F /N .x1 ; : : : ; xN / D .b .Ub ' / FN .ƒis x1 ; : : : ; ƒis xN /: P

(22.43)

Since the one-particle amplitude (the vector of state) in the configuration space has the form (see (5.47)) F1 .x/ D

1 .2/3=2

Z

p e ik x Cikx F1 .k/d k; k 0 D k2 C m2 ; p 2k 0 0

0

(22.44)

it is easy to verify that the transformation Ub replaces the state with momentum k P p k2 C m2 by the state with momentum k and with the same

and with energy k 0 D



D U 1 D U b , i.e., it is energy k 0 . In addition, since U 2 D 1, the operator Ub P P b b P P unitary and self-adjoint. But if the operator Ub is given by formula (22.43) with ƒi t instead of ƒis , it folT p lows from (22.44) that the vector of a state with the positive energy k 0 D k2 C m2 is transformed into the vector of a state with the negative energy .k 0 /. To avoid such situation, we should define the representation operator of time reflection by the antiunitary transformation T N F /N .x1 ; : : : ; xN / D .b .Ub ' / FN .ƒi t x1 ; : : : ; ƒi t xN /: T

This transformation was first introduced by E. Wigner [221]. The additional information about these transformations can be found in [181], Part 7, Chapter 3.

22.3

Representation of Anticommutation Relations of Spinor Fields

22.3.1 Representation of Anticommutation Relations of the Operators of Creation and Annihilation of Fermions and Antifermions As in the case of a complex scalar field, we have two types of particles, i.e., particles and antiparticles. But, in a spinor field describing particles and antiparticles with spin 1=2, each particle (antiparticle) can have two values of the spin projection on the direction of motion k (see Section 6.3.2). Therefore, the corresponding operators of creation and annihilation have the additional index s D 1; 2 corresponding to the spin values of 1=2 and C1=2: Their anticommutation relations are given by formulas (19.4). The corresponding vector of the Fock space FF has structure (22.37) but with Fm;n D Fm;n . 1 ; k1 I : : : I m ; km j 1 ; p1 I : : : I n ; pn /;

Section 22.3 Representation of Anticommutation Relations of Spinor Fields

263

and the functions Fmn are antisymmetric under a permutation of the couple of variables i , ki with the couple j , kj , i ¤ j and l , pl with k , pk , l ¤ k. The scalar product of two vectors F and G has the form .F; G/ D

1 X

2 X

Z .d k/m

n;mD0 ./m ;./n D1 3m R

Z

.d p/n Fm;n .. ; k/m j.; p/n /  Gm;n .. ; k/m j.; p/n /: R3n

In order to satisfy relation (19.4), we need to define the action of the operator-valued  ˙ distributions a˙ s .k/ and as 0 .p/ in the following way: 

0 .aC s .k/F /m;n ..s; k/m j .s ; p/n / m 1 X Dp .1/j 1 ıssj ı.k  kj / m j D1

 Fm1;n ..s; k/m n ¹.sj ; kj /º j .s 0 ; p/n /; (22.45) p .as0 .p/F /m;n ..s; k/m j.s 0 ; p/n / D m C 1FmC1;n ..s 0 ; p/I .s; k/m j.s 0 ; p/n /; (22.46) .asC0 .p/F /m;n ..s; k/m j .s 0 ; p/n / n 1 X Dp .1/mCj 1 ıs 0 sj0 ı.p  pj / n j D1

 Fm;n1 ..s; k/m j .s 0 ; p/n n ¹.sj0 ; pj /º/; p  0 .a n C 1.1/m s .k/F /m;n ..s; k/m j .s ; p/n / D  Fm;nC1 ..s; k/m j .s; k/I .s 0 ; p/n /:

(22.47)

(22.48)

The additional factor .1/m in formulas (22.47) and (22.48) ensures the validity of the anticommutation relations 



 0  0 ˙ Œa˙ s .p/; as 0 .p /C D Œas .p/; as 0 .p /C D 0

of the operators acting on different groups of variables. From relations (22.45)–(22.48), we can easily determine the action of the true op erators as˙ .f / and a˙ s 0 .g/ for f; g 2 D0 and verify that they satisfy the relations 



C  0 Œas .f /; aC s 0 .g/C D Œas .f /; as 0 .g/C D ıss .f; g/;

(22.49)

 Œa˙ s .f

(22.50)

 /; a s 0 .g/C

D Œas˙ .f /; as0 .g/C D 0:

264

Chapter 22

Representations of Commutation and Anticommutation Relations

As in the case of a scalar field, these operators are defined on the set of everywhere  ˙ dense vectors D0 , and the norms of the vectors a˙ s .f /F and as .f /F at each s D 1; 2 for F 2 D0 satisfy inequalities (22.21) and (22.22). But these estimates are rather rough. It turns out that the operators of fermion fields are bounded operators due to the anticommutation relations (22.49) and (22.50). This result can be formulated by the following theorem. 

˙ Theorem 22.1 ([14]). The operators a˙ s .f / and as .f / are bounded operators in FF , and their norms satisfy the relations 

˙ ka ˙ s .f /kFF D kas .f /kFF D kf kH1 :

(22.51)



C Proof. We prove the theorem for the operators a s .f / and as .f /. In (22.49) and 0 (22.50), we set s D s and f D g and write the anticommutation relations between these operators: 



C C  2 a s .f /as .f / C as .f /as .f / D kf k ; 







   a s .f /as .f / C as .f /as .f / D 0;

(22.52)

asC .f /asC .f / C asC .f /asC .f / D 0: Two last relations give



2 C 2 a s .f / D as .f / D 0;

which presents, in fact, the Pauli principle in the operator form. Consider the operators C b s D a  P s .f /as .f /

and

b s D asC .f /a  Q s .f /:

b s and Q b s are defined and are symmetric on D0 . Using formula The operators P (22.52), we have   C C 2 C 2 2b b 2s D a  P s .f /as .f /as .f /as .f / D as .f / as .f / C kf k P s ;

i.e.,

b 2s D kf k2 P bs : P

In the same way, we obtain and

b 2s D kf k2 Q bs Q

bs C Q b s D kf k2H  1; P 1

where 1 is the identity operator.

bs D Q b s D 0; bs Q bsP P

(22.53)

265

Section 22.3 Representation of Anticommutation Relations of Spinor Fields

b s and Q b s for arbitrary F 2 D0 These relations and the symmetry of the operators P yield b s F k2 D kf k4 kF k2 : b F k 2 C kQ kP FF FF H1 FF Whence we have and

b s F kFF kf k2H kF kFF kP 1

(22.54)

b s F kFF kf k2 kF k2 : kQ H1 FF

(22.55)

Since D0 is everywhere dense in FF , we obtain b s kFF D kf k2 kP H1

and

b s kFF D kf k2 : kQ H1

(22.56)

Relations (22.54) and (22.55) become the equalities on the vectors F D 0 and F D asC .f / 0 ; respectively. b s and their boundedness that the operab s and Q It follows from the symmetry of P b b tors P and Q are self-adjoint. In view of the self-adjointness and equalities (22.56) and (22.53), we have b s kFF kf k2H1 D kP D

sup

b s F; P bs F / .P FF

1=2

kF kFF D1

D kf kH1

sup

b s F; F /1=2 .P

kF kFF D1

D kf kH1 sup .asC .f /F; asC .f /F / D kf kH1 kasC .f /k2FF : kF kFF

This yields equalities (22.51) for the operator asC .f / and the operators as .f /, 

a˙ s .f /. Remark 22.2. The boundedness of Fermi-operators has a very profound content. In Chapter 7, we show that the boundedness of smoothed operators of the Fermi-field enables one to prove the convergence of the series of perturbation theory for a regularized matrix of scattering for some models similar to the Yukawa model or in the quantum electrodynamics.

22.3.2 Representation of Anticommutation Relations in the Configuration Space 

In the configuration space, 4-component operators-spinors .x/ and .x/ (or  N .x/ D .x/ 0 ) can be given in terms of the operators as˙ .k/ and a ˙ s .k/ by rela-

266

Chapter 22

Representations of Commutation and Anticommutation Relations

tions (19.2) and (19.3). The anticommutation relations of their positive- and negativefrequency parts take the form (see also (19.5)) Œ

 ˛ .x/;

N ˙ .y/ D 1 S  .x  y/: ˇ 2 ˛ˇ

(22.57)

In Section 6.5.1, we constructed the Hilbert space of states of classical particles and antiparticles. This space was constructed on the negative-frequency (solutions with positive energy) solutions of the Dirac equations for the spinors .x/ and c .x/, i.e., on the total system of solutions ¹ i .x/º and ¹ jc; º (here, i , j are not the indices of spinors, but their numbers in the infinite set of solutions, i.e., i; j 2 N). Their connection with the corresponding Fourier-transforms is given by formulas (6.79) and (6.103). Therefore, as in the scalar case, the vectors of states of the Fock space FF in the coordinate representation can be presented in terms of the corresponding vectors in the momentum space as follows: Fm;n ..˛; x/m j .ˇ; y/n / Z Z P Pn 1 m n i m iD1 ki xi i j D1 pj yj .d k/ .d p/ e D .2/3=2.mCn/ 2 2 m X n X Y Y c;s 0 ;  v˛sii; .ki / vˇ j .ps /Fm;n ..s; k/m j .s 0 ; p/n /: iD1 si D1

j D1 sj0 D1

j

(22.58)

Let us multiply Equations (22.45)–(22.48) by the corresponding spinors v, N v; and v c . Then, according to Definitions (6.79), (6.80), and (22.58), formulas (6.84), (6.85), (6.103), and (6.104), and the definitions of the functions S ˙ (see (19.6), (19.8), and (19.9)), we obtain the rule for the action of the symbols ˙ .x/ and N ˙ .y/: . N ˛C .x/F /m;n ..˛; x/m j .ˇ; y/n / m 1 1 X Dp .1/j 1 S˛./ .xj  x/ i j˛ m j D1

 Fm1;n ..˛; x/m n ¹.˛j ; xj /º j .ˇ; y/n /;  0 ˛ 0 .x /F /m;n ..˛; x/m j .ˇ; y/n / p D m C 1FmC1;n ..˛ 0 ; x 0 I ˛1 /I .˛; x/m

(22.59)

.

n ¹.˛j ; xj /º j .ˇ; y/n /;

C .x/F /m;n ..˛; x/m j .ˇ; y/n / ˇ n 1 1 X Dp .1/mCk1 SˇC˛0 .x i n kD1

(22.60)

.

 yk /

 C˛0 ˇk Fm;n1 ..˛; x/m j .ˇ; y/n n ¹.ˇk ; yk /º/;

(22.61)

267

Section 22.4 Space of States of a Free Electromagnetic Field

. N ˇ0 .x 0 /F /m;n ..˛; x/m j .ˇ; y/n / p D .1/m n C 1Cˇ 0 ˇ 00 Fm;nC1 ..˛; x/m j .ˇ 00 ; x 0 /I .ˇ; y/n /:

(22.62)

We recall: as above, we sum over the repeated indices (i.e., over ˛ 0 in (22.61) and over ˇ 00 in (22.62)). With regard for formulas .C D C T / and .C 2 D 1/; relations (22.59) and (22.60) yield (22.57) (upper sign), and relations (22.61) and (22.62) yield, respectively, (22.57) (lower sign). The factor .1/m ensures that all last anticommutators are equal to zero. In conclusion, we verify the validity of axiom AQ4, i.e., we check the law of transformation (17.13). By virtue of the consistency, the action of the operators UL and 

U L in the Fock space remains the same as that in the classical case, i.e., it is described by formula (5.49). We rewrite the condition of invariance (17.13) for the spinor .x/ in the form 

UL

˛ .x/UL

D S.ƒ/˛˛0

1 ˛ 0 .ƒ .x

 a//:

(22.63)

Let us verify (22.63) separately for ˛C .x/ and ˛ .x/ under orthochronous Lorentz transformations. Using formulas (5.49) and (22.61), it is easy to see that (22.63) holds for the operator ˛C .x/, if the equality S.ƒ/S C .ƒ1 .x  a/  yk /C D S C .x  a  ƒyk /C S 1 .ƒ/T is valid. Or, in view of property (6.90), we obtain S.ƒ/S C .ƒ1 .x  a/  yk / D S C .x  a  ƒyk / S.ƒ/:

(22.64)

Using the Fourier transformation (19.9), we see that (22.64) follows from the identity c S.ƒ/ S.ƒ/kO D ƒk which holds due to (6.22) and (1.12). For ˛ .x/; equality (22.63) is almost trivial. Analogously, we can verify the law of transformation (17.13) for fields N .x/ and c .x/ with regard for the equality C 1 S C .x/C D S  .x/T which is a consequence of (6.95) and (6.36).

22.4

Space of States of a Free Electromagnetic Field

If we omit the difficulties related to the positive definiteness of the operator of energy P 0 , then the construction of the Fock space for a system composed from arbitrary

268

Chapter 22

Representations of Commutation and Anticommutation Relations

numbers of temporal, longitudinal, and transversal photons is the same as that in Section 22.1. A vector of the N -particle Hilbert space FN has the form FN D FN .1 ; k1 I : : : ; N ; kN /;

N D N0 C N1 C N2 C N3 ;

where N0 is the number of temporal photons (with j D 0), N3 is the number of longitudinal photons j D 3, and N1 and N2 are the numbers of photons with transverse polarization (right and left). The scalar product in the space F WD FE takes the form .F; G/ D

1 X

Z

X

N D0 1 ;:::;N

d k    d kN FN .1 ; k1 I : : : I N ; kN / R3N

 GN .1 ; k1 I : : : I N ; kN /: ˙ .k/ which By direct calculations, we can verify that the action of the operators a satisfy the commutation relations (20.26) is given by the formulas  .a .k/F /N .; ki ; : : : ; N ; kN / p D N C 1FN C1 .; k; 1 k1 I : : : I N ; kN /;

(22.65)

C .k/F /N .; ki I : : : I N ; kN / .a N 1 X Dp .g j /ı.k  kj / N j D1

 FN 1 .1 ; k1 I : : : I j 1 ; kj 1 I j C1 ; kj C1 I : : : I N ; kN /:

(22.66)

According to the Gupta–Bleuler formalism (see Section 20.2.2), the following step is the construction of a set of physically admissible vectors P in the space FE . We define the operator  (see (20.31)) by the formula .F /N .1 ; k1 I : : : I N ; kN / D

N Y

0

.g j j /FN .01 ; k1 I : : : I 0N ; kN /: (22.67)

j D1

We can verify that  satisfies condition (20.31), and the space FE relative to the bilinear form (20.32) is transformed in the Hilbert space with indefinite metric. We now construct the set of physically admissible vectors P which was defined by formula (20.36). The definition of operators of annihilation yields  .k/ 0 D 0: a

Hence, 0 2 P. It becomes clear now that the set P can be constructed as a linear span of vectors of the form b1C .f1 /    bnC .fn / 0 ;

269

Section 22.4 Space of States of a Free Electromagnetic Field

R where bjC .fj / D fi .k/bjC .k/d k, fj .k/ 2 H1 , and the operators bjC .k/ are arbitrary operators of creation commuting with the operators L .k/ (see (20.35)). It follows from the commutation relations (20.26) that such operators are a1C .k/ and a2C .k/. But, in view of the same relations (20.26), it is easy to calculate that the opC  C erators L .k/ D k  A  .k/ also commute with the operators L .p/ D p A .p/. 2

Indeed, with regard for relation (20.24) and equality p 2 D p 0  p2 D 0, we obtain C 2 ŒL .k/; LC .p/ D k  p  ŒA  .k/; A .p/ D p ı.k  p/ D 0:

This allows us to construct the vector F 2 P in the form (see (22.32)) 1 X

F D

n0 ;n1 ;n2

1 p n0 Šn1 Šn2 Š D0

Z .d k/n0 .d p/n1 .d q/n2 R3.n0 Cn1 Cn2 /

 Fn0 ;n1 ;n2 .k1 ; : : : ; kn0 ; p1 ; : : : ; pn1 ; q1 ; : : : ; qn2 / 

n0 Y

n1 n2 Y Y a1C .pj / a2C .qk / 0 : LC .ki / j D1

iD1

(22.68)

kD1

Using relations (20.26), (20.30), and (20.31), it is seen that, for vectors (22.68) such that F0;n1 ;n2  0, and Fn0 ;n1 ;n2 6 0 at n0 ¤ 0, we have hF; F i D .F; F / D 0: Whereas for vectors (22.68) such that Fn0 ;n1 ;n2 D 0 for all n0  1; hF; F i D .F; F / D .F; F / > 0: In other words, for the vectors of physically admissible states that correspond only to photons with transverse polarization, the bilinear form (20.32) is identical to the scalar product in the space FE : Hence, hF; F i D kF k2 > 0:

(22.69)

Then, for an arbitrary vector F 2 P; we have hF; F i D .F; F / D

1 X n1 ;n2 D0

Z .d p/n1 .d q/n2 jF0;n1 ;n2

R3.n1 Cn2 /

.p1 ; : : : ; pn1 ; q1 ; : : : ; qn2 /j2 > 0: Thus, for the vectors from P; the probabilistic interpretation is valid, and equality (20.40) is satisfied.

270

Chapter 22

Representations of Commutation and Anticommutation Relations

Remark 22.3. By P0 P; we denote a subset of vectors-states, for which equality (22.69) holds, i.e., states corresponding only to photons with transverse polarization. Every vector F 2 P of the form (22.68) can be associated with a vector F 0 2 P0 , by setting Fn0 ;n1 ;n2 D 0 in (22.68) for all n0  1. We call vectors F and F0 the equivalent states if they lead to the same physical consequences. From the mathematical viewpoint, this means that the mean values of an arbitrary operator A corresponding to some physical quantity in the states F and F0 are identical, i.e., (22.70) hF; AF i D hF0 ; AF0 i: To prove relation (22.70), we firstly calculate the mean values for the field operator A .x/. In (22.68), we choose Fn0 ;n1 ;n2 in the form Fn0 ;n1 ;n2 .k1 ; : : : ; kn0 I p1 ; : : : ; pn1 I q1 ; : : : ; qn2 / D fn0 .k1 ; : : : ; kn0 /Fn01 ;n2 .p1 ; : : : ; pn1 I q1 ; : : : ; qn2 /: Then, in view of the definition of vectors F 2 P and F 0 2 P0 , it is easy to calculate (Problem 22.3) that C hF; AC  .k/F i D hF0 ; .A .k/  k f1 .k//F0 i;  hF; A  .k/F i D hF0 ; .A .k/  k f1 .k//F0 i:

(22.71)

Definitions (8.15) yield hF; A .x/F i D hF0 ; .A .x/  @ f0 .x//F0 i; where i f0 .x/ D .2/3=2

Z

e ikx i dk p .f1 .k// C 0 .2/3=2 2k

Z

e ikx dk p f1 .k/; 2k 0

(22.72)

(22.73)

which yields f0 .x/ D 0: It follows directly from formula (22.72) that hF; F .x/F i D hF0 ; F .x/F0 i:

(22.74)

In other words, the mean values of the observable F in the states F and F0 are identical. The more general proposition (22.70) follows from the fact that the observable quantities are gauge-invariant. Clearly, the direct verification of equality (22.70) can be quite awkward. Equalities (22.71)–(22.74) yield equality (22.70), if we take into account that the physically observable quantities depend on the tensor F D @ A  @ A . In conclusion, we note that the construction of representations of the commutation relations (20.25) in the coordinate Fock space is analogous to that in the scalar case (see Section 22.2.5).

271

Section 22.5 Space of Occupation Numbers

22.5

Space of Occupation Numbers

Sometimes, it is convenient to work with the so-called basis of occupation numbers. This means that we fix some basis related to pure physical states of the system in the space of states H . Then every vector of the space of states can be expanded in this basis, and the expansion coefficients are the probability amplitudes to find, for example, n1 particles in the pure state ˆ1 , n2 particles in the pure state ˆ2 ; etc. In this case, the action of operators can be given on the expansion coefficients. To demonstrate how such construction can be realized, we consider the simple example of the space of states of a free scalar field, i.e., the Fock space FB . First, we need to construct a basis in the space: H1 D L2 .R3 / D L2 .R1 / ˝ L2 .R1 / ˝ L2 .R1 /: The basis in the space L2 .R1 / can be chosen in the form of a sequence of orthonormalized Chebyshev–Hermite functions  eQi .t / D

1 p i  2 iŠ

1=2

1 2

Hi .t /e  2 t ; t 2 R1 ; i D 0; 1; : : :

We now introduce the basis in L2 .R3 / constructed by eQi .t /: ek .p/ D eQk .1/ .p .1/ /eQk .2/ .p .2/ /eQk .3/ .p .3/ /; k D .k .1/ ; k .2/ ; k .3/ /; k .i/ D 0; 1; 2; : : : As for the basis in the space of n particles, we define the functions Gk1 ;:::;kn .p1 ; : : : ; pn / D ek1 .p1 / : : : ekn .pn /:

(22.75)

System (22.75) is a system of orthogonal normalized functions. But they are not symmetric, because if ki ¤ kj ; then the permutation of pi and pj gives the different functions. We now construct symmetric functions with the help of the operator of symmetrization b S: S Gk1 ;:::;kn .p1 ; : : : ; pn / GkS1 ;:::;kn .p1 ; : : : ; pn / D b 1 X D Gk1 ;:::;kn .p.1/ ; : : : ; p.n/ /; nŠ

(22.76)

2Sn

where Sn is the group of permutations from n elements. But these functions are not normalized. To normalize the functions, we order them as follows.

272

Chapter 22

Representations of Commutation and Anticommutation Relations

First of all, we order vectors of the basis of the one-particle space F1 D L2 .R3 /. In other words, let ¹ek .p/º WD ¹ej .p/ºj1D1 : For example, e0;0;0 .p/ WD e1 .p/, e1;0;0 .p/ WD e2 .p/, e0;1;0 .p/ WD e3 .p/, etc. It is clear that, at the construction of functions (22.76), the identical basis vectors can be repeated some times. Let, for example, the vector e1 .p/ be repeated n1 times, e2 .p/ be repeated n2 times, etc. It is clear that n1 C n2 C    C nj C    D n, and nj D 0; 1; 2; : : : . By n D .n1 , n2 ; : : : ; nj : : :/, we denote a finite sequence, i.e., only a finite number of numbers nj is not equal to zero. For the given n; we define the vector S GnS .p1 ; : : : ; pn / D G1; : : : ; 1 ; 2; : : : ; 2 ; : : : .p1 ; : : : ; pn / D „ƒ‚… „ƒ‚… n1

n2

Db S.e1 .p1 / : : : e1 .pn1 /e2 .pn1 C1 / : : : e2 .pn1 Cn2 / : : : /: Let the basis in the n-particle space be defined by the system of vectors en .p1 ; : : : ; pn / D Nn GnS .p1 ; : : : ; pn /; where the normalization factor Nn is determined from the condition .en ; en / D 1:

(22.77)

It is seen from relation (22.77) that if the multiplicity of vectors ei .p/ is 1 (i.e., .ni D 1/), then we have (22.78) .GnS ; GnS / D .nŠ/1 ; since the contributions of terms of the sum over  in (22.76) to the left-hand side of (22.78) which correspond to different permutations become zero due to the conditions of orthogonality: .ei ; ej / D 0 at i ¤ j: Hence, only nŠ terms corresponding to the same permutation  are nonzero. If the contribution to GnS is made by n1 vectors e1 .p/, n2 vectors e2 .p/; etc., then the permutations of momenta in the group of vectors e1 .p/, e2 .p/; etc. do not lead to orthogonal terms, because they do not change, in fact, the functions Gn . The number of identical terms is n1 Š n2 Š : : : nk Š : : : , and the sum retains nŠ n1 Š n2 Š : : : nk Š : : : terms which are orthogonal to one another. Hence, in this case, we obtain .GnS ; GnS / D

n1 Š : : : nk Š : : : ; nŠ

273

Section 22.5 Space of Occupation Numbers

i.e.,

1=2 nŠ : n1 Š : : : nk Š : : : Finally, we obtain that the basis in FB is a sequence of vectors 0 1 0 :: B C : B C B C 0 B C eOn D B C: Ben .p1 ; : : : ; pn /C B C 0 @ A :: : 

Nn D

The expansion of an arbitrary vector F 2 FB in the basis ¹eOn º takes the form X X F D fn eOn D f .n1 ; n2 ; : : :/eOn : (22.79) n1 ;n2 ;:::

¹nº

In order to obtain the representations of commutation relations for the operators corresponding to the operators ' ˙ .p/ and act in the space of coefficients of the expansion f .n1 ; n2 ; : : :/, we expand the operators ' ˙ .p/ in the one-particle basis ei .p/ i D 1; 2; : : : : 1 X ei .p/ai˙ : ' ˙ .p/ D iD1

Z

Whence we have ai˙

˙

D .' ; ei / D

Using formulas (22.12), we easily find aiC f .n1 ; n2 ; : : : ; ni ; : : :/ D ai f .n1 ; n2 ; : : : ; ni ; : : :/

d p' ˙ .p/ei .p/:

p

ni C 1 f .n1 ; n2 ; : : : ; ni C 1; : : :/; p D ni f .n1 ; n2 ; : : : ; ni  1; : : :/:

It is easy to prove that Œai ; ajC  D ıij ; Œai˙ ; aj˙  D 0: These operators act in the space l 2 of sequences fO D ¹fn º such that, for fO 2 l 2 , X X kf kl 2 D jfn j2 D jf .n1 ; n2 ; : : :/j2 < 1: ¹nº

n1 ;n2 ;:::

The Hilbert space l 2 is unitarily equivalent to the Fock space FB . The operator which introduces the indicated unitary equivalence is given by relation (22.79) which realizes, in fact, the corresponding Fourier transformation of the functions F 2 FB in the sequence fO 2 l 2 .

Chapter 23

Green Functions

The Green functions were introduced in mathematical physics as an auxiliary object for the determination of solutions of differential equations. Every Green function can be connected with its “own” differential operator Dx and can be found as a solution of the partial differential equation: Dx G.x/ D ı.x/:

(23.1)

This solution is also called the fundamental solution of the differential equation. It is determined to within an arbitrary function satisfying the homogeneous equation (23.1) (see [207], Section 10). However, in the quantum field theory, this notion acquires a broader meaning. In what follows, it is shown that the Green functions of quantized fields describe the causal relationships between the processes of creation and annihilation of particles at different points of the Minkowski space. These functions are closely connected with the commutator functions of the free quantized fields. In the present chapter, we consider the Green functions corresponding to free noninteracting fields. The convenience of investigation of the Green functions in the quantum field theory was first demonstrated by Gell-Mann and Low [64] and Schwinger [182]. Schwinger used the Green functions as the foundation of his approach to the construction of the quantum field theory.

23.1

Green Functions of the Scalar Field

In the case of a scalar field, the operator Dx coincides with the Klein–Gordon–Fock operator (5.6) and Equation (23.1) takes the form .x  m2 /G.x/ D ı.x/:

(23.2)

As in Section 5.3, it is convenient to seek the solution in the form of a Fourier integral Z 1 Q G.x/ D (23.3) d k e ikx G.k/: .2/4 By using the representation for the ı-function, Z 1 ı.x/ D d k e ikx ; .2/4

275

Section 23.1 Green Functions of the Scalar Field

Q we obtain the equation for G.k/: Q .m2  k 2 /G.k/ D 1: Since the formula

1 Q G.k/ D 2 (23.4) m  k2 p has two singular points k 0 D ˙ k2 C m2 , integral (23.3) (as the integral with respect to the variable k 0 ) has two poles and can be calculated within the method of residues. But such calculation is not single-valued, since its result depends on the rule of bypass of the poles by a contour. Such ambiguity is quite clear, because the partial differential equation (23.2) has the unique solution only if the boundary conditions are fixed. Of significant importance for practical applications are retarded, advanced, and causal Green functions. The retarded Green function is defined by the boundary condition G.x/ D 0 at

x 0 < 0:

(23.5)

Then we have G.x/ D D ret .x/ D .x 0 /D.x/;

(23.6)

where D.x/ is the Pauli–Jordan commutator function defined by Equation (18.8). The advanced Green function is defined by the boundary condition G.x/ D 0 at

x 0 > 0:

(23.7)

Then we obtain G.x/ D D adv .x/ D .x 0 /D.x/:

(23.8)

We can easily obtain Equations (23.6) and (23.8) from relation (23.3), by substituting (23.4) there and integrating with respect to the variable k 0 with the help of of the theory of residues (Problem 23.1). In this case, the rule of bypass of the poles in the complex plane of the variable k 0 is determined by the corresponding boundary condition (23.5) or (23.7) (see the instruction to Problem 23.1). In quantum field theory, the causal Green function D c .x  y/ is of the greatest importance. It describes the causal connection of the processes of creation and annihilation of particles at different points of the space-time x and y. According to the interpretation of the operators ' ˙ .x/; the process of creation of a particle at a point x 2 M and its annihilation at a point y 2 M (i.e., y 0 > x 0 ) is obviously described by the matrix element . 0 , '  .y/' C .x/ 0 /. Then Equations (18.5) and (18.6) yield 1 . 0 ; '  .y/' C .x/ 0 / D D  .y  x/ D iD C .x  y/: i

276

Chapter 23

Green Functions

Conversely, at x 0 > y 0 ; the particle is created at a point y and disappears at a point x, i.e., 1 . 0 ; '  .y/' C .x/ 0 / D D  .x  y/: i Therefore, the causal connection of these processes should be described by such function D c .x y/ which is proportional to D C .x y/ at x 0 < y 0 and to D  .x y/ at x 0 > y 0 : It is easy to verify, by performing simple calculations, that such conditions are satisfied by the function Z e ik.xy/ 1 c d k: (23.9) D .x  y/ D .2/4 m2  k 2  i " The integral on the right-hand side should be considered as an pimproper integral with respect to k 0 with the rule of bypass of the poles k 0 D ˙ m2 C k2 . This rule is given by the introduction of an infinitely small parameter " p which shifts the poles in the complex plane of the variable k 0 into the points k 0 D ˙ m2 C k2  i ". Hence, the contour of integration Im k 0 D 0 bypasses the right pole above and the left one below. After the calculation of the integral, the parameter " must be turned to zero. Sometimes, the symbol i 0 is used in definition (23.9) instead of i ". Let us verify that the function D c .x  y/ chosen in such manner satisfies the necessary requirements. Consider, for example, the case x 0 > y 0 . Then we have e ik

0 .x 0 y 0 /

D e i.Re k

0 /.x 0 y 0 /.Im k 0 /.x 0 y 0 /

:

(23.10)

Function (23.10) decreases if Im k 0 > 0. In order to calculate the integral with respect to k 0 in (23.9), we must close the integration contour in the upper half-plane Im k 0  0 and to determine the value of the integral by the left pole (Problem 23.2). Then we obtain Z p e ik.xy/ i 0 d k ; k D k2 C m2 : (23.11) D c .x  y/ D .2/3 2k 0 Hence, at x 0 > y 0 ; the causal function is identical to D  .x y/ (see (18.6)). Quite analogously, we can verify that, at x 0 < y 0 ; D c .x  y/ D D C .x  y/: Finally, we obtain D c .x  y/ D .x 0  y 0 /D  .x  y/  .y 0  x 0 /D C .x  y/:

(23.12)

Remark 23.1. It is easy to verify that the function D c .x  y/ is not defined at x 0 D y 0 . At x ¤ y; such indeterminacy can be eliminated, since the left and right limits are the same at x 0 ! y 0 : D c .0; x  y/ D D  .0; x  y/ D D C .0; x  y/:

Section 23.2 The Green Functions of Spinor, Vector, and Electromagnetic Fields

277

But, for x D y; all three functions are singular, and their connection with formula (23.12) can no longer be regarded as exact because this is, in fact, the problem of definition of the product of distributions.

23.2

The Green Functions of Spinor, Vector, and Electromagnetic Fields

In Chapters 16 and 17, we calculated the commutator functions of spinor, vector, and electromagnetic fields. It is seen from formulas (19.7) and (20.12) that these functions can be obtained from the commutator function of a scalar field (the Pauli– Jordan function) by the action of the corresponding differential operators, and the Green function of the electromagnetic field coincides (to within g  ) with the Pauli– Jordan function at m D 0. It is seen from the definition of the causal Green function for a scalar field that these relations also hold for the causal functions of the spinor, vector, and electromagnetic fields(photon Green function) (Problem 23.3). Hence, we can define the causal functions of these fields by the formulas Z .m C p/ O ˛ˇ ipx 1 c c O e dp; (23.13) S˛ˇ .x/ D .i @ C m/˛ D .x/ D 4 2 2 .2/ m  p  i"   @2 1 c .x/ D g C 2   D c .x/ D m @x @x   Z g  k k 2 1 m e ikx d k; (23.14) D .2/4 m2  p 2  i " c;el. .x  y/ D g D0c .x  y/ D g  D c .x  y/ jmD0 D Z e ikx g d k: (23.15) D 4 .2/ k2 C i " The function D c .x/ and, hence, the causal Green functions of other fields can be calculated exactly (see [26], Section 15). This distribution is singular and has the form p p 1 m ı.x 2 /  D c .x/ D p .x 2 /¹J1 .m x 2 /  iN1 .m x 2 /º 4 8 x 2 p mi .x 2 /K1 .m x 2 /; (23.16) C p 4 2 x 2 where J1 .z/ is the Bessel function of the first kind, N1 .z/ is the Neumann function of the first order, and K1 .z/ is the zero-order Hankel function of the imaginary argument. Remark 23.2. In this section, we introduced the Green functions as fundamental solutions (see [207], Section 10) of the partial differential equations which are defined by the differential operator on the left-hand side of the equation for the relevant

278

Chapter 23

Green Functions

field. The uniqueness of a representation can be ensured, if we seek the solutions of Equation (23.1) in a certain class of functions, by imposing certain boundary conditions. But since these solutions are singular distributions, their calculation by formulas (23.13)–(23.15) should be executed with a certain care. For example, the calculation of the Green function of a scalar field D c .x  y/ is carried out for x 0 ¤ y 0 , Therefore, this function is not defined for x 0 D y 0 (see (23.12) and Remark 23.1), and the calculation of functions (23.13)–(23.15) with the help of formula (23.12) can lead to the appearance of additional terms which are the distributions concentrated at the points x 0 D y 0 . Remark 23.3. In the calculation of the Green functions of vector fields and the electromagnetic field, we must consider the invariance of the theory under gradient transformations. Such invariance allows one to fix a gauge and to obtain different equations (see, e.g., (8.6) and (8.9) for the electromagnetic field) for the same field. This means that the differential operators present in Equation (23.1) can be different in the definition of a Green function. This means that the form of the Green functions of the same field for different gauges will be different. This question will be discussed in Chapter 8 in more detail. We only note that the Green function of the electromagnetic field D0c .x  y/ is the Green function of the d’Alembert operator (5.5).

23.3

Time-Ordered Product and Green Functions

To generalize the notion of Green functions to the case of interacting fields, it is convenient to write the Green functions of free fields in the form of vacuum averages of a time-ordered product (T -product) of operators of the free field. To this end, we define the operation of time-ordered product of two field operators ui .x/, uj .y/ as follows: ´ ui .x/uj .y/ at x 0 > y 0 ; T .ui .x/uj .y// D (23.17) ˙uj .y/ui .x/ at x 0 < y 0 ; where we choose the sign “C” if both operators are Bose-operators or one Boseoperator and one Fermi-operator and the sign “” if both operators are Fermi-operators. For a scalar real field, it is easy to find that ² . 0 ; '.x/'.y/ 0 / D 1i D  .x  y/ at x 0 > y 0 ; . 0 ; T .'.x/'.y// 0 / D . 0 ; '.y/'.x/ 0 / D iD C .x  y/ at x 0 < y 0 : Hence, i. 0 ; T .'.x/'.y// 0 / D D c .x  y/:

(23.18)

279

Section 23.4 Wick Theorems

Analogous relations can be written for the Green functions of spinor, vector, and electromagnetic fields: i. 0 ; T .

˛ .x/ N ˇ .y// 0 /

c D S˛ˇ .x  y/;

c i. 0 ; T .V .x/V .y// 0 / D D .x  y/;

i. 0 ; T .A .x/A .y// 0 / D g

23.4



D0c .x

 y/:

(23.19) (23.20) (23.21)

Wick Theorems

23.4.1 Wick Theorem for Normal Products The notion of normal product was given in the formulation of axiom AQ9. For two operators of a free field ui .x/ and uj .y/; the ordinary product ui .x/uj .y/ differ from the normal product W ui .x/uj .y/ W only by the permutation of the operators u i .x/ and C uj .y/ (see formula (17.32)). In other words, C C  ui .x/uj .y/ D W ui .x/uj .y/ W Cu i .x/uj .y/  "uj .y/ui .x/ C D W ui .x/uj .y/ W CŒu i .x/; uj .y/˙ ;

(23.22)

where we take the commutator ." D 1/ in the case where at least one of the fields is a Bose-field and the anticommutator ." D 1/ if both fields are Fermi-fields. The operation of commutation in the last term of relation (23.22) is called the pairing operation of the fields ui .x/ and uj .y/, and its result is called by the pairing and is denoted by a square bracket under symbols. Since the vacuum mean of a normal product is always equal to zero, the pairing can be also defined as the vacuum mean of an ordinary product of two operators. In other words, C ui .x/uj .y/ D Œu i .x/; uj .y/˙ D . 0 ; ui .x/uj .y/ 0 /:

(23.23)

Now, a partial case of the Wick theorem for two operators of free fields can be written in the form u1 .x1 /u2 .x2 / D W u1 .x1 /u2 .x2 / W C u1 .x1 /u2 .x2 / : In order to formulate the Wick theorem for the product of an arbitrary number of fields, we introduce the notion of a normal product with pairing. But we note firstly that the definition of a normal product yields directly W u1 .x1 /    un .xn / W D .1/pF W ui1 .xi1 /    uin .xin / W; where pF is the number of permutations of Fermi-operators at the transition from the order 1, 2; : : :, n to the order i1 , i2 ; : : :, in . By definition, we write W u1 .x1 /    uj .xj /    uk .xn /    un .xn / W D .1/pF .j;k/ uj .xj /uk .xk /  W u1 .x1 /    uj 1 .xj 1 /uj C1 .xj C1 /    uk1 .xk1 /ukC1 .xkC1 /    u.xn / W;

280

Chapter 23

Green Functions

where pF .j; k/ is the number of permutations of Fermi-operators at the transition from the order 1; : : : ; n to the order j , k, 1; : : : , j  1, j C 1; : : : , k  1, k C 1; : : : ; n. The normal products with any number of pairings are defined analogously. Theorem 23.4. The product of an arbitrary number of the operators of free quantized fields is equal to the sum, in which each term is the normal product of these fields with all possible pairings, including the term of the normal product without pairing: X .1/pF .j1 ;j2 / W u1 .x1 / : : : un .xn / D W u1 .x1 / : : : un .xn / W C uj1 .xj1 /uj2 .xj2 X C

/ W u .x /    u1 .x

¹j1 ;j2 º¹1;:::;nº

/    u1 .x

1 1 j1 j1 j2 j2 /    un .xn / W h .1/pF .j1 ;j2 ;j3 ;j4 / uj1 .xj1 /uj2 .xj2 / uj3 .xj3 /uj4 .xj4 /

¹j1 ;:::;j4 º¹1;:::;nº

C .1/pF .j1 ;j3 ;j2 ;j4 / uj1 .xj1 /uj3 .xj3 / uj2 .xj2 /uj4 .xj4 / C .1/pF .j1 ;j4 ;j2 ;j3 / uj1 .xj1 /uj4 .xj4 / uj2 .xj2 /uj3 .xj3 /

1

i

1

 W u1 .x1 /    uj1 .xj1 /    uj4 .xj4 /    un .xn / W C    : Here after, the mark b u means the absence of the field u in the corresponding product. Proof. of the theorem can be easily performed by the method of mathematical induction. Remark 23.5. Theorem 23.4 is valid also in the case where it is necessary to reduce a product of operators, each of which is a normal product of some number of operators, i.e., a product of the form W u1 .x1 /    un1 .xn1 / W W un1 C1 .xn1 C1 /    un1 Cn2 .xn1 Cn2 / W    W un1 CCnk1 C1 .xn1 CCnk1 C1 /    un1 CCnk .xn1 CCnk / W to the normal form (i.e., to the form of a normal product). But, in this case, we must not consider the pairings of operators which belong to the same normal product. Example 23.6. W u1 .x1 /u2 .x2 / W W u3 .x3 /u4 .x4 / W D W u1 .x1 /u2 .x2 /u3 .x3 /u4 .x4 / W C W u1 .x1 /u2 .x2 /u3 .x3 /u4 .x4 / W C W u1 .x1 /u2 .x2 /u3 .x3 /u4 .x4 / W C W u1 .x1 /u2 .x2 /u3 .x3 /u4 .x4 / W C W u1 .x1 /u2 .x2 /u3 .x3 /u4 .x4 / W C W u1 .x1 /u2 .x2 /u3 .x3 /u4 .x4 / W C W u1 .x1 /u2 .x2 /u3 .x3 /u4 .x4 / W : (23.24)

281

Section 23.4 Wick Theorems

Concluding this subsection, we give the form of pairing for the basic fields. Definition (23.23) and relations (18.6), (18.14), (19.5), (20.10), and (20.25) yield:  for a real scalar field, 1 '.x/'.y/ D D  .x  y/I i

(23.25)

 for a complex scalar field, 1   '.x/'.y/ D '.y/'.x/ D D  .x  y/; i

(23.26)

'.x/'.y/ D '.x/'.y/ D 0I

(23.27)





 for spinor fields, 1  D S˛ˇ .x  y/; i N ˛ .x/ ˇ .y/ D 1 S C .y  x/; i ˇ˛ ˛ .x/ ˇ .y/ D N ˛ .x/ N ˇ .y/ D 0I ˛ .x/ N ˇ .y/

(23.28) (23.29) (23.30)

 for vector fields,  1  .x  y/; V .x/V  .y/ D D i

(23.31)

V .x/V .y/ D V  .x/ V  .y/ D 0I

(23.32)





 for an electromagnetic field, A .x/A .y/ D ig  D0 .x  y/:

(23.33)

We note also that all pairings of fields of different types are identically zero.

23.4.2 Wick Theorem for a Time-Ordered Product In Section 23.3, we defined the operation of time-ordered multiplication for two fields. It is easy to generalize formula (23.17) to the case of an arbitrary number of fields: T .u1 .x1 /    un .xn // D .1/pF ui1 .xi1 /    uin .xin /; xi01

>

xi02

>  >

(23.34)

xi0n :

Here, pF is the number of permutations of the Fermi-operators at the transition from the order 1; : : : ; n to i1 ; : : : ; in . Since the ordinary products of fields can be written in terms of normal products by Theorem 23.4, we can rewrite T -products in the form of a sum of normal products

282

Chapter 23

Green Functions

with all possible pairings. But, instead of the use of an ordinary pairing at fixed time variables, we can introduce the so-called chronological pairing. To this end, let us consider the time-ordered product of two operators: ´ u1 .x1 /u2 .x2 /; if x10 > x20 ; T .u1 .x1 /u2 .x2 // D (23.35) "u2 .x2 /u1 .x1 /; if x20 > x10 ; " D C1 for Bose-fields, and " D 1 for Fermi-fields. By Theorem 23.4, we have u1 .x1 /u2 .x2 / D W u1 .x1 /u2 .x2 / W C u1 .x1 /u2 .x2 /

(23.36)

and "u2 .x2 /u1 .x1 / D " W u2 .x2 /u1 .x1 / W C " u2 .x2 /u1 .x1 / D W u1 .x1 /u2 .x2 / W C " u2 .x2 /u1 .x1 / : We now define the time-ordered pairing ) by the formula ´ at x10 > x20 ; u1 .x1 /u2 .x2 / ; u1 .x1 /u2 .x2 / D " u2 .x2 /u1 .x1 / ; at x20 > x10 :

(23.37)

(23.38)

Then it follows from relations (23.35)–(23.38) that T .u1 .x1 /u2 .x2 // D W u1 .x1 /u2 .x2 / W C u1 .x1 /u2 .x2 / :

(23.39)

Definition (23.38) yields u1 .x1 /u2 .x2 / D " u2 .x2 /u1 .x1 / :

(23.40)

Since the average of a normal product of fields over vacuum is equal to zero, we obtain another definition of pairing: u1 .x/u2 .x/ D . 0 ; T .u1 .x1 /u2 .x2 // 0 /:

(23.41)

It is clear that Theorem 23.4 can be reformulated for T -products. Theorem 23.7. The T -product of operators of free fields (or their linear combinations) is equal to a sum of the normal products of these fields with all possible time-

283

Section 23.4 Wick Theorems

ordered pairings, including the term without pairing. In other words, T .u1 .x1 /    un .xn // D W u1 .x1 /    un .xn / W X .1/pF .j1 ;j2 / W uj1 .xj1 /uj2 .xj2 / C ¹j1 ;j2 º¹1;:::;nº

1

1

W u1 .x1 /    uj1 .xj1 /    uj2 .xj2 /    un .xn / W X Œ.1/pF .j1 ;j2 ;j3 ;j4 / uj1 .xj1 /uj2 .xj2 /uj3 .xj3 /uj4 .xj4 / C ¹j1 ;:::;j4 º¹1;:::;nº

C .1/pF .j1 ;j3 ;j2 ;j4 / uj1 .xj1 /uj3 .xj3 /uj2 .xj2 /uj4 .xj4 / C .1/pF .j1 ;j4 ;j2 ;j3 / uj1 .xj1 /uj4 .xj4 /uj2 .xj2 /uj3 .xj3 / 

1

1

 W u1 .x1 /    uj1 .xj1 /    uj4 .xj4 /    un .xn / W C    : For the T -products of operators which are normal products of some number of operators, a remark and a formula similar, respectively, to Remark 23.1 and formula (23.24) with time-ordered pairings instead of ordinary ones are valid. Concluding this subsection, we note that, by Definitions (23.38)–(23.41) and formulas (23.18)–(23.21), the time-ordered pairings of free fields coincide with the causal Green functions of these fields. Therefore, according to formulas (23.25)– (23.33), we have:  for a real scalar field,

1 '.x/ '.y/ D D c .x  y/; i

(23.42)

1   '.x/'.y/ D '.y/ '.x/ D D c .x  y/; i

(23.43)

 for a complex scalar field,





'.x/'.y/ D '.y/ '.y/ D 0;

(23.44)

 for spinor fields, N .y/ D 1 S c .x  y/; i ˛ˇ ˛ .x/ ˇ .y/ D N ˛ .x/ N ˇ .y/ D 0; ˛ .x/

(23.45) (23.46)

 for vector fields,  1 c V .x/V  .y/ D D .x  y/; i 



V .x/V .y/ D V  .x/V  .y/ D 0;

(23.47) (23.48)

284

Chapter 23

Green Functions

 for an electromagnetic field, A .x/A .y/ D ig  D0c .x  y/:

(23.49)

The time-ordered pairings of fields of different types are identically zero.

23.4.3 Generalized Wick Theorem Theorem 23.7 yields one more useful relation for the vacuum average of a T -product of free fields. Theorem 23.8. . 0 ; T .u.x/u1 .x1 / : : : un .xn // 0 / D

n X

.1/pF .j / u.x/uj .xj /

j D1

 . 0 ; T .u1 .x1 / : : : uj 1 .xj 1 /uj C1 .xj C1 / : : : un .xn // 0 /;

(23.50)

where pF .j / is the number of permutations of Fermi-operators at the transition from the order 1; : : : ; n to j; 1; : : : ; j  1; j C 1; : : : ; n. Theorem 23.8 follows from Theorem 23.7 and the fact that the vacuum averages of the normal products of the operators of free fields are zero (see Problem 23.4). Remark 23.9. The definition of a T -product implies that the operators can be permuted under the sign of a T -product with regard for their statistics. In other words, T .u1 .x1 /    un .xn // D .1/pF T .ui1 .xi1 /    uin .xin //; where pF is the number of permutations of Fermi-operators at the transition from the order 1; : : : ; n to i1 ; : : :, in . Remark 23.10. It is easy to see from (23.35) and (23.38) that the operation of T product is not single-valued in the case where the time variables of operators coincide with one another. On the other hand, using the Wick theorem, we see that the T -products of operators can be given in terms of normal products accompanied by ordinary products of singular causal functions (see formula (23.16)). It is seen from (23.16) that the singularities are concentrated at the points, where the arguments coincide with one another.

23.5

Operation of Multiplication and the Regularization of Distributions

In the following subsequent sections, we consider the theory of interacting fields. The most constructive method of the theory is perturbation theory. From the mathematical

285

Section 23.6 N -Point Green Functions of Free Fields

viewpoint, the application of perturbation theory to quantized fields is not substantiated up to now even in the case where the interaction constant is extremely small. For example, there is no rigorous proof of the convergence of series of perturbation theory after the elimination of ultraviolet divergences in quantum electrodynamics, where the results of calculations of some physical effects (Lamb shift, etc.) fit magnificently the experimental data. But even if we do not take the convergence of the whole series into account, we are faced with mathematical problems related to the operation of multiplication of singular functions of the form D c .x  y/, S c .x  y/; etc. In the theory of distributions, the operation of multiplication of distributions with identical arguments is not defined (see, e.g., [207], Section 5.9). In fact, axiom AQ9 ensures the correct definition of expressions which represent basic physical quantities and include the products of operator-valued distributions at a single point. In other words, the Wick product is a version of the regularization of such products. But the subsequent application of perturbation theory leads to the appearance of new products of singular distributions with identical arguments, which requires the new methods of regularization difficult to be substantiated from the physical viewpoint. Some general aspects of the regularization of the Green functions and their products are considered in [26, Chapters 15 and 16].

23.6

N -Point Green Functions of Free Fields

Formulas (23.18)–(23.21) define 2-point Green functions as the vacuum averages of a T -product of corresponding fields. In what follows, we call the quantities 1i D c .xy/, 1 c 1 c  c i S˛ˇ .x  y/, i D .x  y/; and ig Dg .x  y/ as 2-point Green functions of free fields for convenience. In the theory of interacting fields, the Green functions play the important role in the study of various physical phenomena. Let us consider their generalization to the case of a great number of variables. By definition, we have:  for a real scalar field, 0 .x1 ; : : : ; xN / WD . 0 ; T .'.x1 / : : : '.xN // 0 /I GN

(23.51)

 for a complex scalar field, 0 GN .x1 ; : : : ; xN I y1 ; : : : ; yN / 



D . 0 ; T .'.x1 / : : : '.xN /'.y1 / : : : '.yN // 0 /I

(23.52)

 for spinor fields, 0 GN .˛1 ; x1 I : : : I ˛n ; xn j ˇ1 ; y1 I : : : I ˇn ; yn / D . 0 ; T . ˛1 .x1 / : : : ˛n .xn / N ˇ1 .y1 / : : : N ˇn .yn // 0 /I

(23.53)

286

Chapter 23

Green Functions

 for vector fields, Gn0 .1 ; x1 I : : : I n ; xn / D . 0 ; T .V1 .x1 /    Vn .xn // 0 /:

(23.54)

Analogous relations can be written for a complex vector field and an electromagnetic field. The Wick Theorem 23.7 allows one to calculate the Green functions of free fields. Using this theorem and taking into account that the vacuum average of the normal products of fields is zero, we obtain, for example, the N -point Green function of a free scalar field (for N D 2n) in the form 0 .x1 ; : : : ; xN / D GN

0 X .i1 ;:::;iN /

1 c 1 D .xi1  xi2 /    D c .xiN 1  xiN /; i i

(23.55)

where the sum over the indices (i1 ; : : : ; iN ) is taken over all possible different collections of pairs .xi1 ; xi2 /    .xiN 1 ; xiN / from the collection x1 : : : xN . Relation (23.55) also follows from the recurrence relation 0 .x1 ; : : : ; xN / GN

N X 1 c 0 D .x1  xk /GN D 2 .x2 ; : : : ; xk1 ; xkC1 ; : : : ; xN /; i kD2

which can readily be obtained from relation (23.51) by applying the generalized Wick theorem (Theorem 23.8) to the operator '.x1 /. Similar relations can easily be deduced for the Green functions of the other fields by using Definitions (23.52)–(23.54), Theorem 23.8, and Equations (23.42)–(23.49).

Chapter 24

Problems to Part IV

Problem 18.1.

Show that the operators of the free scalar field satisfy the axioms of quantization AQ3–AQ7.

Problem 19.1.

Show that the operators of the free electron-positron field satisfy the axioms of quantization AQ3–AQ7.

Problem 20.1.

Starting from Lagrangian (20.15), deduce the equal-time commutation relations between the generalized coordinate (field) V .x/ and the generalized momentum   .x/, i.e., prove the validity of axiom AQ7.

Problem 20.2.

Deduce the equal-time commutation relations by using the Dirac method of quantization for the systems with additional constraints.

Problem 20.3.

Prove that the Heisenberg equation for the electromagnetic field is consistent with the Maxwell equations.

Problem 20.4.

Determine the matrix of connections for the electromagnetic field with Coulomb gauge.

Problem 21.1.

Establish the law of transformation of the charge-conjugate spinor c under the transformation of space inversion [relation (21.16)].

Problem 21.2.

[relations (21.21)]. Prove the antilinearity of the operator Ub T

Problem 21.3.

bC b -transformed free neutrino field satisfies the addiProve that the P tional condition (6.148).

Problem 21.4.

Is the Lagrangian of the free neutrino field (6.153) invariant under b, C b , and P bC b -transformations? P

Problem 21.5.

Check relations (21.30)–(21.32).

Problem 22.1.

Prove that the operator of energy of free real scalar bosons is a symmetric nonnegative operator.

Problem 22.2.

Prove formula (22.35).

288

Chapter 24

Problems to Part IV

Problem 22.3.

Starting from Definitions (20.32), (20.35), and (22.68), check Equations (22.71).

Problem 23.1.

Deduce the integral representations for the retarded and advanced Green functions. Write the expressions for these representations with the help of the Pauli–Jordan function.

Problem 23.2.

Take the integral with respect to k 0 in relation (23.9) and represent the function D c .x  y/ in the form (23.11).

Problem 23.3.

Write the causal Green functions of the spinor, vector, and electromagnetic fields in the form of Fourier transforms.

Problem 23.4.

Prove relation (23.50).

Part V

Quantum Theory of Interacting Fields. General Problems

Chapter 25

Construction of Quantum Interacting Fields and Problems of This Construction In the previous part, we have considered operators of free quantized fields that satisfy all conditions (axioms) of quantum relativistic theory. Here we describe the construction of the theory of interacting fields. Note that the construction of these fields depends on the form of interaction. For example, the form of the operator of an electron–positron field that takes part in the electromagnetic interaction differs from the operator of the same field for electrons taking part in processes of weak interaction. Therefore, for the construction of a unique quantum field corresponding to a specific elementary particle, a theory that combines all known interactions between elementary particles is required. The construction of this theory is a matter of the future; to a large extent, it depends on experimental and theoretical investigations in all directions of mathematics and physics. In the present part, we consider the main principles of this construction and the problems arising in the course of its realization. The main problems are general and do not depend on the form of interaction. The specific form of an interaction affects only the level of complexity of these problems. For simplicity, we take the self-interaction of a real scalar field. The more realistic models introduced in Part III are considered in Part VII.

25.1

Formal Construction of a Quantum Field

Consider the model of a self-interacting real scalar field with Lagrangian L .x/ D L0 .x/ C LI .x/;

(25.1)

where L0 .x/ is the Lagrangian of the free field: L0 .x/ D

1 W @ '.x/@ '.x/ W m2 W ' 2 .x/ W : 2

(25.2)

The quantity LI .x/ describes a self-interaction of the form LI .x/ D  W ' 4 .x/ W

(25.3)

and the constant  > 0 is called a coupling constant. The Lagrange–Euler equation corresponding to Lagrangian (25.1)–(25.3) has the form (25.4) .  m2 /'.x/ D 4 W ' 3 .x/ W :

292

Chapter 25 Construction of Quantum Interacting Fields

In formulating Axiom AQ9 (see Remark 17.1), we substantiated the introduction of a normal product as the process of regularization of the operation of multiplication of operator-valued generalized functions with the same arguments. In (25.3) and (25.4), this means, respectively, the following: W ' 4 .x/ W D ' 4 .x/  6' 2 .x/.ˆ0 ; ' 2 .x/ˆ0 /; W ' 3 .x/ W D ' 3 .x/  3'.x/.ˆ0 ; ' 2 .x/ˆ0 /:

(25.5)

The right-hand sides of (25.3) and (25.4) should be understood in the sense of a limit transition of the form (17.35), i.e., it is necessary to define operations (25.5) for the normal products W '.x1 /'.x2 /'.x3 /'.x4 / W and W '.x1 /'.x2 /'.x3 / W using the combinatorial Wick theorem and substituting the vacuum average .ˆ0 ; '.xi /'.xj /ˆ0 / for the pairing of two fields and then pass to the limit in the obtained expression as xi ! x; i D 1; 2; 3; 4: Also note that the infinite constant 1 .ˆ0 ; ' 2 .x/ˆ0 / D . 0 ; '02 .x/ 0 / D D  .0/; i

(25.6)

where '0 .x/ is the operator of the free field and 0 is the free vacuum. Identity (25.6) follows from the spectral representation of the 2-point Wightman function .ˆ0 ; '.x1 /'.x2 /ˆ0 / (see [181], Chapter 17, Section 2). In Section 17.4.1, we have established that the Heisenberg equation for a free scalar field coincides with the Klein–Fock–Gordon equation. It is obvious that an analogous statement must be true for an interacting field, i.e., Equation (25.4) is equivalent to the Heisenberg equation @'.x/ D Œ'.x/; H ; (25.7) i @x 0 where H D P 0 D H0 C HI ; Z 1 ŒW @ '0 .x/@ '0 .x/ W Cm2 W '02 .x/ Wd x H0 D 2

(25.8) (25.9)

x 0 D0

is the free Hamiltonian, and Z W '0 .0; x/4 W d x

HI D

(25.10)

x 0 D0

is the interaction Hamiltonian. Indeed, assuming that '.0; x/ D '0 .0; x/ at the initial time x 0 D 0; we rewrite a solution of Equation (25.7) in the form '.x/ D e ix

0H



'0 .0; x/e ix

0H



(25.11)

293

Section 25.1 Formal Construction of a Quantum Field

and show that this solution satisfies Equation (25.4). Using relation (25.11), we get @k @k '.x/ D e ix and

0H



@k @k '0 .0; x/e ix

0H



(25.12)

  @0 @0 '.x/ D i @0 e ix0 H ŒH ; '0 .0; x/ e x0 H D i e ix0 H ŒH ; @0 '.x/;x 0 D0 e ix0 H ;

(25.13)

where we have used the relation ŒH ; '0 .0; x/ D i @0 '.x/jx 0 D0 ;

(25.14)

which is a corollary of the equal-time commutation relations (17.20) for a scalar field. By analogy, we establish that ŒH ; @0 '.x/;x 0 D0 D i @k @k '0 .0; x/ C i m2 '0 .0; x/ C i 4 W '03 .0; x/ W : (25.15) It is clear that (25.13), together with (25.15) and (25.11), is identical to (25.4). Taking, in addition, into account the free evolution of a free scalar field, i.e., Equation (17.40), we obtain '0 .x/ D e ix

0H

0

'0 .0; x/e ix

0H

0

;

(25.16)

and rewrite (25.11) in the form '.x/ D e ix

0H



e ix

0H

0

'0 .x/e ix

0H

0

e ix

0H



D U 1 .x 0 ; 0/'0 .x/U.x 0 ; 0/; (25.17)

where the evolution operator U.x 0 ; 0/ WD e ix

0H

0

e ix

0H



(25.18)

U.0; 0/ D 1;

(25.19)

is formally a unitary operator, i

@ U.x 0 ; 0/ D HI .x 0 /U.x 0 ; 0/; @x 0

and HI .x 0 / WD e ix

0H

0

HI e ix

0H

0

:

(25.20)

Solution (25.17), (25.18) is formal. In order that it have a rigorous mathematical meaning, it is necessary to define this operator in a certain Hilbert space of states, prove the self-adjointness of the operators H0 and H ; show that the spectrum of H satisfies the conditions of Axiom AQ5, and construct the ground state of the Hamiltonian H ; i.e., the physical vacuum state ˆ0 : H ˆ0 D 0;

ˆ0 ¤ 0:

This is an extremely difficult mathematical problem. There exist only few models almost equivalent to the model without interaction for which this problem can be solved. In the solution of this problem, the following three main difficulties are encountered:

294

Chapter 25 Construction of Quantum Interacting Fields

(i) volume divergences (Haag theorem); (ii) ultraviolet catastrophe; (iii) instability of vacuum. These problems are briefly discussed in what follows. Note that, at the formal level, the quantization of an interacting scalar field is reduced, within the framework of self-interaction (25.3), to the quantization of the free field and to the construction of a Heisenberg field according to relation (25.11).

25.2

Mathematical Problems of Construction of a Quantum Interacting Field

In Subsection 22.2.3, we have investigated in detail the Hamiltonian of the free scalar field H0 that acts in the Fock space F D FB and is a positive self-adjoint operator in this space. For the construction of a quantum field, it is necessary to determine the total Hamiltonian H : Thus, it remains to define an interaction Hamiltonian of the form (25.10) in the space FB : To this end, we use relation (18.2) and the identity 4

W '0 .0; x/ W D

4 X

C4 ' C .0; x/j '  .0; x/4j ; j

j D0 j

where C4 D 4Š=j Š.4  j /Š: Further, taking (5.58) into account, we easily obtain (Problem 25.2) 4 X j HI D C4 Hj;4j ; (25.21) j D0

where

Z Hj;4j D

d k1    d k 4 R4s

.2/s ı.k1 C    C kj  kj C1      k4 / q Q4 s=2 4 k2 C m2 .2/ iD1 i

' C .k1 /    ' C .kj /'  .kj C1 /    '  .k4 /:

(25.22)

For reasons that will become obvious below, we rewrite the operator Hj;4j using the variables k 2 Rs ; s D 1; 2; 3: As an example, we consider the case j D 4: Then, for fO 2 D loc [see (22.27)], using relations (22.12) and (22.120 ) we obtain 0

1 .H4;0 fO/N .p1 ; : : : ; pN / D p N.N  1/.N  2/.N  3/ X .2/s ı.pj1 C    C pj4 / fN 4 .p1 ; : : : ; pOj1 ; : : : ; pOj4 ; : : : ; pN /:  q Q4 s=2 4 p2 C m2 .2/ jl ¤jk ; k¤l iD1 i (25.23)

Section 25.2 Mathematical Problems of Construction of a Quantum Interacting Field

295

Here, pOj1 means that the function fN 4 does not have the argument pj1 : We see that the right-hand side of (25.23) is not a component of a vector of the Fock space because Z (25.24) d pjf .p/j2 ı.p/2 D ı.0/s jf .0/j2 D C1: Thus, this is the first fundamental mathematical difficulty, which can be called the volume divergence. It follows from relations (25.23) and (25.24) that the interaction Hamiltonian cannot be consistently defined in the same space as the free Hamiltonian H0 : However, this difficulty is more serious and is connected with the Haag theorem. According to this theorem, it is impossible to construct an interaction representation within the framework of the axioms formulated in Chapter 17. We formulate this theorem as follows: Haag theorem (see, e.g., [191]). Let '0 .x; x 0 / be a free field of mass m and let '.x; x 0 / be a field that satisfies the conditions of Axioms AQ1–AQ9. Namely, for f 2 S .Rs / (a test function of the Schwartz space), there exist two irreducible collections of operators at time t D x 0 ; namely Z 0 '0 .f; x / D f .x/'0 .x; x 0 /d x and '.f; x 0 /; that act in the Hilbert spaces H1 and H2 ; respectively (it is possible that H1  H2 ). In each of these two spaces, the corresponding unitary representation of the Lorentz group Uj acts. Suppose that, for each representation Uj ; there exists a unique .j / invariant state ˆ0 ; i.e., .j / .j / Uj ˆ0 D ˆ0 : Let there exist a unitary operator V D V .x 0 / such that '.f; x 0 / D V 1 '0 .f; x 0 /V: Then '.x; x 0 / is a free field of mass m: The main conclusion of this theorem is the impossibility of the construction of an interacting field on the basis of relation (25.17). In this connection, Haag and Kastler [85] proposed to change the system of axioms and restrict the consideration to observables that belong only to algebras of local observables U.O/; where O is a certain open domain of the Minkowski space Md D MsC1 : In a simplified form, this idea lies in the consideration of local dynamics instead of the global dynamics (25.17) under the assumption that the interaction holds only in a certain relatively small space– time “box” T  ƒ M; T R1 ; ƒ Rs ; s D 1; 2; 3:

296

Chapter 25 Construction of Quantum Interacting Fields

Then, instead of the Hamiltonian H defined by (25.8)–(25.10), we can introduce the new Hamiltonian H .g/ D H0 C HI .g/; Z HI .g/ D g.x/ W '0 .0; x/4 W;

(25.25) (25.26)

x 0 D0

where g.x/ is called the function of interaction intensity, and, furthermore, supp g.x/ ƒ;

(25.27)

i.e., g.x/ D 0 for x 62 ƒ and g.x/ D 1 for x 2 ƒn@ı ƒ; where @ı ƒ is a certain domain of thickness ı that lies inside ƒ and encloses the entire surface @ƒ of the domain ƒ: This domain (and the function g.x/) is defined differently depending on the specific problem. We assume that 0 < g.x/ < 1 in the domain @ı ƒ and that g.x/ is an infinitely differentiable function of its arguments x .1/ ; : : : ; x .s/ : We will also sometimes use the space–time “switching” of interaction. In this case, instead of the function g.x/; we consider a switching function g.x/ of the form g.x/ D g 0 .x 0 /g.x/

(25.28)

with supp g 0 .x 0 / T and the same properties on @ı T: Hamiltonian (25.26) also admits representation (25.21), (25.22) with Z gQ .k1 C kj  kj C1      k4 / d k1    d k 4 Q Hj;4j .g/ D q 4 s=2 4 k2 C m2 .2/ 4s iD1 i R  ' C .k1 /    ' C .kj /'  .kj C1 /    '  .k4 /; where gQ is the Fourier transform of the function g in the space Rs : Z 1 g.x/ D e ixk gQ .k/ d k: .2/s

(25.29)

(25.30)

Rs

Using the definition of norm (22.6) in the N -particle space FN ; one can easily verify that k.H4;0 fO/N k2FN 6 const K.g/N.N  1/.N  2/.N  3/ kfON 4 kFN 4 ; (25.31) where

Z K.g/ D

d k1    d k 4 q R4s

jQg.k1 C    C k4 /j2 : q k21 C m2    k24 C m2

(25.32)

Section 25.2 Mathematical Problems of Construction of a Quantum Interacting Field

Using the inequality

n Y

aj 6

j D1

n Y X

297

n

ajn1

j D1 j ¤i

with aj D .kj2 C m2 /1=2 and n D 4; we get 0 K.g/ 6 4 @

Z

Rs

13

dk A 2 .k C m2 /2=3

Z jQg.k/j2 d k:

The second integral is finite by virtue of the chosen properties of the function g.x/; whereas the first integral is finite only in the case where k 2 R1 (i.e., s D 1). Thus, in the real physical .3C1/-dimensional space–time, we again obtain an infinite norm for the vector H4;0 fO due to the divergence of the integrals with respect to the momentum variables for large values of k: These divergences are called ultraviolet divergences. The renormalization theory, which is presented in what follows, eliminates these divergences. In conclusion, we only note that, at the level of the construction of a Hamiltonian, this means that, instead of the operator HI .g/; one should consider a regularized operator HI~ .g/: This operator has the same form (25.21), (25.29), but the integration with respect to the momenta kj ; j D 1; : : : ; 4; is carried out not over the .i/ entire space Rs but under the restrictions jkj j 6 ~; j D 1; : : : ; 4; i D 1; 2; 3: Further, it is necessary to change the form of interaction by adding a certain operator counterterm ıH ~ .g; m/ specially chosen for this model to the total Hamiltonian. Thus, it is necessary that the total Hamiltonian H .gI ~/ D H0 C HI~ .g/ C ıH ~ .g; ; m/ converge in the sense of operator convergence as ~ ! 1 to a certain self-adjoint positive operator in the space of states. This is a fairly complicated mathematical problem, which, at present, is solved only for few simple models in two- or three-dimensional space–time (see [74]). In this connection, the problem of the existence of the ground state (physical vacuum) necessary for the construction of perturbation theory, which plays the key role in theoretical calculations, also remains unsolved.

Chapter 26

Scattering Theory. Scattering Matrix

The main sources of information for the study of physical properties of interaction of elementary particles and investigation of their structure are experimental and theoretical data on scattering processes. Scattering theory is the main tool for the analysis of experimental data. For this reason, the construction of scattering theory is the most important branch of quantum field theory. Unfortunately, the construction of relativistic invariant scattering theory in the most general form is the matter of future. In this connection, one distinguishes relativistic and nonrelativistic scattering theories. Actually, the latter is the nonrelativistic approximation of the former; in fact, it studies reactions without creation and annihilation of particles, i.e., particles do not change before and after an interaction (the so-called single-channel scattering theory). This theory is the most developed one from the mathematical point of view. Note that multichannel processes (in this case, scattering can lead to the creation of completely different particles or to the same particles with new particles created, etc.) can also be studied within the framework of the nonrelativistic approach. However, this theory is much more complicated and encounters considerable mathematical difficulties. In this chapter, we describe problems related to the construction of relativistic scattering theory. Note that the main principles of the construction of classical scattering theory, quantum mechanical (or nonrelativistic) theory, or quantum field (or relativistic) theory are the same (for details, see [118, 194, 197]).

26.1

Quantum Description of Scattering. Definition of Scattering Operator

Unlike the classical theory based on the notion of coordinates and momenta of particles, a system in the quantum theory is defined by its state ‰: According to the Schrödinger representation, this state depends on time, and its evolution is described by the Schrödinger equation d (26.1) i ‰.t / D H ‰.t /: dt A solution of this equation (under the condition that the Hamiltonian H is a welldefined self-adjoint operator in the Hilbert space of states H ) is determined by the unitary Schrödinger evolution operator US .t / D U.t /: ‰.t / D U.t /‰.0/ D e iH t ‰.0/;

(26.2)

where ‰.0/ is a certain normalized vector from H corresponding to the initial time.

Section 26.1 Quantum Description of Scattering. Definition of Scattering Operator

299

If we use this approach for the description of a scattering process, i.e., assume that the vector ‰.t / D U.t /‰.0/ describes the evolution of a certain scattering process, then it is convenient to suppose that the scattering itself occurs at the time t D 0: In this case, it is no longer suitable to call ‰.0/ the initial state. For this reason, we call it the state vector at the scattering time and denote it by ‰ D ‰.0/:

(26.3)

Assume that, in the distant past .t ! 1/; particles moved as free ones, i.e., their evolution is described by the vector (asymptotic state) ‰in .t / D U0 .t /‰in D e iH0 t ‰in :

(26.4)

Similarly, we assume that, in the distant future .t ! C1/; the scattered particles will also move as free ones: ‰out .t / D U0 .t /‰out D e iH0 t ‰out :

(26.5)

It is worth noting that, in the general case, the free Hamiltonian of incoming particles H0in and the free Hamiltonian of scattered particles H0out are not identical: H0in ¤ H0out :

(26.6)

In this case, to preserve relations (26.4) and (26.5), it is necessary to set H0 D H0in C H0out :

(26.7)

If, among the incoming and scattered particles, there are particles of the same type, then the corresponding Hamiltonian in the sum in (26.7) is taken only once. For relations (26.1)–(26.7) to correspond to the same scattering process, it is necessary that the following asymptotic conditions be satisfied: lim .‰.t /  ‰in .t // D 0;

t!1

lim .‰.t /  ‰out .t // D 0:

(26.8)

t!C1

We do not discuss here the problem of the meaning of limits (26.8) because we consider only formal constructions. The asymptotic conditions (26.8) guarantee the existence of the vector state ‰.t / for any vectors ˆin and ˆout from the Hilbert space of states H : This means that a pair of so-called wave operators ˙ (called Møller wave operators) can be defined by using Eqs (26.2)–(26.5) and (26.8) according to the relations 

‰ D lim t!1 U .t /U0 .t /‰in WD C ‰in ; 

‰ D lim t!C1 U .t /U0 .t /‰out WD  ‰out ;

(26.9)

300

Chapter 26

or, in the operator form,

Scattering Theory. Scattering Matrix



˙ WD lim U .t /U0 .t /: t!1

(26.10)

Of interest is the following inverse problem: Does every vector ‰ 2 H define a state ‰.t / D U.t /‰ that has i n- and out -asymptotics? From the viewpoint of physics, the answer is negative because the Hamiltonian H D H0 C Hint can have bound states. In this case, if ‰ corresponds to a bound state, then U.t /‰ is a stationary state in which a particle remains localized in a certain bounded domain. This implies that the Hilbert space of states has at least three components: Hin ; Hout ; and Hb : It turns out that (see, e.g., [197], Chapter 2, Section 4), for asymptotically complete theories, one has Hin D Hout ;

(26.11)

Hin ?Hb ;

(26.12)

Hout ?Hb ;

and H D Hin ˚ Hb D Hout ˚ Hb :

(26.13)

To determine the scattering operator we note that the wave operators ˙ are isometric operators in H : Indeed, since the operators U.t / and U0 .t / are unitary, we have kU.t / U0 .t /‰in kH D k‰in kH :

(26.14)

It is obvious that this relation remains true in the limit (if it exists), i.e., k ˙ ‰in kH D k‰in kH :

(26.15)

Equality (26.15) guarantees the isometricity of the operators ˙ and the relation

˙ ˙ D 1:

(26.16a)

However, in the general case, one has ˙ ˙ ¤ 1; i.e., the operators ˙ can be nonunitary. It turns out that the nonunitarity of wave operators is connected with the existence of a nonempty subspace of bound states Hb : The dimension of this subspace is the measure of deviation of the isometric operators ˙ from their unitarity. It can be shown (see, e.g., [181], Chapter 11, Section 4) that

˙ ˙ D 1  Pb ;

(26.16b)

where Pb is the operator of projection to Hb : We now define a scattering operator as a mapping of the vectors ‰in into the vectors ‰out : (26.17) ‰out D S‰in :

301

Section 26.1 Quantum Description of Scattering. Definition of Scattering Operator

For the first time, the scattering matrix was introduced by Wheeler [216] in 1937 as the mean value of the operator S between two physical states. Later, in 1943, Heisenberg [88] defined the S -matrix in the context of problems of scattering theory of symmetric particles and used it as a basis of contemporary quantum scattering theory. Using relations (26.9) and (26.16a), we easily obtain ‰out D  ‰ D  C ‰in :

(26.18)

Thus, the S-operator can be expressed in terms of wave operators by the relation S D  C :

(26.19)

For the construction of the operators S and ˙ by methods of perturbation theory, it is customary to use the interaction picture, which has been described in detail in Section 17.2. For this purpose, we define a shift operator with respect to the time variable by the formula U.t2 ; t1 /ˆ.t1 / D ˆ.t2 /;

ˆ.ti / 2 H ;

i D 1; 2:

(26.20)

Using (26.20) and the fact that ˆ.t / is a normalized physical state in the space of states, we obtain the following important properties of the operator U.t2 ; t1 / (Problem 26.1): U.t; t / D 1;

(26.21)

U.t2 ; t0 /U.t0 ; t1 / D U.t2 ; t1 /; and



U.t2 ; t1 / D U 1 .t1 ; t2 / D U .t1 ; t2 /:

(26.22)

(26.23)

Recall (see Section 17.2) that the state vector ˆ.t / in the interaction picture is related to the state vector ‰.t / in the Schrödinger picture as follows: ˆ.t / D e iH0 t ‰.t /:

(26.24)

Rewriting a solution of Equation (26.1) in the form (26.2) but with the initial condition

we obtain

‰.t /j tDt1 D ‰.t1 /;

(26.25)

‰.t2 / D e iH.t2 t1 / ‰.t1 /:

(26.26)

Taking (26.24) and (26.26) into account, we get ˆ.t2 / D e iH0 t2 ‰.t2 / D e iH0 t2 e iH.t2 t1 / ‰.t1 / D e iH0 t2 e iH.t2 t1 / e iH0 t1 ˆ.t1 /:

(26.27)

302

Chapter 26

Scattering Theory. Scattering Matrix

Comparing (26.20) and (26.27), we obtain the following explicit relation for the evolution operator U.t2 ; t1 / (see also (25.18)): U.t2 ; t1 / D e iH0 t2 e iH.t2 t1 / e iH0 t1 :

(26.28)

It is easy to verify that the operator U.t; t0 / satisfies the differential equation i @ t U.t; t0 / D HI .t /U.t; t0 /;

(26.29)

where HI .t / is defined by (25.20). The problem of the determination of the evolution operator U.t; t0 / from the differential equation (26.29) can be considered as the operator Cauchy problem with the initial condition (26.30) U.t0 ; t0 / D 1: We consider this problem in the next section. It follows from the definition of vectors in the interaction representation (26.24) and relation (26.4) that the vectors of i n- or out -states .t / D e iH0 t ‰ in .t / D ‰ in ˆ in out out out and time-independent. Therefore, we can assume that D ˆ in .1/: ‰ in out out Comparing (26.27) and (26.17), we can define the scattering operator as follows: SD

lim U.t2 ; t1 /:

t1 !1 t2 !C1

(26.31)

Using the definitions of the wave operators ˙ (see (26.9)) and the evolution operator U.t2 ; t1 / (see (26.28)), one can easily verify that

˙ D lim U.0; t /: t!1

(26.32)

In conclusion of this section, note that all constructions presented here are formal, and a rigorous substantiation is required for each specific form of the Hamiltonians H0 and Hin : Unfortunately, at present, this theory can be constructed only for a narrow class of interactions.

26.2

Formal Construction of the Scattering Operator by the Method of Perturbation Theory

Perturbation theory is one of the main theoretical methods for the calculation of specific physical processes. Of course, the applicability range of perturbation theory is

303

Section 26.2 Method of Perturbation Theory

bounded and depends on a small parameter, the proper choice of the ground state for which a perturbation is considered, and other factors. In this chapter, we consider the construction of a formal perturbation theory for the scattering operator without analyzing problems of convergence of the perturbation series and the mathematical correctness of the terms of this series. Despite the mathematical difficulties caused by the application of the interaction representation (Dirac picture), the construction of perturbation theory is most convenient in exactly this representation because the scattering operator is determined with the use of the evolution operator U.t; t0 / that satisfies the differential equation (26.29) and initial condition (26.30). These equations are equivalent to the following integral equation for U.t; t0 /: Zt U.t; t0 / D 1  i  HI . /U.; t0 /d : (26.33) t0

Applying the method of successive approximations to Equation (26.33), one can represent the evolution operator in the form of the infinite series U.t; t0 / D

1 X

Un .t; t0 /;

(26.34)

nD0

where Zt n

Un .t; t0 / D .i /

Zt1 dt2   

dt1 t0

tZn1

t0

dtn HI .t1 /    HI .tn /:

(26.35)

t0

In [42, 43], Dyson noted that, rewriting all integrals in relation (26.35) as integrals along the segment Œt0 ; t ; inserting the product of the Heaviside functions .ti  tj / into the integrand (see the definition in Section 5.3), and using the definition of T product (23.34), one can transform expression (26.35) as follows: Zt n

Un .t; t0 / D .i /

Zt dt2   

dt1 t0

Zt

t0

dtn .t1  t2 /.t2  t3 /    .tn1  tn / t0

 HI .t1 /HI .t2 /    HI .tn / Zt Zt .i /n X D dt1    dtn .t.1/  t.2/ / nŠ 2Pn t0

t0

   .t.n1/  t.n/ /HI .t.1/ /    HI .t.n/ / Zt Zt .i /n D dt1    dtn T .HI .t1 /    HI .tn //; nŠ t0

tn

where Pn is the permutation group of the numbers 1; 2; : : : ; n:

304

Chapter 26

Scattering Theory. Scattering Matrix

Using definition (26.31), we now represent the perturbation series of the scattering operator in the final form. To rewrite this series in the covariant form, we take into account that, for the theories whose interaction Lagrangian does not contain derivatives, according to (4.60) and (4.61) one has Z Z 0 00 HI .t / D HI .x / D d x TI .x/ D  d x LI .x/: We finally get SD

1 X

Sn D 1 C

nD0

Z 1 X 1 Sn .x1 ; : : : ; xn /dx1    dxn ; (26.36) nŠ

nD1

M4n

n

Sn .x1 ; : : : ; xn / D .i / T .LI .x1 /    LI .xn //:

(26.37)

Expressions (26.36) and (26.37) are sometimes represented in the form of the socalled T -exponent. Formally taking the sign of the T -product outside the integral sign, we rewrite the expression for Sn in the form  Z n 1 (26.38) Sn D T i  LI .x/dx ; nŠ and the expression for the S-operator as follows: S D T .e i

R

LI .x/dx

/:

(26.39)

Despite the fact that expression (26.39) is formal, its combinatorial properties are the same as for an ordinary exponent. Therefore, it can be used for the proof of the main properties of the S-operator. Remark 26.1. It follows from relation (4.60) that the density of the interaction Hamiltonian HI .x/ D TI00 .x/ does not coincide with the density of the interaction Lagrangian LI .x/ for the theories that have derivatives in the interaction Lagrangian. In this case, series (26.34) does not coincide with series (26.36) for t0 D 1 and t D C1: However, it turns out (see, e.g., [181], Chapter 14, Section 3) that expansion (26.36) leads to correct physical conclusions. Also note that relation (26.36) for the S-operator can also be obtained from the most general (axiomatic) principles of relativistic invariance, unitarity, and causality. This scheme of construction was proposed by Stueckelberg in [192, 193] and completed by Bogoliubov in [21]. This scheme is described in detail in [26], Chapter 17. Remark 26.2. Due to the indeterminate form of the T -product for xi0 D xj0 ;

i ¤ j;

305

Section 26.3 Main Properties of the S -Operator

the expression for Sn is not uniquely defined. The quantity Sn is defined up to a certain anti-Hermitian quasilocal (i.e., vanishing for xi ¤ xj ) operator ƒn .x1 ; : : : ; xn /: In what follows, this fact is used in the theory of elimination of divergences in the expression for Sn . In conclusion, we write the expression for the S -operator in the case of interaction with intensity g.x/ (see relation (25.28)): Z 1 X 1 Sn .x1 ; : : : ; xn /g.x1 /    g.xn /dx1    dxn : (26.40) S.g/ D 1 C nŠ nD1

The expression for S.g/ coincides with (26.36), (26.37) if the Lagrangian LI .x/ is replaced by the Lagrangian LI .xI g/ D LI .x/g.x/; where the function g.x/ is defined by (25.28). By analogy with relation (26.39), the operator S.g/ can also be represented in the form of the T -exponent as follows: S.g/ D T .e i

26.3

R

LI .xIg/dx

/:

(26.41)

Main Properties of the S -Operator

In this section, we briefly discuss some main properties of the scattering operator, namely, relativistic invariance, unitarity, and microcausality. First, we obtain one more important representation of the S -operator in the form of decomposition in normal products of operators of free fields.

26.3.1 Normal Form of the Operator S Definition (26.31) of the operator S and its representation in terms of the wave operators (26.19) and (26.32) is rather general. For the detailed analysis of scattering processes, one needs a more specific analytic expression for the S -operator. We give this expression as an additional axiom, which is substantiated in what follows in the process of investigation of the theory of the S -operator. This expression is based on the Berezin theorem ( [14], Chapter 2, Section 1.10) on the representation of a bounded operator acting in the Fock space F in the form of a decomposition in creation and annihilation operators of free fields. A relativistic-invariant analog of this .k/ decomposition in the free fields uj .x/; where x 2 M4 ; k D 1; 2; : : : ; t; t is the number of types of fields, and j ranges over the values of components of a field of the type k; has the form X 1 X Z SD NŠ ND0

j

M4jNj

FN .xjj/N W

N Y nD1

ujn .xn / W .dx/jNj ;

(26.42)

306

Chapter 26

Scattering Theory. Scattering Matrix

where N WD .N1 ; : : : ; N t /; NŠ WD N1 Š    N t Š; jNj WD N1 C    C N t ;   .1/ .1/ .1/ .1/ .t/ .t/ .t/ .t/ .x j j/N WD x1 ; : : : ; xN1 j j1 ; : : : ; jN1 I : : : I x1 ; : : : ; xN t j j1 ; : : : ; jN t ; N Y

.1/

ujn .xn / WD u

.1/

j1

nD1

.1/

.1/

.x1 /    u

.1/ 1

jN

.1/

.t/

.xN1 /    u

.t /

j1

.t/

.t/

.x1 /    u

.t / t

jN

.t/

.xN t /;

.dx/N WD dx1    dxN1    dx1    dxN t : .1/

.1/

.t/

.t/

The coefficient functions FN are either symmetric functions with respect to the arguments of the same field type (for bosons) or antisymmetric functions (for fermions). Formally, one can express the operator S in the form (26.42) by using its representation (26.36), (26.37). To show this we restrict ourselves to a specific interaction. For simplicity, we again consider the case of self-interaction of the scalar field (25.3). In this case, representation (26.42) takes the form SD

Z 1 X 1 NŠ

N D0

FN .x1 ; : : : ; xN / W '.x1 /    ; '.xN / W dx    dxN :

(26.43)

M4N

The first step in the derivation of (26.43) is the reduction of each term of the perturbation series (26.36), (26.37) to the normal form (i.e., the representation in the form of normal (Wick) products). We begin with the first order .n D 1/: Z Z T .W ' 4 .x1 / W/dx1 D i  W ' 4 .x1 / W dx1 : (26.44) S1 D i  M4

M4

To compare (26.44) with the corresponding contribution to expression (26.43), we rewrite S1 in the form Z Z Œ.i / dy1 ı.x1  y1 /ı.x2  y1 /ı.x3  y1 /ı.x4  y1 / S1 D M44

M4

 W '.x1 /    '.x4 / W dx1    dx4 3 2 Z Z X 1 7 6 D dy1 ı.x.1/  y1 /    ı.x.4/  y1 /5 4.i / 4Š M44

2P4

M4

 W '.x1 /    '.x4 / W dx1    dx4 :

(26.45)

Comparing (26.43) and (26.45), we conclude that the expression in brackets is the first term of the expansion of the coefficient function F4 .x1 ; x2 ; x3 ; x4 / in the pertur-

307

Section 26.3 Main Properties of the S -Operator

bation series in the coupling constant : Therefore, if we take FN .x1 ; : : : ; xN / D

1 X

.n/

n FN .x1 ; : : : ; xN /;

(26.46)

nDn0

then .1/ F4 .x1 ; : : : ; x4 /

X Z

D .i /

2P4

dy1 ı.x.1/  y1 /    ı.x.4/  y1 /:

(26.47)

M4

The first term of the expansion corresponding to n D n0 depends on N; i.e., n0 D n0 .N /: For N D 4; we have n0 D 1: Further, for S2 ; we get Z

.i /2 S2 D 2Š

 T W ' 4 .x1 / W W ' 4 .x2 / W dx1 dx2

M42

D

Z

.i /2 2Š



W ' 4 .x1 /' 4 .x2 / W C16 W ' 3 .x1 / '.x1 /'.x2 / ' 3 .x2 / W

M42

 2 C 72 W ' 2 .x1 / '.x1 /'.x2 / ' 2 .x2 / W  3  4  dx1 dx2 : C 96 W '.x1 / '.x1 /'.x2 / '.x2 / W C24 '.x1 /'.x2 / (26.48) In the last equality, we have used the Wick theorem for the chronological product (Theorem 23.7). It is now easy to verify that S2 D

2 8Š C

Z

.2/

F8 .x1 ; : : : ; x8 / W '.x1 /    '.x8 / W dx1    dx8

M48 Z 2

6Š

.2/

F6 .x1 ; : : : ; x6 / W '.x1 /    '.x6 / W dx1    dx6

M46

C

2 4Š

Z

.2/

F4 .x1 ; : : : ; x4 / W '.x1 /    '.x4 / W dx1    dx4 M44

C

2

Z

.2/

F2 .x1 ; x2 / W '.x1 /'.x2 / W dx1 dx2 C

2Š M42

2 .2/ F ; 0Š 0

308

Chapter 26

Scattering Theory. Scattering Matrix .2/

where, with regard for (23.42), (26.43), and (26.46), the expressions for FN ; N D 0; 2; 4; 6; 8; can be written in the form Z .i /2 X .2/ F8 .x1 ; : : : ; x8 / D dy1 dy2 ı.x.1/  y1 /    ı.x.4/  y1 / 2Š 2P8 42 M

ı.x.5/  y2 /    ı.x.8/  y2 /; .2/ F6 .x1 ; : : : ; x6 /

Z .i /2  16 X D 2Š 2P6

(26.49)

dy1 dy2 ı.x.1/  y1 /ı.x.2/  y1 /

M42

1  ı.x.3/  y1 / D c .y1  y2 /ı.x.4/  y2 /ı.y2  x.5/ /ı.y2  x.6/ /; i (26.50) Z 2 .i /  72 X .2/ dy1 dy2 ı.x.1/  y1 / F4 .x1 ; : : : ; x4 / D 2Š 2P4



M42

2 1 c D .y1  y2 / ı.y2  x.3/ / ı.y2  x.4/ /:  ı.x.2/  y1 / i Z .i /2  96 X .2/ F2 .x1 ; x2 / D dy1 dy2 ı.x.1/  y1 / 2Š  

2P2

(26.51)

M42

3 1 c D .y1  y2 / ı.y2  x.2/ /; i

(26.52)

and, finally, .2/ F0

.i /2  24 D 2Š



Z dy1 dy2

4 1 c D .y1  y2 / : i

(26.53)

M42

It is clear that, for N D 8; 6; 2; one has n0 D 2; whereas for N D 0 one should set .0/

F0

D 1;

i.e., n0 .0/ D 0: Performing analogous operations for S3 ; S4 ; etc., and formally summing up the terms with the same N; we obtain expansion (26.43) with FN in the form (26.46). The procedure described above is called the reduction of the S -operator to the normal form. Remark 26.3. It is easy to verify that the coefficient functions are singular generalized functions. Furthermore, due to the translational invariance of the function .2/ D c .x  y/; integral (26.53) is divergent, i.e., F0 is proportional to an unbounded

Section 26.3 Main Properties of the S -Operator

309

constant. However, it will be established in what follows that the sum of all vacuum contributions F0 (series (26.46) for N D 0) is the multiplier of each coefficient function, i.e., (26.54) FN .x1 ; : : : ; xN / D F0  FN0 .x1 ; : : : ; xN /; and F0 is the total amplitude of vacuum–vacuum transition. For most theories, it can be shown that F0 D e i'0 ; where '0 is a real unbounded constant, i.e., F0 is imaginary and does not affect the transition probability (see e.g. [181], Chapter 14, Section 2). To eliminate the divergences caused by the vacuum contributions, it is necessary to consider the operator S.g/ defined by (26.41) in intermediate calculations.

26.3.2 Invariance of the Scattering Matrix under Lorentz Transformations and Transformations of Charge Conjugation An important physical condition is the relativistic covariance. It means that relation (26.17) for the scattering operator remains true for the transformed amplitudes of states, i.e., 0 0 D S‰in or ˆ0out .C1/ D Sˆ0in .1/; (26.55) ‰out where

ˆ0 .˙1/ D UL ˆ.˙1/

and UL is the linear unitary operator corresponding to the Lorentz transformation L (1.20) that defines a representation of the Lorentz group in the space of states. Comparing (26.17) and (26.55), we obtain the following law of relativistic invariance: S D UL1 SUL :

(26.56)

We also write the law of transformation in the case of interaction with intensity g.x/: In this case, it is necessary to take into account that the function g.x/ changes under the Lorentz transformation (1.20). It follows from physical arguments that the domain of interaction remains the same in the coordinate system ¹x 0 º; i.e., g.x/ ! g 0 .x 0 / D g.x/; or

Lg: g 0 .x/ D g.L1 x/ WD b

Thus, the law of transformation of amplitude has the form ˆ.g/ ! ˆ0 .g 0 / D UL ˆ.g/: In this case, the definition of the operator S.g/ has the form ˆ0out .g 0 / D S.g 0 /ˆ0in .g 0 /;

(26.57)

310

Chapter 26

Scattering Theory. Scattering Matrix

or ˆout .g/ D S.g/ˆin .g/: Hence, taking (26.57) into account, we obtain S.b Lg/ D UL S.g/UL1 : Using condition (26.56) and the condition of the relativistic invariance of the field operators (17.13), we easily obtain the following conditions for the relativistic invariance of the coefficient functions (Problem 26.2): .det ƒ/jNj FN .Lx j j0 /.SL /j0 j D FN .x j j/; where SL D S.ƒ; a/ is defined by (1.24). We also write the following condition for the invariance of the scattering operator under the operation of charge conjugation: D S: U 1 SUb C b C

(26.58)

This equality is a corollary of (21.10) and (26.39).

26.3.3 Unitarity of the Scattering Operator One of the most important properties of the S-operator is its unitarity. This property can easily be proved with the use of representation (26.19) in terms of the wave operators and their isometric properties (26.16a) and (26.16b). Indeed, S  S D .  C /  C D C   C D C C  C Pb C D 1  0 D 1

(26.59a)

and S S  D  C C  D     Pb  D 1  0 D 1:

(26.59b)

The identity ˙ Pb ˙  0 is a corollary of the orthogonality condition (26.12). The unitarity of the operator S means that, for every normalized vector ‰in ; there exists a unique normalized vector ‰out and vice versa. The unitarity of the S -operator plays a very important role in the construction of the renormalized scattering matrix by the perturbation method and in the proof of dispersion relations. In this connection, it is necessary to rewrite the unitarity condition 



S.g/S .g/ D S.g/S.g/ D 1

(26.60)

Section 26.3 Main Properties of the S -Operator

311

in terms of expansion coefficients (operators Sn .x/n ). Substituting expansion (26.40) into condition (26.60) and using the known formula of resummation of a series, we obtain the relation Z 1 X n X  1 Sk .x1 ; : : : ; xk /S nk .xkC1 ; : : : ; xn / kŠ.n  k/Š nD0 kD0

 g.x1 /    g.xn /dx1    dxn D 1;

(26.61)

which is true for an arbitrary function g.x/: To obtain the unitarity conditions for the operators Sn .x/n ; we rewrite (26.61) in the form Z X 1 n X X  1 Sk .x1 ; : : : ; xk /S nk ..x/n n ¹x1 ; : : : ; xk º/ nŠ  nD1

kD0

 g.x1 /    g.xn /dx1    dxn D 0; where the term with n D 0 has been canceled by the identity operator on the righthand side, and the factor nŠ=kŠ.nk/Š has been rewritten using the sum of all possible mappings of the set of indices ¹1; : : : ; kº to the set ¹1 ; : : : ; k º with regard for the 

symmetry of Sk .x1 ; : : : ; xk / and S nk .xkC1 ; : : : ; xn / in their arguments. Since the function g.x/ is arbitrary, we finally obtain the infinite system of equations (n > 1) 

Sn .x1 ; : : : ; xn / C S n .x1 ; : : : ; xn /C C

n1 XX



Sk .x1 ; : : : ; xk /S nk ..x/n n ¹x1 ; : : : ; xk º/ D 0:

(26.62)

kD1 

The next problem is to establish a relationship between experimentally measured quantities (differential cross-sections) and the S -operator. Prior to passing to this problem, we establish one more important property of the operator S according to which it preserves the energy of the incoming beam.

26.3.4 Law of Conservation of Energy In the construction of the S-operator, we have assumed that the particles of incoming and outgoing beams are free, i.e., the interaction between them is absent. Thus, their energy is the kinetic energy of free particles: Ein D .‰in ; H0 ; ‰in /;

Eout D .‰out ; H0 ; ‰out /:

The main statement of this chapter is the following: Proposition 26.4. If the particles that hit the target and the scattered particles are free, then (26.63) Ein D Eout :

312

Chapter 26

Scattering Theory. Scattering Matrix

Proof. It follows from definition (26.17) and the unitarity of the operator S (see (26.59a) and (26.59b)) that equality (26.63) is equivalent to the statement that S commutes with H0 : (26.64) SH0 D H0 S: To prove equality (26.64), we consider the identity e iH e iH t e iH0 t D e iH.tC / e iH0 .tC / e iH0

or, in terms of the operators U.t / and U0 .t / [see (26.2) and (26.4)], 



e iH U .t /U0 .t / D U .t C  /U0 .t C  /e iH0 :

(26.65)

It is easy to verify that if limits (26.10) exist, then we can also pass to the limits as t ! 1 in (26.65). Using this statement, we get e iH ˙ D ˙ e iH0 : Differentiating this relation with respect to  and setting  D 0; we get H ˙ D ˙ H0 :

(26.66)

Of course, equality (26.66) holds in this case not for all vectors from H but only for vectors from the domain of definition D.H / \ D.H0 /: We now see that relation (26.64) follows directly from definition (26.19), relation (26.66), and the self-adjointness of the operators H and H0 (Problem 22.3). Remark 26.5. An important condition for physical processes, including scattering processes, is the causality condition. According to this condition, any event that takes place at time x 0 at a point x can affect the event that takes place at time y 0 > x 0 at a point y only if .x  y/2 > 0: In terms of field operators, this condition is represented in the form of axiom AQ6 (see Section 17.3). It can also be rewritten in the form of a certain operator equation in variational derivatives9 for the scattering matrix S.g/: It has the form   ıS.g/  ı S.g/ D 0 for x . y; (26.67) ıg.x/ ıg.y/ where x . y means that x 0 < y 0 and .x  y/2 < 0: Therefore, events that take place at points x; y 2 M4 are separated by a space-like interval, and the event at the point x has occurred earlier. For the detailed substantiation of Equation (26.67), see Section 17.5 in [26]. The conditions of relativistic invariance, unitarity, and causality are sufficient for the construction of a perturbation series for the operator S.g/ in the form (26.39). Exactly this method was used in [26] (see Section 18). 9

For the definition of variational derivative, see Chapter 28 (Remark 28.2).

Section 26.3 Main Properties of the S -Operator

313

26.3.5 Matrix Elements of the S -Operator and the Scattering Amplitude To have a connection between the theory and experiment, it is necessary to obtain working formulas that associate the effective cross-sections of scattering processes and the matrix elements of the scattering operator between the initial and final states. Prior to deducing these formulas, it should be noted that we use certain assumptions that are, generally speaking, incompatible from the physical point of view. For example, it is assumed that all particles of the incoming beam on a target are free (the state of the system before the process of interaction (scattering) is an eigenstate of the operator of free energy H0 ). However, we cannot exclude the interaction between particles of the beam. The same assumption is also used in the determination of the scattering matrix S: Let the initial state of the system be described by the amplitude ˆs ; where s is the number of types of particles that move as free ones at t D 1: For calculations to be mathematically correct and physically consistent, we assume that, in the distant past, at large distances from a target (or the place where a scattering process takes place), particles move as free ones and their interaction occurs in the e D V T /: This situation is guaranteed by the introduce M4 .jƒj space–time cube ƒ tion of the function g.x/ into the interaction Hamiltonian (see (25.25)–(25.27)) and the function g.x/ (see (25.28)) into the expression for the action of the scattering matrix. For the amplitudes of the initial and final states, it is convenient to use a periodic approximation of field operators (see, e.g., [26], Section 3.3 or [158], Section 12.1). Using this approximation, we define the initial amplitude ˆs : In the case of a oneparticle state, this amplitude can be written in the form Z ƒ C 1=2 C a;ƒ .k/ 0 ; ˆ1 D Q ƒ k; .p/a .p/ d p 0 D V where aC .p/ is the creation operator of a free particle with momentum p and discrete quantum numbers : The function Q ƒ k; .p/ D

V k;ƒ .p/; .2/3=2

where k;ƒ .p/ is the indicator function of the cube Ik;ƒ ; plays the role of the wave function of a particle in the momentum representation (see [26], Section 22). As 3=2 ı.p  k/: ƒ % R3 ; we have Q ƒ k; ! .2/ By virtue of the commutation (or anticommutation) relations, we have 2 kˆƒ 1 k D V;

(26.68)

i.e., the amplitude is a normalized vector in a unit volume. Thus, in the limit of infinite volume, the one-particle state can be written in the form ˆ1  ˆk1 D .2/3=2 aC .k1 / 0 :

314

Chapter 26

Scattering Theory. Scattering Matrix

Then the amplitude of the state that consists of L1 particles of the first type with .1/ .1/ .1/ .1/ momenta k1 ; : : : ; kL1 and quantum numbers j1 ; : : : ; jL1 ; : : : ; Ls particles of the .s/

.s/

.s/

.s/

type s with momenta k1 ; : : : ; kLs and quantum numbers j1 ; : : : ; jLs has the form jLj=2 ˆƒ s DV

Ln s Y Y

aC.n/

nD1 mn D1

mn ;ƒ

  k.n/ m n 0 :

(26.69)

The state of the f -type of particles after scattering has an analogous form, namely ˆfƒ D V jM j=2

f Mn Y Y

aC0.n/

nD1 mn D1



mn ;ƒ

 k0.n/ m n 0 ;

(26.70)

or, in the limit to infinite volume, Ln s Y Y

ˆs D .2/3jLj=2

nD1 mn D1

ˆf D .2/3jM j=2

f Y Mn Y nD1 mn

aC.n/ .k.n/ mn / 0 ;  mn

aC.n/ .k0.n/ mn / 0 :  mn

(26.71)

(26.72)

We use either relations (26.69) and (26.70) or relations (26.71) and (26.72) depending on the situation. In calculations related to probability interpretation, amplitudes normalized to unity are used. For this reason, from the mathematical point of view, it is more correct to use first the finite-volume approximation and then pass to the thermodynamic limit .V ! 1/ only in the final function. Further, it is necessary to write the matrix element .ˆf ; S.g/ˆs /;

(26.73)

where S.g/ is the scattering matrix with interaction intensity g: The scattering matrix S.g/ is expressed in terms of the operator U.t2 ; t1 I g/; which is constructed with the use of the operator H.g/ and relations (26.28) or (26.34), (26.35): t1 D min ¹t j t 2 T º;

t2 D max ¹t j t 2 T º:

It should be noted that the switching of the interaction on and off at certain given times t1 and t2 leads to the energy indeterminacy of incoming and outgoing particles by virtue of the Heisenberg indeterminacy principle Et „: The gradual (also called adiabatic) switching interaction is more consistent. According to this switching, HI .t / in the equation for U.t; t0 / is replaced by e " jtj HI .t / (or by the switching function g 0 as in (25.28)) and then the limit transition as " tends to zero is performed in the final expressions (adiabatic limit).

315

Section 26.3 Main Properties of the S -Operator

However, function (25.28) is more convenient for calculations. In this case, the operator S.g/ has the form (26.42) with coefficient functions FN .gI .x j j/N /; and, furthermore, g!1

FN .gI .x j j/N / ! FN .x j j/N : Technically, it is more convenient to write the matrix element in the adiabatic limit g ! 1 because, in this case, FN .x j j/N is a translation-invariant function with respect to the variables .x/N : To determine the matrix element (26.73), note that, according to (18.2) and (19.2), the free fields uj .x/ can be represented in the form luj .x/ D ujC .x/ C uj .x/; Z 1 ˙ d pe ˙ipx uQj .p/a˙ .p/; uj .x/ D .2/3=2 and their commutators (or anticommutators) with the operators a˙ .k/ have the form Œuj˙ .x/; a .k/" D .1/ı1;" e ˙ikx uQj;˙ .k/;

(26.74)

where, for " D 1 (also denoted simply by ), one takes the commutator and ı1;" D C1; whereas for " D C1; one takes the anticommutator and ı1;" D 0: The quantities uQj .k/ are ordinary functions:  for a scalar field: p 1 ; k 0 D k2 C m2 I (26.75) uQj;˙ .k/ D p .2/3=2 2k 0  for spinor fields [see (19.2)]: uQj;˙ .k/ D

1 v ;˙ .k/ .2/3=2 j

 for vector fields: uQ a;˙  .k/ D

or

1 vN ;˙ .k/I .2/3=2 j

a .k/ e ; p .2/3=2 2k 0

(26.76)

(26.77)

a .k/ are the amplitudes of the corresponding polarizations. where e We discuss these functions in more detail when considering specific models of QED and QCD. Substituting relations (26.42), (26.71), and (26.72) into (26.73) for g D 1; taking into account that only the term with jNj D jLj C jMj remains in sum (26.42), and using the operators corresponding to incoming and outgoing particles, we obtain the

316

Chapter 26

Scattering Theory. Scattering Matrix

expression .ˆf ; S.1/ˆs / D .2/

3N 2

Z

0 jMj

.dx /

jLj

.dx/

nD1 m0 D1

M4jNj

FM;L ..x 0 j j 0 /M I .x j j /L /

f Mn Y Y

Ln s Y Y nD1

m0 D1



.n/

0.n/

 m0

uQ

.n/ jm0

.n/

mn ikm uQj .n/ .k.n/ m /e

0.n/

0.n/

.km0 /e ikm0 .n/

xm

:

0.n/

xm0

(26.78)

Remark 26.6. The matrix element .ˆf ; S.1/ˆs / is also nonzero for jLj C jMj > jNj: However, in this case, one or several operators of creation in the amplitude ˆs coincide with analogous operators in the amplitude ˆf and have the same momenta. In fact, this means that the particles corresponding to these operators with probability one fly through the target and do not take part in the interaction. For this reason, we assume in what follows that the particles of the same type have different momenta before and after scattering. By virtue of condition (26.56), the coefficient functions are translation-invariant generalized functions. This means that the matrix moment (26.3.5) has the structure 1 0 f X Ln Mn s X X X 0.n/ .n/A FML ..k 0 jj 0 /M I .kjj /L /; km0 C km .ˆf ; S.1/ˆs / D ı @ nD1 mD1

nD1 m0 D1

(26.79) where FML are the Fourier transforms of the functions FML multiplied by uQj : The functions FML are called the scattering amplitudes, and the ı-function to the left of the amplitude FML represents the law of conservation of energy-momentum of the beam of incoming particles on a target. The structure of the functions FML becomes clear if we calculate them by using perturbation theory. Also note that the term “amplitude” is sometimes used for the function that is the ordinary Fourier transform of the coefficient function FML (after the separation of the ı-function) and does not contain the factors uj .k/:

26.4

Feynman Diagrams

In previous chapters, we have written analytic expressions for the scattering matrix in each order of perturbation theory in the case where the interaction Lagrangian is given. The main technical procedure is the reduction of the perturbation series (26.36), (26.37) to the normal products of free fields using the Wick theorem for the T -product. Using this theorem, one can easily establish that the nth term of the perturbation series Sn .x1 ; : : : ; xn / (see (26.37)) is equal to the sum of normal products of Lagrangians L .x1 /; : : : ; L .xn / with all possible chronological pairings of the operators appearing in the definition of Lagrangian (including the term without pairing) multiplied by

317

Section 26.4 Feynman Diagrams

.i /n : However, it is possible to avoid the multiple application of the Wick theorem if we use the method of graphic representation of awkward analytic expressions, introducing the correspondence rules known as Feynman diagrams. For the first time, these rules were proposed by Feynman in 1949 [54, 55]. First, we write these rules for the S-operator, i.e., for its expansion in a perturbation series. Then we establish analogous rules for the matrix elements corresponding to specific scattering processes.

26.4.1 Feynman Diagrams for the S -Operator It follows from the arguments presented above that the correspondence rules for the analytic expressions obtained by using the Wick theorem and Feynman diagrams depend on the form of the Lagrangian L .x/; i.e., the number and type of fields in the definition of Lagrangian. In what follows, we consider all possible types of fields. Here, for simplicity, we again focus on the example of the simplest field, i.e., the real scalar field with self-interaction (25.3). In this case, it suffices to introduce only three correspondence rules. We associate each pairing of two operators '.x/'.y/ whose analytic expression is given by (23.42) with the solid line (or a dotted line in the case of several types of fields, where the solid line denotes the pairing of other fields) y ;

x

(26.80)

which is an internal line of the diagram. We associate each free operator with the line x

;

(26.81)

which is an external line of the diagram, and each vertex with the factor .i /: It is easy to see that the operator-valued generalized function S1 .x1 / (see (26.44)) is associated with the diagram x1 ;

(26.82)

and the operator-valued generalized function S2 .x1 ; x2 / (see (26.48)) is associated with the diagrams presented in Fig. 26.1 (a–e). It should be noted that the multiplicity of each diagram in the corresponding expression for Sn .x1 ; : : : ; xn / is combinatorially determined according to the rules of the Wick theorem.

26.4.2 Feynman Diagrams for Coefficient Functions of the S -Operator The coefficient functions are associated with specific scattering processes, namely, with the scattering amplitudes of the corresponding processes. In fact, their Fourier transforms, up to certain factors, coincide with the matrix elements of the S-operator

318

Chapter 26

x1

x2

x1

(a)

Scattering Theory. Scattering Matrix

x2

x1

(b) x1

x2

x2

(c) x1

(d)

x2

(e)

Figure 26.1. Second-order Feynman diagrams for ' 4 -interaction.

between the initial and final states of the corresponding scattering process. For completeness, we also give the correspondence rules for the coefficient functions .n/ FN .x1 ; : : : ; xN / in the nth order of perturbation theory. Each function of this type is associated with the sum of diagrams (with corresponding multiplicities) whose con.n/ tributions are calculated as follows: All diagrams corresponding to FN .x1 ; : : : ; xN / have n internal vertices y1 ; : : : ; yn and N external vertices x1 ; : : : ; xN : A vertex connected with a diagram by a single line is called an external vertex. Each line that connects two internal vertices, i.e., y1 

 y2 ;

is associated with the casual function 1i D c .y1  y2 /: Each line that connects an external vertex with an internal one, i.e., y1 

 x1 ;

is associated with the Dirac ı-function ı.y1  x1 / (i.e., with their product with respect to each coordinate component). Each internal vertex is associated with the factor .i /; and each external vertex is associated with the factor 1: At each internal vertex, integration over the entire space M4 is carried out. Thus, .2/ according to these rules, the expression for F0 (see (26.53)) is associated with the .2/ diagram in Fig. 26.1 (e), the expression for F2 .x1 ; x2 / (see (26.52)) is associated with the diagrams x 1 y1

y2 x 2

x 2 y1

+

y2 x 1

y1

:=

y2

, (26.83)

319

Section 26.4 Feynman Diagrams .1/

.2/

the expressions for F4 .x1 ; : : : ; x4 / and F4 .x1 ; : : : ; x4 / (see (26.47) and (26.51)) are associated with the diagrams y1

;

y1

y2

(a)

;

(26.84)

(b)

.2/

the expression for F6 .x1 ; : : : ; x6 / (see (26.50)) is associated with the diagram y1

y2

;

(26.85)

.2/

and the expression for F8 .x1 ; : : : ; x8 / [see (26.49)] is associated with the diagram y1

y2

:

(26.86)

The circles on the external lines mean that this diagram corresponds to the sum of contributions taken with all possible permutations of external variables x1 ; : : : ; xN to guarantee the symmetry of the coefficient functions (for external lines that correspond to fermions, a circle means an antisymmetrization operator).

26.4.3 Feynman Diagrams for Matrix Elements of the S -Operator For the calculation of the amplitudes of specific scattering processes of elementary particles, the method of Feynman diagrams can be generalized to the description of the matrix elements of the S-operator with the use of perturbation theory. In fact, all preliminary calculations have been made in Subsection 26.3.5. It has been established that if the initial and final states are defined with the use of amplitudes (26.71) and (26.72), then the determination of the matrix elements reduces to the calculation of commutators (or anticommutators) of all creation operators appearing in the definition of the amplitudes ˆs and ˆf (see (26.71) and (26.72)) and annihilation operators appearing in (26.42). The result of these commutations is given by (26.74), where the functions uQj .k/ depend on the types of fields and are defined by relations (26.75)–(26.77) for the main types of fields. For a real scalar field, uQj .k/ is defined by (26.75). By analogy with the case where the operator-valued expressions Sn .x1 ; : : : ; xn /; n D 1; 2; are associated with diagram (26.82) and the diagrams presented in Fig. 26.1, we can introduce correspondence rules between the analytic expressions for the matrix elements of the S-operator (obtained by using perturbation theory) and their graphical representations. The important specific feature of these

320

Chapter 26

Scattering Theory. Scattering Matrix

representations is the correspondence of each external line of this diagram to a real particle in the initial or final state of the scattering process. The internal lines of the diagram correspond to so-called virtual states of particles, i.e., states that cannot be observed. Therefore, the Feynman diagrams for matrix elements can be regarded as a representation of actual scattering processes, i.e., actual channels of interconversion of particles, each of which gives the corresponding contribution to the total probability of a specific process. To clarify these correspondence rules, we calculate the matrix element of the S operator .ˆs ; S.1/ˆf / in the second order of perturbation theory, i.e., the quantity .ˆs ; S2 ˆf /

(26.87)

in the case where the initial state ˆs and final state ˆf are the states corresponding to two free particles with momenta k1 ; k2 and k10 ; k20 ; respectively, for which ki ¤ kj0

for all

i; j D 1; 2:

As in the previous section, we consider the case of self-interaction (25.3). Furthermore, we are not interested here in the problem of normalization of the amplitudes of the initial and final states, which should be considered depending on conditions of an individual problem. Assume that the initial and final amplitudes have the form ˆs D .2/3 ' C .k1 /' C .k2 / 0 ; 3 C

ˆf D .2/ '

.k01 /' C .k02 / 0 :

(26.88) (26.89)

Taking into account relations (26.88), (26.89), and (26.48) and the definition of normal product (see, e.g., (17.32)), we obtain the following expression for the matrix element (26.87): 

Z 2

6

.ˆf ; S2 ˆs / D 36.i / .2/

dx1 dx2

2 1 c D .x1  x2 / i

M42 0 C 0  .' .k1 /' .k2 / 0 Œ2' C .x1 /' C .x1 /'  .x2 /'  .x2 / C 4' C .x1 /' C .x2 /'  .x1 /'  .x2 /' C .k1 /' C .k2 / 0 /: C

We now use the commutation relations (26.74), which take the following form in the case considered: p 1 e ikx ; k 0 D k2 C m2 : (26.90) Œ'  .x/; ' C .k/ D p .2/3=2 2k 0 Using (26.90), we interchange the operators '  .x/ and ' C .kj / until the operators '  .xi / act on the vacuum vector 0 ; which gives the zero contribution. Finally, we

321

Section 26.4 Feynman Diagrams

k1

k1

k1

k2

p1

p2

p1

p2

k2

k2

k2

k1

k1

p1

k2

p2

k2

k1

(a)

(b)

(c)

Figure 26.2. Feynman diagras for the matrix element (26.87).

obtain 288.i /2 .ˆf ; S2 ˆs / D q 4.2/6 k100 k200 k10 k20 

dp1 dp2 .2/4 ı.k1 C k2  p1  p2 / M42

1 1 .2/4 ı.p1 C p2  k10  k20 / 2 4 2  p1  i "/ .2/ i.m  p22  i "/ 1 dp1 dp2 .2/4 ı.k1  k10  p1  p2 / 4 2 .2/ i.m  p12  i "/

.2/4 i.m2 Z

C M42



1 .2/4 .k2  k20 C p1 C p2 /  p22  i "/

.2/4 i.m2 Z

dp1 dp2 .2/4 ı.k1  k2  p1  p2 /

C M42



" Z

1  p12  i "/ #

.2/4 i.m2

1 .2/4 ı.k2  k10 C p1 C p2 / :  p22  i "/

.2/4 i.m2

(26.91)

In the derivation of expression (26.91), we have used the representation (23.9) of the casual function D c .x1  x2 / and the relation 1 .2/4

Z M4

dxe ipx D ı.p/ D

4 Y

ı.p  /:

(26.92)

D0

It is easy to see that, up to the factor 288; the analytic expression (26.91) can be obtained by summing up the contributions of the Feynman diagrams presented in Fig. 26.2.

322

Chapter 26

Scattering Theory. Scattering Matrix

To this end, we use the following correspondence rules: p Each external line of the diagram with momentum k D .k 0 ; k/.k 0 D k2 C m2 / is associated with the factor k k

”

1 .2/3=2

; p 2k 0

(26.93)

each internal line is associated with the propagator p

”

1 ;  p 2  i "/

(26.94)

.2/4 i.m2

and each vertex is associated with ı-function, which indicates the law of conservation of energy-momentum k1

p1

k2

p2

.i /.2/4 ı.k1 C k2  p1  p2 / ”

:

(26.95)

The integration is carried out along each internal line with respect to the 4-momentum “associated” with this line. Note that the multiplicity of contributions, i.e., the number factor, must be combinatorially calculated with regard for the Wick theorem and the number of commutations of the operators appearing in the amplitudes of the initial and final states with the operators of the expression for Sn .x1 ; : : : ; xn /: Remark 26.7. It is easy to see that the contributions (26.91) of the diagrams presented in Fig. 26.2 (a–c) are infinite because the integrals in the definitions of these contributions are divergent for large values of momenta. For this reason, these divergences are called “ultraviolet divergences.” Problems connected with the emergence of ultraviolet divergences and methods for their elimination are considered in Chapter 29.

26.5

Effective Cross-Sections and Scattering Matrix

In this chapter, we give and substantiate main relations that associate results of theoretical analysis of physical processes with characteristics of these processes that can be measured experimentally. For scattering processes of elementary particles, these characteristics are differential and total effective cross-sections. To form a general idea of them that most adequately reflects the actual picture, we begin with the classical theory.

Section 26.5 Effective Cross-Sections and Scattering Matrix

323

26.5.1 Classical Picture First, we assume that a beam of particles, which are regarded as small balls, hits a disk (target) of very small radius. Let the area of the target be small (smaller than a unit area, say, 1 cm2 ) and let nN be the number of particles that hit a unit area for 1 s. Then a part of the beam flies through the target, and the other part deviates due to the collision with the target. By definition, the share of particles that deviate and are scattered at an angle  from the direction of motion into a certain solid angle  for 1 s is proportional to n: N N D . I  /nN D  . /n ; N

(26.96)

where the coefficient of proportionality  . / has the dimension of area because the dimensions of nN and N are as follows: Œn N D cm2  s1 and ŒN  D s1 : The quantity  . / or d D  . /d (26.97) is called the differential cross-section of a scattering process. It is clear that the total cross-section is given by Z t D

 . /d :

(26.98)

The total cross-section determines what part of incoming particles is scattered under the condition that a single particle hits a unit area of the target per unit time. In the above definition,  t coincides with the area of the disk. In the case where the target is an isolated atom or even an elementary particle, the scattering of particles occurs not due to direct collisions of incoming particles with an atom or elementary particle but due to their interaction. The room in which the deviation of incoming particles takes place is called the effective scattering cross-section. Of course, a single isolated atom (or elementary particle) cannot be a target. As a target, one may take, e.g., a very thin plate, say, of thickness d and area S: If we denote the number of atoms (regarded as “elementary targets”) in a unit volume by n; then the number of these “elementary targets” in a unit area of the plate is equal to nd; and the number of particles that deviate due to scattering, according to (26.96)–(26.98), is equal to N n t nd:

(26.99)

How can one measure the differential and total effective cross-sections of the scattering process? Let n.0/ N be the number of particles that hit a unit area of the target for 1 s and let n.x/ N be the corresponding value measured inside the target at a distance from its surface. Then n.x/ N  n.x N C x/ is the number of particles scattered as they fly the distance x: According to (26.99), we have N n.x/ N  n.x N C x/ D n t n.x/x:

(26.100)

324

Chapter 26

Scattering Theory. Scattering Matrix

As x ! 0; relation (26.100) reduces to an ordinary differential equation, which has the solution n t x : (26.101) n.x/ N D n.0/e N It should be noted that relation (26.101) corresponds to the assumption that each incoming particle “collides” with an “elementary target” only once, i.e., there is no multiple scattering. To realize this situation, one needs a very thin target. Thus, for N for very thin targets (with given n and d ), one can determine  t by measuring n.0/ the incoming beam and n.d N / for the beam moving through the target. The differential cross-section  . / can be determined by calculating the ratio of the density of the flow of scattered particles to the density of particles that hit a unit area for different scattering angles : Further, it is necessary to find relations that associate  t and  . / with physical characteristics of particles. These relations enable one to reconcile theory and experiment. It is worth noting that the best idea of the derivation of expressions for differential cross-sections is to find them for every individual process under given experimental conditions. The expressions thus obtained may differ by certain factors due to different normalizations of the wave functions of the initial state. However, in all cases, the differential cross-section is proportional to the squared scattering amplitude, which is defined by a matrix element of the S-matrix between the initial and final states. The simplest way to demonstrate this is to consider the case of scattering of plane waves .1/ ): moving along the OX -axis with velocity v D „k m (k D k 'k .x/ D

1 e ikx : .2/3=2

Then the number of particles passing through a unit area per unit time is equal to the density of the flow of particles: !  v i „ @' k  @'k 'k  ' k D : (26.102) Nin D jx D 2m @x @x .2/3 

The last relation is a quantum-mechanical analog (see (4.35)), ' k p=m' O k ; of the classical flow v; where is the density of incident particles and v is their velocity. To calculate the number of particles Nout that pass per unit time through the surface element S located at a distance r from the scattering center at an angle ; it is necessary to take the spherical part of the scattered wave (see [115], Part XVII) instead of 'k in relation (26.102): k .r/ D

e ikr 1 f . /; .2/3=2 r

where f . / is called the scattering amplitude.

Section 26.5 Effective Cross-Sections and Scattering Matrix

325

Taking into account only the terms that decrease not faster than 1=r 2 ; we get Nout D jr S D

v jf . /j2  ; .2/3

where

S r2 is the solid angle at which the plane S is seen from the scattering point. Taking into account relations (26.96) and (26.97), we obtain the final relation for the differential cross-section:  D

d D  . /d D jf . /j2 d : Of course, this relation holds only for elastic scattering (the velocities of the incoming and outgoing waves are equal). However, it will be shown in what follows that, in the general case, the dependence on the amplitude remains the same.

26.5.2 Quantum Picture Using relation (26.96), we can easily verify the following: if one observes the scattering of a single particle, i.e., if one checks whether it appears in a certain solid angle  located at an angle  with the direction of motion of the incoming particle, then the quantities  . / (for experiments repeated many times) determine the probability of this scattering of the particle. According to quantum-mechanical ideas, the number of scattered particles in a unit 0.1/ 0.f / volume per unit time, as well as particles that have random momenta k1 ; : : : ; kMf 0.1/

0.f /

in the intervals d k1    d kMf ; can be calculated by using the matrix elements of the S -operator. For the process of calculation to be mathematically correct, it is necessary to use again the finite-volume approximation for the amplitudes of the initial and final states and the S-operator S.g/; i.e., to consider the case where, in a certain finite box ƒ.jƒj D V / and in the time interval T D jŒt1 ; t2 j; the interaction has the intensity g: Thus, the probability density for the unit time interval and unit volume is calculated as follows: 2 j.ˆfƒ ; S.g/ˆƒ s /j : wfƒs D L1    Ls 2 V T kˆfƒ k2 kˆƒ s k It is also necessary to take into account that only the particles participating in scattering, i.e., the particles that change their momenta, are considered. This means that the diagonal matrix element of the S-matrix is equal to zero, i.e., in fact, we calculate the matrix element of the operator S.g/  1: Taking (26.68)–(26.70) into account, we get 2 j.ˆfƒ ; ŒS.g/  1ˆƒ s /j ; wfƒs D n1    ns V T V jNj

326

Chapter 26

Scattering Theory. Scattering Matrix

where n1 ; : : : ; ns are the numbers of particles in a unit volume for all types of beams hitting the target. To simplify calculations, we note that, in real experiments, as a rule, one or two types of initial beams are used. In the final state, one, two, or (at most) three types of particles are obtained. For simplicity, we take s D f D 2: Further, we take into account that, according to (26.79), the matrix element has the form Q 1 C k2  k10  k20 /F2I2 .k1 ; k2 ; k10 ; k20 /; .ˆf ; ŒS.g/  1ˆs / D g.k where g.k/ Q is the Fourier transform of the function g.x/: In the limit as g.x/ ! 1 2 ! ı.k/ı.k/: Q (ƒ % R3 ; t1 ! 1; t2 ! C1), we have g.k/ ! ı.k/ and jg.k/j Assuming that the volume ƒ is very large but not infinite, we replace ı 2 .k/ by Z 1 VT ı.k/g.0/ Q D ı.k/ g.x/dx  ı.k/ : 4 .2/ .2/4 We also take into account that the number of final states of one quantum particle in the phase volume d k  ƒ is equal to Vd kh3 D Vd k.2/3 .„ D 1/: Then the number of scattered particles in the momentum interval d k01 d k02 is equal to n1 n2 .2/4 ı.k1 C k2  k10  k20 / j F22 .k1 ; k2 ; k10 ; k20 /

d k01 d k02 : .2/3 .2/3

(26.103)

On the other hand, taking into account the definition of effective differential crosssection (26.96) and the fact that the number of particles hitting a unit area of a target per unit time (in the case considered, a target is a beam of particles of the second type) can be represented in terms of the relative velocity of beams 1 and 2, we obtain another expression for N (see (26.99)), namely, n1 n2 jv1  v2 jd:

(26.104)

Comparing (26.103) and (26.104), we obtain the following relation for the effective differential cross-section and the scattering amplitude: d D .2/4

ı.k1 C k2  k10  k20 / d k1 d k2 jF22 .k1 ; k2 ; k10 ; k20 /j2 : jv1  v2 j .2/3 .2/3

(26.105)

It can be shown that this relation is relativistic-invariant (see [2], Section 18).

Chapter 27

Equations for Coefficient Functions of the S -Matrix In the previous chapter, we have defined the scattering operator S and constructed its perturbation series (26.36), (26.37) (or, in the brief form, (26.39)) for a given interaction Lagrangian. However, even for the simplest model of a self-interacting real scalar field (25.3), the analysis of contributions of the second-order terms of the expansion of the operator S in the coupling constant  shows that the integrals corresponding to these contributions are divergent for large values of momenta if the space-time dimension satisfies the condition d D s C 1 > 3 (see Remark 26.7). However, even if one artificially removes these divergences by setting the space dimension s D 1 (i.e., d D 2) or restricts the integration to a finite interval, there arises another difficulty caused by the divergence of the entire series even for very small values of the coupling constant : This is explained by the fact that the number of possible Feynman diagrams increases with the order of perturbation faster than nŠ (as .nŠ/2 for (25.3)). In these cases, we can speak only of the asymptotic behavior of this series (see [172], Chapter XII.4) and the corresponding summation method. This explains the importance of studying the S-operator beyond the framework of perturbation theory. One of the most efficient methods here is the construction and investigation of various types of equations for the coefficient functions FN ; which completely define the S-operator. For the first time, these equations were proposed in [156] on the basis of analysis of Feynman diagrams. Later, in [95] and [158], they were analytically derived from the Schwinger equations for the Green function and, independently, from formal equations for the S -operator. In this chapter, we briefly describe the derivation of these equations, establish their relationship with the generally accepted perturbation theory, and briefly analyze methods that enable one to go beyond the framework of perturbation theory. For a detailed analysis of these equations and their investigation, see [158]. Prior to passing to the derivation of equations for coefficient functions, we consider two formal operations over arbitrary Wick ordered functionals of the form (26.42) and (26.43). These operations are associated with two formal “operators” that act on the coefficient functions of Wick ordered functionals. This action is similar to the action of the creation and annihilation operators of free fields in the Fock space (see relations (22.40a,b)). Due to this analogy, they are called creation and annihilation operators of external lines of Feynman diagrams (see [158], Section 11).

328

27.1

Chapter 27

Equations for Coefficient Functions of the S -Matrix

Creation and Annihilation Operators of External Lines of Feynman Diagrams

For simplicity, we again consider the case of a real scalar field. An arbitrary Wick ordered functional is defined by the relation Z 1 X 1 W D NŠ N D0

fN .x1 ; : : : ; xN / W '.x1 /    '.xN / W .dx/N ;

(27.1)

M4N

where '.x/ is the operator of the free scalar field and fN is a sequence of coefficient functions, which uniquely define W in the space of states H : Note that, in the case of interaction (25.3), the coefficient functions fN .x1 ; : : : ; xN / are symmetric functions of their arguments, i.e., fN .x1 ; : : : ; xN / D fN .xi1 ; : : : ; xiN /;

(27.2)

where ¹i1 ; : : : ; iN º is an arbitrary permutation of indices ¹1; : : : ; N º: We define two new Wick ordered functionals of the form WC .x/ D W '.x/W W;

W .x/ D W '.x/W W :

A pairing of the free field '.x/ with W is understood as the sum of all possible pairings of the operator '.x/ with each operator of the expansion for W: Let us calculate the coefficient functions of the new functionals. We have Z 1 X 1 fN .x/N W '.x/'.x1 /    '.xN / W .dx1 /N WC .x/ D NŠ N D0

D

1 X N D0

M4N

1 NŠ

Z

M4.N C1/

"

# N C1 X 1 ı.x  xi /fN ..x/N C1 n ¹xi º/ N C1 iD1

N C1

 '.x1 /    '.xN C1 /.dx/ Z 1 X 1 .C/ D fN .x; .x/N / W '.x1 /    '.xN / W .dx/N : NŠ N D0

(27.3)

M4N

In the second equality of (27.3), we have used the symmetry of the coefficient functions fN .x1 ; : : : ; xN / and some abbreviations (see Notation, part 1). In the third equality of (27.3), we have first passed to summation from N D 1 and then, setting f0C D 0; again to summation from N D 0:

329

Section 27.1 Operators of External Lines of Feynman Diagrams .C/

The new sequence of coefficient functions fN .C/

f0

.C/ fN .x; .x/N /

is defined by the relations

D 0; D .aC .x/f /N .x1 ; : : : ; xN / D

N X

ı.x  xi /fN 1 ..x/N n ¹xi º/;

(27.4) N > 1:

iD1

By analogy, for W .x/ we get Z 1 X 1 W .x/ D NŠ N D1

D

1 X N D1

fN .x/N

W '.x/'.x1 /    '.xj /    '.xN /.dx/N

j D1

M4N

1 .N  1/Š

N X

2

Z

6 4

M4.N 1/

Z

3 1 c 7 D .x  y/fN .y; .x/N 1 /dy 5 i

M4 N 1

 W '.x1 /    '.xN 1 / W .dx/ Z 1 X 1 ./ D fN .x; .x/N / W '.x1 /    '.xN / W .dx/N : NŠ N D0

(27.5)

M4N ./

The new sequence of coefficient functions fN is defined by the relation Z 1 c ./ D .x  y/fN C1 .y; .x/N /dy: (27.6) fN .x; .x/N / D .a .x/f /N .x/N D i M4

The “operators” a˙ .x/ act in the space of sequences f D ¹fN .x1 ; : : : ; xN /º1 N D0 : After the renormalization

1 fN ! p fN ; NŠ

the action of the “operators” a˙ .x/ resembles the action of the operator-valued generalized functions of the free scalar field (22.40a,b) with the only difference that, for the field operator '  .x/; the function 1i D  .x/ is used instead of 1i D c .x/: In the next chapter, it will be shown that the perturbation series for the coefficient functions FN can be rewritten with the use of the action of powers of the “operators” a˙ .x/ on the vacuum vector 0 : According to the Feynman rules formulated in Section 26.4.2, the contribution of the operator aC .x/ corresponds to an external line of the Feynman diagram, and the contribution of the “operator” a .x/ corresponds to an

330

Chapter 27

Equations for Coefficient Functions of the S -Matrix

internal line of the Feynman diagram, i.e., it annihilates an external line by transforming it into an internal one. To deduce equations, it is necessary to consider a WOF of the form T .'.x/W /: By virtue of the Wick theorem for the T -product, we get T .'.x/W / DW '.x/W W C W '.x/W W :

(27.7)

According to (27.3) and (27.5), functional (27.7) is associated with the following “operator” in the space of coefficient functions: a.x/f D aC .x/f C a .x/f;

(27.8)

where f is the sequence of coefficient functions of the WOF W: It is easy to generalize this correspondence rule to more complicated WOF. For example, by analogy with (27.7), we obtain T .W ' 2 .x/ W W / DW ' 2 .x/W W C2 W '.x/'.x/W W C W '.x/ '.x/W W : Then the sequence of the corresponding coefficient functions has the form aC .x/aC .x/f C 2aC .x/a .x/f C a .x/a .x/f DW a2 .x/ W f; where the operation of normal product for the “operators” a˙ .x/ is defined in the same way as for the free fields ' ˙ .x/: Finally, generalizing the previous calculations, we formulate the following statement: Proposition 27.1. Let Pn .'.x// be an arbitrary Wick polynomial of degree n in the free field '.x/: Then the WOF T .Pn .'.x//W / is associated with the sequence of coefficient functions (27.9) Pn .a.x//f; where f is the sequence of coefficient functions of the WOF W: In conclusion, note that, by using relations (27.4) and (27.6), one can easily obtain the following formal commutation relations for the “operators” a˙ .x/: 1 Œa .x/; aC .y/ D D c .x  y/: i By virtue of the parity of the function D c .x/; the “operators” a.x/ and a.y/ commute, i.e., Œa.x/; a.y/ D 0; unlike the operators of a free field, which commute only for spacelike points, i.e., for .x  y/2 < 0:

331

Section 27.2 Equations of the Resolvent Type

27.2

Equations of the Resolvent Type

Using (26.39), we easily obtain the formal relation   @LI ıS D i T S : ıui .x/ @ui .x/

(27.10)

The functional derivatives of the S-operator with respect to quantum fields are defined as the limits of the corresponding functional derivatives with respect to the additive classical terms .x/ (see Remark 28.2) of the operator-valued functions ui .x/ for .x/  0 (for details, see [26], Part IX). For interaction (25.3), relation (27.10) has the form ıS D 4i T .W ' 3 .x/ W S /: ı'.x/ Substituting expression (26.43) for S in this relation, we obtain, on its left-hand side, a WOF of the form (27.1) with coefficient functions fN D FN C1 .x; x1 ; : : : ; xN /I on the right-hand side, we obtain a WOF with coefficient functions determined by (27.9) according to Proposition 27.1. Hence, we get FN C1 .x; x1 ; : : : ; xN / D 4i .W a3 .x/ W F /N .x1 ; : : : ; xN /; or FN .x; x1 ; : : : ; xN 1 / D 4i .W a3 .x/ W F /N 1 .x1 ; : : : ; xN 1 /:

(27.11)

Setting x D xj and xj D xN and using (27.2), we rewrite Equation (27.11) in the form FN .x1 ; : : : ; xN / D 4i .W a3 .xj / W F /N 1 .x1 ; : : : ; xj 1 ; xj C1 ; : : : ; xN /: Taking the sum over j D 1; 2; : : : ; N; we obtain the following equation (for N > 1): Z FN .x/N D 4i  M4

N

 1 X dx ı.x  xj /.W a3 .x/ W F /N 1 .x/N n ¹xj º : (27.12) N j D1

Taking into account the definition of the “operator” aC .x/ [relation (27.4)] and definb by the relation ing the operator of the number of particles N b 1 f /N .x1 ; : : : ; xN / D .N

1 fN .x1 ; : : : ; xN /; N

N > 0;

332

Chapter 27

Equations for Coefficient Functions of the S -Matrix

we rewrite (27.12) in the form (N > 1) Z b 1 aC .x/ W a3 .x/ W F /N .x1 ; : : : ; xN /: (27.13) dx.N FN .x1 ; : : : ; xN / D 4i  M4

Taking into account decomposition (27.8), we get W a3 .x/ W D .a .x//3 C 3aC .x/.a .x//2 C 3.aC .x//2 a .x/ C .aC .x//3 : (27.14) Substituting (27.14) into (27.13) and using relations (27.4) and (27.6), we obtain the following infinite system of relations for the coefficient functions FN : 4 4i  X 4k FN .x1 ; : : : ; xN / D  C3 N kD1



4k Y

Z

j D1 4 M

Z

N X

dx

i1 ¤¤ikM4

k Y

ı.xil  x/

(27.15)

lD1

1 dyj D c .x  yj /FN 2kC4 ..y/4k ; .x/N n ¹xi1 ;    ; xik º/; i

where N D 1; 2; 3;    and Fk  0 for k > 1: Furthermore, analyzing the contributions with the use of perturbation theory, we establish that, for the theory with interaction Lagrangian (25.3), there are no Feynman diagrams with an odd number of external lines, i.e., F2k1  0;

k D 1; 2; : : : :

(27.16)

For k D 2; 3; : : : ; identities (27.16) also follow from Equations (23.15). Thus, the system of equations (27.15) holds for N D 2; 4; : : : : One should also take into account that, for N D 0; the expansion of the S operator (26.43) contains the constant F0 ; which is the sum of contributions of all vacuum diagrams: (27.17) F0 D S0 D . 0 ; S 0 /: We can rewrite system (27.15) in the form of a single operator equation by introducing the sequence (27.18) F D .0; F2 .x1 ; x2 /; 0; F4 .x1 ; : : : ; x4 /; : : :/; i.e., F2k1  0

for k D 1; 2; : : : :

Let F 0 denote the sequence of FN0 .x1 ; : : : ; xN /

D ı4N .i /

4 X i1 ¤¤i4

Z dx

4 Y lD1

ı.xil  x/F0 :

333

Section 27.2 Equations of the Resolvent Type

It is easy to verify that the term F40 .x1 ; : : : ; x4 / corresponds to the first term of the expansion of the coefficient function F4 .x1 ; : : : ; x4 / in a perturbation series in  (see (26.47)) multiplied by F0 ; i.e., .1/

F40 .x1 ; : : : ; x4 / D F4 .x1 ; : : : ; x4 /F0 :

(27.19)

The system of equations (27.15) can be rewritten as a single operator equation in the space of sequences (27.18): (27.20) F D BF C F 0 ; where b 1 B D 4i N

Z

dxaC .x/ W a3 .x/ W :

Equation (27.15) is called an equation of the resolvent type because a formal solution of Equation (27.20) can be represented in the form of the action of the resolvent of the operator B on the vector F 0 ; i.e., F D .1  B/1 F 0 ; or in the form of the series of iterations of Equation (27.20): F D

1 X

n B n F 0 :

nD0

Taking (27.19) into account, one can easily represent the sequence of coefficient functions as the formal series F D F0  F 0 ;

F0 D

1 X

nC1 B n F 00 ;

(27.21)

F 00 D .0; 0; 0; F4 .x1 ; : : : ; x4 /; 0; : : :/:

(27.22)

nD0

where

.1/

Remark 27.2. Relations (27.21) and (27.22) completely reproduce the perturbation series without contributions of vacuum diagrams for the coefficient functions FN : Furthermore, the contributions of all possible Feynman diagrams with corresponding multiplicities are obtained by the action of powers of the operator B on the vector F 0 . Remark 27.3. The separation of the factor F0 in (27.15) can be regarded as a formal proof of relation (26.54).

334

Equations for Coefficient Functions of the S -Matrix

Chapter 27

27.3

Equations of the Evolution Type

Consider the evolution of the S -operator with respect to the coupling constant : It is obvious that Sj D0 D 1: By analogy with the previous section, using expansions (26.36)–(26.39) we can easily obtain the formal relation dS D iT .LI S /: d For the interaction Lagrangian (25.3), we get Z dS D iT .W ' 4 .x/ W S /dx: d

(27.23)

M4

We substitute expression (26.43) for S in (27.23). Taking into account Proposition 27.1, we obtain the following equation for the sequence of coefficient functions F (Problem 27.2): dF D AF; d

F j D0 D 0 D .1; 0; 0; : : :/;

(27.24)

Z

where

W a4 .x/ W dx

ADi

(27.25)

M4

and the action of the operators a.x/ D aC .x/ C a .x/ on the coefficient functions is defined by (27.4) and (27.6). Successively performing these operations, we obtain the following infinite system of equations: 4 X d FN .x1 ; : : : ; xN / D i C4k d kD0



4k Y

Z

j D1 4 M

N X

Z

i1 ¤¤ikM4

dx

k Y

ı.xil  x/

lD1

1 dyj D c .x  yj /FN 2kC4 ..y/4k ; .x/N n ¹xi1 ; : : : ; xik º/: (27.26) i

Remark 27.4. Equations (27.26) are written for N > 4: For N 6 3; the form of equations changes as follows: for N D 3 and N D 1; the corresponding coefficient functions are identically equal to zero [see (27.16)]; for N D 2; the terms corresponding to k D 3; 4 are absent; and, for N D 0; the terms corresponding to k D 1; 2; 3; 4 are absent. Remark 27.5. The formal solution of Equation (27.24) F D e  A 0 D

1 X ./n n A 0 nŠ

nD0

(27.27)

Section 27.3 Equations of the Evolution Type

335

completely reconstructs the entire structure of the S-matrix, i.e., the term An 0 corresponds to the contribution of all Feynman n-order diagrams (i.e., diagrams with n vertices). Remark 27.6. For fermion fields, the operator symbols ‰ ˙ .x/ and ‰ ˙ .x/; which are analogs of a˙ .x/ (relations (27.4) and (27.6)), can be written on the basis of the same arguments (see Section 27.1 and Problem 27.3). Euclidean analogs of these operators are constructed in Part VI (Section 34.2.2). In Part VI, we briefly consider this equation in a Euclidean domain and introduce the volume and ultraviolet regularizations to rigorously formulate the problem of the existence of solutions for Equation (27.24) beyond the framework of perturbation theory.

Chapter 28

Green Functions and Scattering Matrix

In previous chapters, we have considered processes of interaction of quantized fields in the case where an interaction occurs in a certain infinitely expanding “effective domain.” This approximation enables one to fairly efficiently describe scattering processes under the assumption that the particles can be regarded as free ones before and after scattering. Despite the fact that this assumption does not correspond to the real situation, it gives a fairly accurate theoretical description of these processes. However, this does not exhaust all problems of the theory. One of its important directions is the description of the bound states of an interacting system: the determination of energy levels, the lifetime of excited states, or effective cross-sections in the case where the initial and final states have bounded systems of elementary particles. For the solution of these problems, it is necessary to consider a Schrödinger-type equation for a wave function of a bounded system of relativistic particles, which is obviously connected with the quantized Heisenberg field for these particles and the physical vacuum (ground) state corresponding to the interacting system. The latter quantities are expressed in terms of Green functions. In Sections 23.1, 23.2, and 23.6, these functions have been determined for free (noninteracting) fields. The method of Green functions is extensively used, e.g., for the calculation of the anomalous magnetic moment of an electron or the hyperfine structure of a hydrogen atom (Lamb shift). These problems were studied in detail in [26] (Section 37 and 38). For the first time, the importance of Green functions in the investigation of the processes indicated above was shown in the works by Gell-Mann and Low [64] and Schwinger [182]. Schwinger used Green functions as a basis of quantum field theory. We begin the investigation of Green functions with the determination of their relationship with the scattering operator.

28.1

Green Functions and the S -Matrix in the Interaction Picture

In Section 23.6, we have defined N -point Green functions as vacuum averages of the T -product of Heisenberg operators of a free field. It is obvious that, in the case of interacting fields, the definition is the same with the free vacuum replaced by the physical vacuum of the system of interacting fields, and the Heisenberg operators of free fields replaced by a Heisenberg field that satisfies the equation for interacting fields. As in the previous chapter, for simplicity, we consider the self-interaction of

Section 28.1 Green Functions and the S-Matrix in the Interaction Picture

337

the real scalar field (25.3). Thus, the N -point Green function is defined by the relation GN .x1 ; : : : ; xN / D .ˆ0 ; T .'.x1 /    '.xN //ˆ0 /:

(28.1)

It follows from properties of the T -product (see (23.34)) and the fact that '.x/ is a Bose field that GN is a symmetric function of its arguments. Furthermore, it follows from the condition of Lorentz invariance of fields (Axiom AQ4) that GN depends on the pseudo-Euclidean distances between all possible pairs of arguments. We now rewrite expression (28.1) in terms of field operators in the interaction picture. Note that all arguments of this section are formal and can be rigorously substantiated only by the introduction of volume and ultraviolet “cutoffs.” First, using relation (25.17) and taking into account properties of the evolution operator U.x20 ; x10 / (see (26.21)–(26.23)), we write '.x/ D U.0; x 0 /'0 .x/U.x 0 ; 0/:

(28.2)

We now express the vacuum vector ˆ0 in terms of the free vacuum 0 : Assume that, at t D ˙1; an interaction between the fields is adiabatically switched off. In addition, assume that bound states of interacting fields are absent. In this case, if ˆn are eigenvectors of stationary states at time x 0 D 0; then the vectors ˆn .˙1/ D U.˙1; 0/ˆn must be eigenvectors of the Hamiltonian of free fields. Then the physical vacuum ˆ0 is associated with the free vacuum 0 by the same relation ˆ0 .˙1/ D U.˙1; 0/ˆ0 :

(28.3)

Thus, the vectors on the left-hand side of equality (28.3) can be chosen as the free vacuum. Taking into account the condition of uniqueness of a vacuum state, we choose

0 D ˆ0 .1/; i.e.,

0 D U.1; 0/ˆ0 : It is clear that ˆ0 .C1/ and ˆ0 .1/ can differ only by a complex number S0 such that (28.4) S0 S 0 D jS0 j2 D 1; i.e., ˆ0 .C1/ D S0 0 D U.C1; 0/ˆ0 : Taking into account property (26.23) of the operator U.x10 ; x20 /; we obtain the following two representations for the vacuum vector ˆ0 : ˆ0 D U.0; 1/ 0 ;

(28.5)

ˆ0 D S0 U.0; C1/ 0 :

(28.6)

338

Chapter 28

Green Functions and Scattering Matrix

To rewrite the Green function (28.1), we assume (without loss of generality) that 0 : x10 > x20 >    > xN

Taking into account the definition of T -product and relations (28.2), (28.5), and (28.6), we reduce expression (28.1) to the form GN .x1 ; : : : ; xN / D .ˆ0 ; '.x1 /    '.xn /ˆ0 / D .S0 U.0; C1/ 0 ; U.0; x10 /'0 .x1 /U.x10 ; 0/U.0; x20 /'0 .x2 /U.x20 ; 0/ 0 0    U.0; xN /'0 .xN /U.xN ; 0/U.0; 1/ 0 /:

Using properties (26.22) and (26.23), the definition of S -operator (26.31), the definition of T -product, and condition (28.4), we finally obtain the relation GN .x1 ; : : : ; xN / D

1 . 0 ; T .'0 .x1 /    '0 .xN /S / 0 /: S0

(28.7)

The normalization constant S0 can be determined from relations (28.5) and (28.6), the definition of S-operator (26.31), and the condition of orthonormality of the vacuum states ˆ0 and 0 : 1 D .ˆ0 ; ˆ0 / D .S0 U.0; C1/ 0 ; U.0; 1/ 0 / D S 0 . 0 ; U.C1; 0/U.0; 1/ 0 /: Using (28.4), (26.22), and (26.31), we obtain (see (27.17)) S0 D . 0 ; S 0 / D F0 : Using the representation of the S-operator in the form of T -product (26.39), we can represent the Green functions (28.7) in the form GN .x/N D

0 ; T .'0 .x1 /    '0 .xN /e i . 0

R

R

LI .x/dx /

; T e i LI .x/dx /

0

:

(28.8)

0

In Part VI, this relation will be rigorously deduced for regularized Euclidean Green functions. Analogous representations can be written for other interactions.

28.2

Schwinger Equation for Green Functions

The equation for Green functions is a direct corollary of the Heisenberg equations (25.4) for operators of a quantized field. First, as an example, we show this for the 2-point Green function G2 .x1 ; x2 /: We represent it in the form GN .x1 ; x2 / D .x10  x20 /.ˆ0 ; '.x1 /'.x2 /ˆ0 / C .x20  x10 /.ˆ0 ; '.x2 /'.x1 /ˆ0 /:

(28.9)

339

Section 28.2 Schwinger Equation for Green Functions

To obtain an equation for G2 .x1 ; x2 /; it is necessary to apply the operator x1  m2 to both sides of (28.9). Note that all mathematical operations should be understood as operations with generalized functions. Here, x1  m2  @x 0 @x 0 C x1 ; where 1 1 x1 is the Laplacian with respect to the space variables x1 : Since the operator x1 does not contain a time derivative, it acts directly on the field '.x/: Let us find the time derivatives. It is easy to see that @x 0 G2 .x1 ; x2 / D ı.x10  x20 /.ˆ0 ; Œ'.x1 /; '.x2 / ˆ0 / 1

C .x10  x20 /.ˆ0 ; @x 0 '.x1 /'.x2 /ˆ0 / 1

C .x20  x10 /.ˆ0 ; '.x2 /@x 0 '.x1 /ˆ0 /: 1

The first term is proportional to ı.x10  x20 /; i.e., it requires that x10 D x20 : According to AQ7 (relation (17.20)), the equal-time commutator of the fields '.x1 / and '.x2 / is identically equal to zero. By analogy, we determine the second derivative: @x 0 @x 0 G2 .x1 ; x2 / D ı.x10  x20 /.ˆ0 ; Œ@x 0 '.x1 /; '.x2 /ˆ0 / 1

1

1

C .ˆ0 ; T .@x 0 @x 0 '.x1 /'.x2 //ˆ0 /: 1

1

Using (18.1) and the normalization condition ˆ0 ; we get .x1  m2 /G2 .x1 ; x2 / D .ˆ0 ; T ..x1  m2 /'.x1 /'.x2 //ˆ0 / C i ı.x1  x2 /: Using relations (25.4)–(25.6) and definition (28.1), we finally obtain the following equation for the 2-particle Green function: .x1  m2 /G2 .x1 ; x2 / D 4G4 .x1 ; x1 ; x1 ; x2 /  12C G2 .x1 ; x2 / C i ı.x1  x2 /; where C D 1i D  .0/ is an infinite constant that compensates for the divergence of the function G4 .x1 ; x1 ; x1 ; x2 / for equal arguments. Performing analogous (but more awkward) calculations in the general case of the N -point Green function (28.1) (see, e.g., [158], Section 3.1), we obtain the equation .x1  m2 /GN .x1 ; : : : ; xN / D 4GN C2 .x1 ; x1 ; x1 ; x2 ; : : : ; xN /  12C GN .x1 ; : : : ; xN / C

N X

i ı.x1  xj /GN 2 .x2 ; : : : ; xj 1 ; xj C1 ; : : : ; xN /:

(28.10)

j D2

Taking into account that the Green function of the operator x  m2 is the causal function D c .x/ (see definition (23.2), (23.9)), one can easily rewrite Equation (28.10)

340

Chapter 28

Green Functions and Scattering Matrix

in the form of an integral equation: Z 1 GN .x1 ; : : : ; xN / D 4i  dy D c .x1  y/GN C2 .y; y; y; x2 ; : : : ; xN / i Z 1 C 12i C dy D c .x1  y/GN .y; x2 ; : : : ; xN / i N X1 D c .x1  xj /GN 2 .x2 ; : : : ; xj 1 ; xj C1 ; : : : ; xN /; C i j D2

(28.11) which can also be derived from representation (28.7) by using the generalized Wick theorem (see Problem 24.1). For simplicity and clarity, it is convenient to represent Equation (28.11) in the graphic form. For this purpose, we use the following notation: 1 c D .x1  x2 / ” x1  i

 x2 ;

x2

GN .x1 ; : : : ; xN / ” x 1

;

GN

(28.12)

xN N X 1 c D .x1  xj / ” x1  i

ı:

j D2

Then Equation (28.11) takes the form x1 x1

GN

= −4i λ

x1 y

G N+2 + 12i λ

x1 y

where the integration is carried out at each internal vertex.

GN

+

G N−2 ,

(28.13)

Section 28.3 Green Functions and Coefficient Functions of S -Operator

28.3

341

On the Relationship between the Green Functions and the Coefficient Functions of the Scattering S -Operator

To establish a relationship between the Green functions and the coefficient functions, we first define so-called dressed coefficient functions, namely 0 1 Z N Y 1 e N .x1 ; : : : ; xN / D @ D c .xj  yj /A FN .y1 ; : : : ; yN /dy1    dyN ; F i j D1

(28.14) i.e., for each variable, the convolution with the causal two-point Green function is carried out. This operation may be called an extension of the coefficient functions e e N .p1 ; : : : ; pN / depends beyond the mass surface because their Fourier transform F on the 4-momenta p1 ; : : : ; pN that are not connected by the relations pj2 D m2 ; j D 1; 2; : : : ; N: Using the formal operator transformations introduced in the previous chapter (see relations (27.4) and (27.6)), we rewrite (28.14) in the form e N .x1 ; : : : ; xN / D .a .x1 /    a .xN /F /0 F D . 0 ; a .x1 /    a .xN /F / D . 0 W a.x1 /    a.xN / W F /;

(28.15)

where F D ¹FN º1 N D0 : Using Proposition 23.1, we rewrite relations (28.7) and (28.8) for Green functions in terms of the operations a.x/ as follows: GN .x1 ; : : : ; xN / D . 0 ; a.x1 /    a.xN /F /;

(28.16)

where F is the sequence of coefficient functions. According to the results of Sections 27.2 and 27.3, this sequence can be rewritten in the form F D . 0 ; e i

R

LI .xIa/dx

0 /1 e i

R

LI .xIa/dx

0 :

(28.17)

In relation (28.17), LI .xI a/ has the same form as LI .x/ with a.x/ instead of '.x/: We now apply the Wick theorem for normal products to the product in relation (28.16). According to (28.15), we obtain .N D 2k/ e N .x1 ; : : : ; xN / GN .x1 ; : : : ; xN / D F X 1 c e N 2 .x1 ; : : : ; xOj1 ; : : : ; xOj2 ; : : : ; xN / D .xj1  xj2 /F C i ¹j1 ;j2 º¹1;:::;N º h1 X 1 C D C .xj1  xj2 / D c .xj3  xj4 / i i ¹j1 ;:::;j4 º¹1;:::;N º i 1 1 1 1 C D c .xj1  xj3 / D c .xj2  xj4 / C D c .xj1  xj4 / D c .xj2  xj3 / i i i i e N 4 .x1 ; : : : ; xOj1 ; : : : ; xOj4 ; : : : ; xN / C : : : : F (28.18)

342

Chapter 28

Green Functions and Scattering Matrix

e N and In conclusion, we write the following formal relations for the functions F GN in terms of the operator symbols a.x/ for model (25.3): R e N .x1 ; : : : ; xm / D 1 . 0 ; W a.x1 /    a.xN / W e i Wa4 .x/Wdx 0 /; F F0 R 4 1 . 0 ; a.x1 /    a.xN /e i Wa .x/Wdx 0 /; GN .x1 ; : : : ; xm / D F0

F0 D S0 D . 0 ; e i

R

Wa4 .x/Wdx

0 /:

(28.19) (28.20) (28.21)

Remark 28.1. Representations (27.21), (27.22), and (27.27) for the coefficient funce N ) and (28.20) and (28.21) for the tions of the S-matrix (or (28.19) for the functions F Green functions can be regarded as a formal method of quantization of fields. This method is equivalent to the quantization by the method of path integrals, which is described in detail in Chapter 30.

28.4

Equations for Green Functions in Terms of Functional Derivatives

The method of functional derivatives is an important technical tool that enables one to considerably simplify the derivation of some equations for Green functions and the investigation of expansions in the coupling constant. For a real smooth function j.x/; x 2 M4 ; that fairly rapidly decreases at infinity, we define a so-called generating functional for the sequence of Green functions GN .x1 ; : : : ; xN / as follows: R

G¹j º WD .ˆ0 ; T e i dxj.x/'.x/ ˆ0 / Z 1 X iN dx1    dxN j.x1 /    j.xN /GN .x1 ; : : : ; xN /: D NŠ

(28.22)

N D0

The Green function GN .x1 ; : : : ; xN / is defined as an N -multiple functional derivative: GN .x1 ; : : : ; xN / D .i /N

ıN G¹j ºjj 0 WD .i /N Gx1 ;:::;xN ¹j ºjj 0 : ıj.x1 /    ıj.xN /

(28.23)

Remark 28.2. Here and in what follows, a variational derivative is understood as a formal operation that acts on functionals of the form (28.22) as an ordinary derivative under the assumption that ıj.x/ WD ı.x  y/: (28.24) ıj.y/

343

Section 28.4 Equations for Green Functions in Terms of Functional Derivatives

Using definitions (28.22) and (28.23), we easily derive an equation for G¹j º from Equations (24.10). To this end, it is necessary to take the first variational derivative of expansion (28.22), act by the operator .x  m2 /; and use the right-hand side of (28.10). As a result, we obtain the equation .x  m2 /Gx ¹j º D 4Gxxx ¹j º  12C Gx ¹j º  ij.x/G¹j º:

(28.25)

It is easy to see that, for  D 0; a solution of Equation (28.25) has the form 1

G 0 ¹j º D e  2

R

j.x/ 1i D c .xy/j.y/dxdy

;

(28.26)

which corresponds to the form of the generating functional for free Green functions. This can be verified by substituting (28.26) into (28.23), performing functional differentiation, and setting j  0: As a result, we obtain expression (23.55). For  ¤ 0; we seek a solution of Equation (28.25) in the form G¹j º D AG 0 ¹j º;

(28.27)

where the operator A is a function of variational derivatives ı=ıj.x/:   ı ADf : ıj Taking into account that the operator A depends only on variational derivatives and ı and acting by the operator A on Equation (28.25) for  D 0; we commutes with ıj.x/ obtain ı AG 0 ¹j º D iA.j.x/G 0 ¹j º/: (28.28) .x  m2 / ıj.x/ For  ¤ 0; we rewrite relation (28.25) in the form .x  m2 /

ı ı3 ı AG 0 ¹j º C 4 AG 0 ¹j º AG 0 ¹j º C 12C 3 ıj.x/ ıj.x/ ıj.x/ D ij.x/AG 0 ¹j º:

(28.29)

Subtracting (28.29) from (28.28), we get i ŒA; j.x/ G 0 ¹j º D 4

ı3 ı AG 0 ¹j º: AG 0 ¹j º  12C 3 ıj.x/ ıj.x/

We seek the operator A as a solution of the following formal operator variational differential equation: ŒA; j.x/ D 4i A

ı3 ı :  12i CA ıj.x/3 ıj.x/

(28.30)

344

Chapter 28

Green Functions and Scattering Matrix

It is convenient to use a resemblance to functions of one variable, for which the following formal relations are true:       d d d ; x D 1; ;x : f Df0 dx dx dx   Then it is easy to see that the operator A that satisfies Equation (28.30) has the form  Z Z ı4 ı2 A D C0 exp i  dx  6i C dx ; ıj.x/4 ıj.x/2 and the generating functional (28.27) can be represented as follows:  Z Z ı4 ı2  6i C dx G¹j º D C0 exp i  dx ıj.x/4 ıj.x/2  Z 1 1  exp  j.x/ D c .x  y/j.y/dxdy : (28.31) 2 i The constant C0 is chosen from the condition G¹0º D 1: Taking into account the first relation in (25.5), we rewrite (28.31) in the general form:    Z ı (28.32) G¹j º D C0 exp i  dxLI x; G 0 ¹j º; ıj.x/ where, as in (28.17), we set LI .xI a/ D LI .x/ D  W a4 .x/ W with a.x/ D ı=ıj.x/: Relation (28.32) holds for any Lagrangian. In the case where LI .x/ depends on spinor fields, the generating functional G also depends on spinor sources .x/ and .x/; N which are Grassmann variables.

28.5

Equations for Truncated Green Functions

It follows from the analysis of Feynman diagrams for Green functions and coefficient functions that the perturbation series corresponding, e.g., to an N -point function is the sum of contributions of connected diagrams with N external lines and all possible disconnected diagrams (i.e., diagrams whose contributions are products of contributions of all their connected parts). Formally summing up the contributions of perturbations of all orders, we establish that the structure of the total function remains the same. According to this structure, one can introduce the corresponding functions, which are called truncated functions. They got this name essentially due to R. Haag [84] (see, also, [101] ). One also uses the name connected functions (see, e.g., [174]). Then the relationship between the Green function and its connected parts can be determined by the relations GN .x1 ; : : : ; xN / D

N X

X

kD1 n1 CCnk DN

X k

GnT1 .xi .1/ ; : : : ; xi .1/ /    GnTk .xi .k/ ; : : : ; xi .k/ /; 1

n1

1

nk

(28.33)

345

Section 28.5 Equations for Truncated Green Functions

where the sum over the variable k means summation over all permutations of indices ¹1; : : : ; N º between groups ¹n1 º; : : : ; ¹nk º such that terms in the sum are not repeated. For example, for N D 4; expansion (28.33) has the form G4 .x1 ; : : : ; xm / D G4T .x1 ; : : : ; xm / C G3T .x1 ; x2 ; x3 /G1T .x4 / C G3T .x1 ; x2 ; x4 /G1T .x3 / C G3T .x1 ; x3 ; x4 /G1T .x2 / C G3T .x2 ; x3 ; x4 /G1T .x1 / C G2T .x1 ; x2 /G2T .x3 ; x4 / C G2T .x1 ; x3 /G2T .x2 ; x4 / C G2T .x1 ; x4 /G2T .x2 ; x3 / C G2T .x1 ; x2 /G1T .x3 /G1T .x4 / C G2T .x1 ; x3 /G1T .x2 /G1T .x4 / C G2T .x1 ; x4 /G1T .x2 /G1T .x3 / C G2T .x2 ; x3 /G1T .x1 /G1T .x4 / C G2T .x2 ; x4 /G1T .x1 /G1T .x3 / C G2T .x3 ; x4 /G1T .x1 /G1T .x2 / C G1T .x1 /G1T .x2 /G1T .x3 /G1T .x4 /:

(28.34)

Note that representations (28.33) and (28.34) are written in the general case regardless of the specific form of an interaction. For example, in the case of interaction (25.3), all odd Green functions are equal to zero. It follows from representation (28.33) that the sequence of truncated Green funcT is completely defined by the sequence GN : For the derivation of equations tions GN T for the functions GN .x1 ; : : : ; xm /; the most convenient is the method of variational equations for generating functionals. The generating functional for the truncated T is defined by the same relation Green functions GN G T ¹j º D DW

1 n Z X i j.x1 /    j.xn /GnT .x1 ; : : : ; xn /dx1    dxn nŠ

nD1 1 X

nD1

in T G ¹j º: nŠ n

(28.35)

To derive an equation for G T ¹j º from Equation (28.25), it is necessary to express the functional G¹j º in terms of the functional G T ¹j º: For this purpose, we substitute decomposition (28.33) into the definition of the functional G¹j º (28.22) and obtain Z 1 X iN dx1    dxN j.x1 /    j.xN / G¹j º D 1 C NŠ N D1



N X

X

X

GnT1 .xi .1/ ; : : : ; xi .1/ /    GnTk .xi .k/ ; : : : ; xi .k/ /

kD1 n1 CCnk DN k 1 X X X

D1C

N D1 l>1

m1 2: Z 1 X 1 ¹˛º D dx1    dxn ˛.x1 /    ˛.xn /n .x1 ; : : : ; xn / nŠ

(28.57)

nD2

and, correspondingly, n .x1 ; : : : ; xn / D

ın ¹˛ºj˛0 : ı˛.x1 /    ı˛.xn /

(28.58)

It turns out that the functions n .x1 ; : : : ; xn / have a perturbation expansion that takes into account only the contributions of strongly connected (one-particle irreducible) Feynman diagrams of the same multiplicity as the functions GnT .x1 ; : : : ; xn /: To show this, we define an operation (which is called the “undressing” (or amputation) of the function GnT .x1 ; : : : ; xn / with respect to variable xj ) inverse to the “dressing” operation defined by (28.14), namely GnT .x1 ; : : : ; xj ; : : : ; xn / Z D dx 0 G21 .xj ; x 0 /GnT .x1 ; : : : ; xj 1 ; x 0 ; xj C1 ; : : : ; xn /:

(28.59)

For example, the completely “undressed” (amputated) 2-point Green function has the form Z G2 .x; y/ D dx 0 dy 0 G21 .x; x 0 /G21 .y; y 0 /G2 .x 0 ; y 0 / D G21 .x; y/: Using relations (28.53) and (28.57), we get j.x/ D i x ¹˛º Di

Z 1 X 1 dx1    dxn ˛.x1 /    ˛.xn /nC1 .x; x1 ; : : : ; xn /: nŠ

nD1

(28.60)

Section 28.6 Equations for One-Particle Irreducible Green Functions. Dyson Equation 351

We substitute expression (28.60) for j.x/ in relation (28.52) and equate the coefficients of the same powers (with equal number of the functions ˛). Then the terms containing only one function ˛ give the equality Z 2 dx 0 dy 0 G2 .x; x 0 /2 .x 0 ; y 0 /˛.y 0 /: ˛.x/ D i Since ˛ is arbitrary, we conclude that the function 2 .x1 ; x2 / D G21 .x1 ; x2 /

(28.61)

satisfies this equality. Taking (28.49) into account, we see that 2 coincides, up to G01 .x; y/; with †.x; y/; which is the sum of the contributions of all strongly connected Feynman diagrams with two external lines. Equating the coefficient of ˛.x2 /˛.x3 / on the right-hand side of (28.52) to zero, we get Z dx10 G2 .x; x10 /3 .x10 ; x2 ; x3 / Z D dx20 dx30 G3T .x; x20 ; x30 /2 .x20 ; x2 /2 .x30 ; x3 /: Applying the integral operator with kernel G21 .x1 ; x/ to both sides of this relation and taking into account (28.50), (28.59), and (28.61), we obtain 3 .x1 ; x2 ; x3 / D G3T .x 1 ; x 2 ; x 3 /:

(28.62)

In the case of interaction (25.3), both functions in (28.62) are identically equal to zero. However, if we consider, e.g., the interaction LI .x/ D  W ' 3 .x/ W;

(28.63)

then the function G3 .x1 ; x2 ; x3 / describes the total vertex Green function, i.e., a 3-point Green function, and the function G3T .x 1 ; x 2 ; x 3 / describes the “undressed” Green function with respect to all three arguments. It is easy to verify that operation (28.59) for G3T .x1 ; x2 ; x3 / means that the perturbation series of the function G3T .x 1 ; x 2 ; x 3 / contains only terms corresponding to strongly connected Feynman diagrams with three external lines. For clarity, we also determine the coefficients of the product of three functions ˛: Taking into account that the expression for 4 is a symmetric function of its arguments, we get 4 .x1 ; : : : ; x4 / D G4T .x 1 ; : : : ; x 4 / XZ  dx 0 G3T .xi1 ; xi2 ; x 0 /3 .x 0 ; xi3 ; xi4 /: 4

(28.64)

352

Chapter 28

Green Functions and Scattering Matrix

Taking into account definition (28.59), equality (28.50) for the function G2 .x; y/; and relation (28.62), we rewrite (28.64) in the form 4 .x1 ; : : : ; x4 / D G4T .x 1 ; : : : ; x 4 / Z  dxdy 3 .x1 ; x2 ; x/G2 .x; y/3 .y; x3 ; x4 / Z  dxdy 3 .x1 ; x3 ; x/G2 .x; y/3 .y; x2 ; x4 / Z  dxdy 3 .x1 ; x4 ; x/G2 .x; y/3 .y; x2 ; x3 /:

(28.65)

It is convenient to represent (28.65) in a graphic form, namely, 3 x2

x1

=

4 x4

x1

x3

x2

x4

x

−

G 4T

, y

x3 3

(28.66) where the bold line denotes the function G2 and the circles denote the sums of the same functions with rearranged arguments. Remark 28.3. The representation (28.6) of the function 4 corresponds to the field theory with interaction (28.63). In the case of interaction (25.3), the second term on the right-hand side of Equation (28.6) vanishes, and one obtains a formula analogous to (28.62) for 4 . Expanding the right-hand side in a perturbation series, one can easily establish that the contributions corresponding to weakly connected diagrams (in the first and the second terms) compensate for each other, and, hence, only the contributions of strongly connected diagrams remain. For a detailed proof of the statement that the functions n .x1 ; : : : ; xn / are the sums of only strongly connected diagrams and a detailed description of this technique, see [205] (see also [30]). To obtain equations for the functions n .x1 ; : : : ; xn /; we again use the method of variational equations for generating functionals. We do this for interaction (25.3). To this end, we rewrite the expression for the functional G T ¹j º in terms of the functional ¹˛º and use Equation (28.38). Using relations (28.52)–(28.56), we get GxT ¹j º D i ˛.x/;

T 1 Gxx ¹j º D xx ¹˛º:

(28.67)

353

Section 28.7 Källén–Lehmann Representation

Further, using (28.67), we obtain T Gxxx ¹j º

ı 1 ¹˛º WD D xx ıj.x/

Z dz

ıxx ¹˛º ı˛.z/ : ı˛.z/ ıj.x/

(28.68)

ı 1 ¹˛º

xx Let us determine the variational derivative ı˛.z/ : Inverse functionals are defined in the same way as inverse functions are defined by relations (28.50), namely, Z 1 ¹˛ºyu ¹˛º D ı.x  u/: (28.69) dy xy

Performing the variation of equality (28.69) with respect to ˛.z/; we get Z dy

1 ¹˛º ıxy

ı˛.z/

Z yu ¹˛º C

1 dy xy ¹˛º

ıyu ¹˛º D 0: ı˛.z/

Using (28.69), we obtain 1 ¹˛º ıxv D ı˛.z/

Z

1 1 dy duxy ¹˛ºyuz ¹˛ºuv ¹˛º:

(28.70)

Substituting (28.70) into (28.68) and taking (28.54) and (28.67) into account, we get Z T 1 1 1 Gxxx ¹j º D i dydudz xy ¹˛ºxu ¹˛ºxz ¹˛ºyuz ¹˛º: Thus, the variational equation for the functional ¹˛º has the form Z 2

.x  m C 12C /˛.x/ D 4

1 1 dydudz xy ¹˛ºxu ¹˛º

1 1  xz ¹˛ºyuz ¹˛º  12xx ¹˛º˛.x/ C 4˛.x/3  i x ¹˛º: (28.71)

To obtain an equation for n .x1 ; : : : ; xn /; it is necessary to perform the variation of (28.71) with respect to the variables ˛.x2 /; : : : ; ˛.xn /; setting x D x1 and using rule (28.70) and definition (28.58) (Problem 28.2).

28.7

Spectral Representation of the 2-Point Green Function (Källén–Lehmann Representation)

In previous sections, we have obtained a series of formal equations for Green functions in the model corresponding to interaction (25.3). Using the method of their derivation, we can write analogous equations for any of the theories considered above. The proof of the existence of solutions of these equations and the determination of the explicit form of Green functions are difficult mathematical problems. By now, these problems has been considered only in terms of a formal perturbation theory, which

354

Chapter 28

Green Functions and Scattering Matrix

leads to divergences described in what follows. However, using the general principles of construction of quantum field theory (Axioms AQ1–AQ5), one can write a representation for the 2-point Green function, which describes certain general properties and is connected with the mass spectrum of the energy-momentum operator. Using (28.1) and the definition of T -product, we get G2 .x1 ; x2 / D .ˆ0 ; T .'.x1 /'.x2 //ˆ0 / D

.x10



x20 /W2 .x1 ; x2 /

C

(28.72) .x20



x10 /W2 .x2 ; x1 /;

(28.73)

where W2 .x; y/ D .ˆ0 ; '.x/'.y/ˆ0 /

(28.74)

is the 2-point Wightman function [219]. It follows from the invariance conditions for field operators (17.13) and vacuum state (17.18) that (28.75) W2 .x; y/ D W2 .ƒx C a; ƒy C a/; where .ƒ; a/ is an arbitrary transformation of the Lorentz group. Equality (28.75) means that (28.76) W2 .x; y/ D w..x  y/2 /; i.e., it depends on the squared distance between the points x and y in the Minkowski space, and equality (28.76) means that the Fourier transform of the generalized func2 tion W2 .x; y/ depends on the squared momentum p 2 D p 0 p2 : For a more detailed investigation of the structure of the function W2 .x; y/; we use Axiom AQ5. Denote the complete system of states corresponding to the spectrum of the energymomentum operator P  by (28.77) ˆn D ˆ.p.n/ ;˛.n/ / ; where p.n/ is the total momentum of the nth state and ˛.n/ are the values of the other quantum characteristics, i.e., 

P  ˆn D p.n/ ˆn :

(28.78)

Note that the system of vectors (28.77) is a complete orthonormal system in the Hilbert space of states H that depends only on its continuous and discrete variables, whereas p.n/ and ˛.n/ are merely the indices of the corresponding states. Therefore, the completeness condition can be rewritten in the form XXZ X   d p.n/ ˆ.p.n/ ;˛.n/ / ˆ.p.n/ ;˛.n/ / D 1 ; ˆn ˆn D (28.79) n

n ˛.n/

where 1 is the identity operator in the space H or the kernel of the identity operator if 

the arguments of the vectors ˆn and ˆn are written (they must be different).10 Using 10 Recall

that the kernel of the identity operator with respect to discrete variables is the Kronecker delta, and, with respect to continuous variables, it is a ı-function.

355

Section 28.7 Källén–Lehmann Representation

the Heisenberg equation (17.39) for the field '.x/ and Equation (28.78), we obtain .ˆm ; '.x/ˆn / D .ˆm ; '.0/ˆn /e i.p.n/ p.m/ /x :

(28.80)

In the definition of the Wightman function (28.74), we insert the identity operator (28.79) between the fields '.x/ and '.y/: As a result, we get XXZ dp.n/ j.ˆ0 ; '.0/ˆ.p.n/ ;˛.n/ / /j2 e ip.n/ .xy/ : .ˆ0 ; '.x/'.y/ˆ0 / D n ˛.n/

(28.81) The Green functions describe actual physical processes, and the support of the Fourier transformation of these functions must lie in the upper light cone of the variable p D .p 0 ; p/; i.e., p 2 > 0 and p 0 > 0: Taking this into account, we get Z 1 W2 .x; y/ D dp .p 2 /.p 0 / .p 2 /e ip.xy/ : (28.82) .2/3 Using the representation Z1 2

d2 ı.p 2  2 /

.p / D

(28.83)

0

of the Heaviside function .p 2 / and taking (18.7) into account, we finally obtain Z1 W2 .x; y/ D

1 d2 .2 / D  .x  yI 2 /; i

(28.84)

0

where 1i D  .x; 2 /  1i D  .x/: The dependence on mass is emphasized here for clarity. Comparing (28.81) and (28.82), one can easily see that the spectral function ./ satisfies the relation XX j.ˆ0 ; '.0/ˆ.p;˛.n/ / /j2 ; (28.85) .p 2 /.p 0 / .p 2 / D .2/3 n ˛.n/

where the sum is taken over all states ˆ.p.n/ ;˛.n/ / with fixed value of momentum p.n/ D p; which restricts the number of terms of the sum by a certain finite number. Furthermore, if the system consists of particles of several types i D 1; : : : ; L; then the isolated points of the spectrum for p 2 D 0 and p 2 D m2i correspond to the vacuum state and stable 1-particle the types i D 1; 2;P : : : ; L: The continuous PL states of 2 2 spectrum starts from p D . iD1 ni mi / ; ni D 0; 1; 2; : : : ; L iD1 ni > 1: For a more detailed proof of relation (28.84), see [22] and also [181], Chapter 17, where the proof of this relation is more formal but more intelligible from the physical point of view.

356

Chapter 28

Green Functions and Scattering Matrix

Taking (23.12) and (28.72) into account, we finally obtain the following representation for the Green function: Z1 d2 .2 /G20 .x; yI 2 /;

G2 .x; y/ D

(28.86)

0

where 1 G20 .x; yI 2 / D D c .x  y/ i is the Green function for a free scalar particle of mass : For the first time, representation (28.86) was used by Källén in [104]. Later, it was proved by Lehmann in [120]. The fine structure of the spectral function .2 / depends on the form of interaction and the corresponding spectrum of physical states (see, e.g., [181], Chapter 17). It is obvious that, in the absence of interaction, one has .2 / D ı.2  m20 /; where m0 is the so-called bare mass, i.e., the mass of a free (noninteracting) scalar particle. In the case of an interacting system, it is also possible to determine the point of the spectrum that corresponds to the stable 1-particle state of a particle of mass m: Then (28.87) .2 / D Z3 ı.2  m2 / C .2 /; where the new distribution function takes into account the spectrum of states that starts from p 2 > m2 : We now give representation (28.86) in the momentum space: e 2 .k/ D G

Z3 C m2  k 2  i "

Z1

m2

d2

.2 / : m2  k 2  i "

e 2 .k/ has a pole for p 2 D m2 ; where m is the It is worth noting that the function G observable mass of the particle. The normalization constant Z3 also depends on interaction and is determined with the use of arguments that guarantee the renormalization of the corresponding theory. Nevertheless, a certain general condition for Z3 can be deduced from the spectral representation of the commutator: Z1 .ˆ0 ; Œ'.x/; '.y/ ˆ0 / D 0

1 d2 .2 / D.x  yI 2 /; i

(28.88)

357

Section 28.7 Källén–Lehmann Representation

where D.x  yI 2 / is the commutator function of the free scalar field of mass  (see (18.8)). Relation (28.88) is a corollary of (28.74), (28.84), (18.7), and (18.8). Differentiating both sides of (28.88) with respect to x 0 ; setting x 0 D y 0 ; and using (17.20) and (18.1), we obtain Z1

Z1 2

2

d2 .2 /:

d . / D Z3 C

1D 0

(28.89)

2

In view of the positivity of the spectral function .2 / (see (28.85)), this yields 0 6 Z3 6 1:

(28.90)

More specific corollaries of the spectral representation follow from the individual form of interaction (e.g., for the interaction of pseudoscalar neutral mesons with nucleons, see [181], Chapter 17, Section 2).

Chapter 29

On Renormalization in Perturbation Theory

In Chapter 26, we have determined the scattering matrix in perturbation theory for a given interaction Lagrangian of fields. In addition, we have noted (see Remark 26.7) that, beginning with the second order of perturbation theory, the analytic expressions corresponding to the Feynman diagrams presented in Fig. 26.1 (c), (d), (e) contain divergent integrals. In this chapter, we analyze the nature of these divergences and their character and study constructive methods for their elimination. First, we investigate the model of a real scalar field with self-interaction (25.3). More realistic models of quantum electrodynamics and quantum chromodynamics are considered in Part 8. The theory of elimination of divergences, also known as renormalization theory, was originated as early as the middle of the last century in the works by Bethe, Dyson, Schwinger, Feynman, and others. Later, it was correctly mathematically formulated by Bogoliubov, Parasiuk, and Hepp. In 1970–1980s, many works devoted to the development of this theory were published. For a detailed survey of methods of contemporary renormalization theory, see [225, 226, 90, 174]. Here, we only briefly consider the main aspects of renormalization theory and try to substantiate the developed mathematical methods from the physical point of view.

29.1

Primitively-Divergent Diagrams. Separation of Divergences by the Pauli–Villars Method

All divergent second-order diagrams are primitively-divergent by definition. We now consider a divergent third-order diagram. If, breaking the external lines of a thirdorder diagram one by one, we obtain diagrams with convergent contributions, then this diagram is called a primitively-divergent diagram of the third order. If at least one of the obtained diagrams is divergent, then the original diagram contains a divergent second-order diagram (as a subblock). Therefore, it is not primitively-divergent and is simply divergent. Thus, A divergent nth-order diagram G is primitively-divergent if all .n  1/th-order diagrams obtained from G by breaking internal lines are convergent.

In the case of interaction (25.3), only three second-order diagrams presented in Fig. 26.1 (c), (d), (e) are primitively-divergent. All divergent diagrams of higher or-

Section 29.1 Primitively-Divergent Diagrams. Pauli–Villars Method

359

ders contain primitively-divergent second-order diagrams (as subblocks). Let us analyze the character of divergences of these three diagrams. We do not consider the vacuum diagram in Fig. 26.1 (e) because the contributions of all vacuum diagrams can be eliminated by the renormalization of the S -operator or a sequence of its coefficient functions as a whole (see (27.21) and Remark 27.3). The contributions of the other two diagrams to the S-operator have the form  4k Z 1 c 2 k D .x1  x2 / dx1 dx2 W ' .x1 / ' k .x2 / W; (29.1) S2 .i / i where k D 1 and k D 2 correspond to the diagrams d and c, respectively. In both cases, relations (29.1) contain the product (power) of two generalized functions of the same arguments. From the mathematical point of view, this operation is not defined. For its definition, an additional regularization is required. One of regularization methods for the definition of the product of singular generalized functions is the Pauli–Villars method(see [26], Section 15). We use this method for the definition of expression (29.1). The first step is to replace the casual function D c .x/ by the c .x/ defined by the relation function DM c .x/ D D c .xI m/  D c .xI M / DM  Z 1 1 1  d k D e ikx .2/4 m2  k 2  i " M 2  k 2  i " Z M 2  m2 e ikx ; D d k .2/4 .m2  k 2  i "/.M 2  k 2  i "/

(29.2)

where the mass M is introduced artificially and does not correspond to actual particles. It follows from the explicit form of the function D c .x; m/ (see (23.16)) that the c .x/ does not contain a term proportional to ı.x 2 /: A more detailed analfunction DM ysis shows that singularities of other types are also absent (see [26], Section 15). In addition, it follows from relation (29.2) that, in the sense of weak convergence, one has c .x/ D D c .x/  D c .xI m/: w  lim DM M !1

First, we consider the expression on the right-hand side of (29.1) for k D 2: This expression is a contribution to the vertex part of the S -operator in the second order of perturbation theory. It also contributes to the scattering amplitude of two particles with momenta k1 and k2 before scattering and k10 and k20 after scattering. Consider one of possible Feynman diagrams that describe these processes, namely k1 k2

q1

k1

q2

k2

:

(29.3)

360

Chapter 29

On Renormalization in Perturbation Theory

According to the Feynman rules (26.93)–(26.95), the corresponding analytic expression has the form F2I2 .k1 ; k2 ; k10 ; k20 / D .2/

where .2/ e  4 .k/ D

1 .2/4

.2/4 ı.k1 C k2  k10  k20 / 2 .2/  4 .k1 C k2 /;  e q 4.2/6 k10 k20 k100 k200

(29.4)

1 : .m2  q 2  i "/Œm2  .q  k/2  i "

(29.5)

Z dq

For large values of q; the integral on the right-hand side of (29.5) is logarithmically divergent. To verify this statement, assume that q 2 is a square in the 4-dimensional Euclidean space. Passing to the spherical coordinate system, we establish that the integral in (29.5) has the following asymptotics for large jqj (the integrals with respect to angular variables are omitted): C1 Z

jqj

3 d D 4

C1 Z

jqj

d ;

i.e., this integral is logarithmically divergent. .2/ After the regularization of (29.2), e  4 .k/ takes the form Z 1 .2/ e c .q/D e c .q  k/; e dq D  4 .kI M / D M M .2/4

(29.6)

where 1 1  2 2  q  i " M  q2  i " M 2  m2 D : .m2  q 2  i "/.M 2  q 2  i "/

e c .q/ D D M

m2

(29.7)

By analogy, one can easily verify that, for M < 1; the integral in (29.5) is convergent. To calculate this integral, we use the relation ." > 0/ 1 Di 2 m  q2  i "

Z1 d˛e i˛.q

2 m2 Ci"/

:

0

Then (29.6) can be rewritten in the form .2/ e  4 .kI M /

1 D .2/4

Z

Z1 " " dM .˛/dM .ˇ/e i˛q

dq 0

2 Ciˇ.qk/2

;

(29.8)

361

Section 29.1 Primitively-Divergent Diagrams. Pauli–Villars Method

where " dM .˛/ D e "˛ .e i˛m  e i˛M /: 2

2

Since the integral of the exponential with respect to dq on the right-hand side of (29.8) does not exist in the ordinary sense, the Fubini theorem cannot be used for changing the order of integration. For this reason, we again use the intermediate regularization by setting Z1 e

i.at 2 C2bt/

Z1 dt D lim

ı!C0 1

1

e i.at

2 C2bt/ıt 2

;

a > 0:

Further, we use the following formula (for detailed calculations, see [26], Section 24): Z  2 ib2 2 e i.ak C2bk/ d k D 2 e  a ; a > 0: (29.9) ia The calculation of Gauss integrals of the form (29.9) yields .2/ e  4 .kI M /

i D 16./2

Z

˛ˇ

" " dM .˛/dM .ˇ/

2

e i ˛Cˇ k : .˛ C ˇ/2

(29.10)

To calculate the integral in (29.10), we use the change of variables (see [26], Section 24) ˛ D ;

ˇ D .1  /:

We get .2/ e  4 .kI M /

i D 16./2

Z1 J" . ; k; M /d ;

(29.11)

0

where

Z1 J" . ; k; M / D

2  d   " Y  iAj .m/ e  e iAj .M / e 

0

(29.12)

j D1

and A1 ./ D .1  /k 2  2 ;

A2 ./ D .1  /2 ;

 2 ¹m; M º:

For the calculation of the integral in (29.12), we use the relation Z1 0

 d   iA B C i" ;  e iB e " D ln e  A C i"

" > 0:

362

Chapter 29

On Renormalization in Perturbation Theory

As a result, we get (see Problem 25.1) .1  /M 2 C m2  .1  /k 2  i " m2  .1  /k 2  i " M 2  .1  /k 2 C .1  /m2  i " C ln : M 2  .1  /k 2  i "

J" . ; k; M / D ln

(29.13)

.2/ Thus, we have the decomposition of e  4 .kI M / into divergent and convergent parts for M ! 1: The convergence should be understood in the weak sense. This decomposition is not unique, and its choice is determined by the condition .2/ R;.2/ 4 .0; M / D 0; reg e  4 .0I M /  e

i.e., the first term of the Maclaurin expansion of the renormalized amplitude must be equal to zero. Remark 29.1. The choice of the zero point of the expansion of the vertex function .2/ 4 is made here for convenience. The ambiguity of the decomposition into convergent and divergent parts allows one to make a finite renormalization, i.e., to subtract or add a finite constant. For the considered model (25.3), the total vertex part 4 must be equal to i  if kj2 D m2 and ki kj D  13 m2 ; where m is the observable mass of particles. The physical substantiation of the choice of the expansion point and an equivalent procedure of finite renormalization are described in what follows. In the case considered, the renormalized amplitude is defined by the relation R;.2/ R;.2/ .2/ .2/ e 4 4 .k/ WD lim e .kI M / WD lim .e  4 .kI M /  e  4 .0; M //: (29.14) M !1

M !1

R;.2/ To determine the amplitude e 4 from relation (29.11), we subtract and add J" . ; 0; M / in relation (29.13). Then .2/ .2/ R;.2/ e 4 .kI M /;  4 .kI M / D a4 .M / C e

where .2/ a4 .M /

i D 8./2

Z1 d ln

.1  /M 2 C m2 ; Mm

0 R;.2/ e 4 .kI M / D

i 16./2

Z1 d ln

Œ.1  /M 2 C m2  .1  /k 2  i "m2 Œ.1  /M 2 C m2 Œm2  .1  /k 2  i "

d ln

Œ M 2  .1  /k 2 C .1  /m2  i "M 2 : Œ M 2 C .1  /m2 ŒM 2  .1  /k 2  i "

0

C

i 16./2

Z1 0

363

Section 29.1 Primitively-Divergent Diagrams. Pauli–Villars Method

It is easy to verify that R;.2/ R;.2/ w  lim e 4 .kI M / D e 4 .k/ M !1

i D 16./2

Z1 d ln

m2

0

m2 ;  .1  /k 2  i "

(29.15)

where the limit is understood in the weak sense. In the configuration space, the expression that corresponds to diagram (29.3) and contributes to S2 .x1 ; x2 / has the form .2/

R;.2/

722 W ' 2 .x1 /Œa4 .M /ı.x1  x2 / C 4

.x1 ; x2 I M /' 2 .x2 / W;

(29.16)

and the expression that contributes to the expansion of the coefficient function F4 .x1 ; : : : ; x4 / [relation (26.51)] in the second order of perturbation theory has the form .2/

F4 .x1 ; : : : ; x4 I M / X Œa42 .M /ı.x.1/  x.2/ /ı.x.1/  x.3/ /ı.x.1/  x.4/ / D 36 2P4 R;.2/

C ı.x.1/  x.3/ /ı.x.2/  x.4/ /4

.x.1/ ; x.2/ I M /:

(29.17)

Thus, the divergent part of the S-matrix is a quasilocal operator, and its support is concentrated at the points where the T -product appearing in the expression for the S-operator (26.36), (26.37) is not defined. By analogy, we determine the contribution of the second primitively-divergent diagram q1 q2

k1

k2 :

(29.18)

q3

The corresponding expression can be represented in the form (see Problem 29.2) .2/ .2/ .2/ R;.2/ e  2 .kI M / D a2 .M 2 / C b2 .M 2 /k 2 C e 2 .kI M /;

(29.19)

where .2/

a2 .M 2 / M 2

.2/

and b2 .M 2 / ln

M2 : m2

Thus, the regularized part can be determined as follows:  R;.2/ .2/ .2/ e 2 .k/ D lim e  2 .0I M /  2 .kI M /  e M !1 .2/ @e  2 .kI M /  jkD0 k  @k 

  2 .kI M / 1 @2 e   j k k : kD0 2Š @k  @k  .2/



(29.20)

364

Chapter 29

On Renormalization in Perturbation Theory

Similarly to (29.17), expression (29.19) has the following form in the coordinate space: .2/

.2/

Œ2 .x1 ; x2 I M / D a2 .M 2 /ı.x1  x2 / .2/

R;.2/

C b2 .M 2 /x1 ı.x1  x2 / C 2

.x1 ; x2 I M /:

(29.21)

.2/ R;.2/ Exactly as the vertex function 4 ; the function e 2 is not uniquely defined, which can be fixed by a finite renormalization, i.e., by using a certain term, which, in the case considered, is an arbitrary polynomial of degree !.G/ D 2 in k:

Remark 29.2. In Section 29.1, we have considered the Pauli–Villas regularization (29.2) for the theory '44 : This enabled us to obtain finite contributions for the ver.2/ .2/ tex function 4 .kI M / and the contribution 2 .kI M / of the self-energy diagram (29.18). However, e.g., for the model '46 ; substitution (29.2) is insufficient to guarantee the .2/ finiteness of 4 .kI M /: It is easy to show that the contribution of the diagram k1

k2

(29.22)

is logarithmically divergent even for M < 1: To guarantee its convergence it is now necessary to introduce two additional masses M1 and M2 : In the general case, the number of additional parameters M D .M1 ; : : : ; MN / can be arbitrary and is chosen so that the contribution of the corresponding diagram is convergent in the case where all Mi < 1; i D 1; 2; : : : ; N: For this purpose, a regularized casual function is chosen in the form 3 2 Z N X 1 Cj 1 c 5 e ikx ; (29.23)  .x/ D dk 4 2 DM 2 2  i" .2/4 m  k2  i " M  k j j D1 and the masses Mj and coefficients Cj must satisfy the conditions N X

Cj Mj2k2 D m2k2

for k D 1; 2; : : : ; N:

j D1 c as follows: Conditions (29.24) enable one to determine the function DM Z e ikx K.M; C/ c ; d k DM .x/ D Q N 2 2 .2/4 j D0 .Mj  k  i "/

(29.24)

365

Section 29.2 Degree of Divergence of Feynman Diagram

where M0 D m and K.M; C/ D

N Y

Mj2

j D1

N X Cj 1m Mj2 iD1 2

! :

Using Equations (29.24), one can easily verify (see [26], Section 16.4) that the coefficients Cj are uniformly (in Mj ; j D 1; 2; : : : ; N ) bounded by a certain constant independent of Mj : jCj j 6 const: Using this, we can prove that w  lim

Mj !1 j D1;2;:::;N

c DM .x/ D D c .x/

and separate the divergent terms by the method described above.

29.2

Degree of Divergence of Feynman Diagram

In this chapter, we consider the general case, which corresponds to a certain arbitrary Lagrangian. Let a model be such that the internal line l of the Feynman diagram G is associated with the propagator e c .pl / D 

m2

Zl .pl / ;  pl2  i "

(29.25)

where Zl is a polynomial in pl of degree rl D degZl .pl /:

(29.26)

The polynomial Zl .pl / may depend on spinor and vector indices, which are omitted here. In the general case, the interaction Lagrangian can also contain derivatives of the field u.x/; i.e., expressions of the form .@ /m u.x/; where m is the order of the derivative @ : Then, in the momentum space, the contribution to a diagram vertex contains the factor .p /m and certain constant matrices or structural constants (which are also not considered here). This situation is typical of Yang–Mills fields considered in what follows. Consider a strongly connected diagram of the nth-order, i.e., a diagram with n vertices, which has L internal lines with momenta p1 ; : : : ; pL and N external lines with momenta k1 ; : : : ; kN : We also consider the general case of (d D s C 1)-dimensional space-time, i.e., dp D dp 0 dp 1    dp s :

366

Chapter 29

On Renormalization in Perturbation Theory

Figure 29.1. Divergent diagrams with !.G/ < 0.

According to the Feynman rules, the contribution of this diagram is as follows: 1 0 Z L n Y Y Y X Zl .pl / #A # @ IG dp1    dpL ı p Œ.pl /j mj .lj / ; l 2 j j m2  pl  i " j D1 lD1

lj 2j

lj

where the sum with respect to lj is taken over all lines that are connected with vertex j; # # pl 2 ¹p1 ; : : : ; pL º if lj is an internal line of the diagram G; and pl 2 ¹k1 ; : : : ; kN º j j if lj is an external line. In the case of a strongly connected diagram, the .n  1/th integration is performed due to the .n  1/th ı-function, and we obtain the following relation for IG : !Z N L X Y Zl .Pl / dp1    dpL.n1/ Pm .p; k/ ki IG ı ; (29.27) 2 m  Pl2  i " iD1 lD1

where Pl are linear combinations of the internal momenta p1 ; : : : ; pL and the external momenta k1 ; : : : ; kN ; Pm .p; k/ is the polynomial of degree m D m1 .l11 / C    C # m1 .lL1 1 /C  Cmn .l1n /C  Cmn .lLnn /; mi .lji / is the power of the momentum .pl /j j of the line lj that enters into the vertex i; and m is the total number of derivatives that enter into all n vertices of the diagram. Applying the same reasoning as in the analysis of the integral in (29.5) to the integral in (29.27), we obtain the following asymptotics for the integral in (29.27) for large values of momenta: Z Z PL d d.LnC1/1 rl 2LCm lD1 (29.28) d D !.G/ ; where !.G/ WD

PL

lD1 .rl

C d  2/ C m  d.n  1/

(29.29)

is called the degree of divergence of the diagram G: If !.G/ > 0; then integral (29.28) is divergent, and these diagrams are called divergent. If !.G/ < 0; then this diagram

367

Section 29.2 Degree of Divergence of Feynman Diagram

is called conditionally convergent. Indeed, the condition !.G/ < 0 is only necessary but not sufficient for the convergence of the integral in (29.27). An example of a divergent diagram with degree !.G/ D 2 < 0 is given in Fig. 29.1 (for d D 4) (Problem 29.4). Each Feynman diagram can be divided into elementary generalized blocks: G D G1  G2      Gk : It is easy to see (Problem 29.7) that !.G/ D

k X

X

!.Gj / C

j D1

.rl C d  2/  d.k  1/;

l2Gn[k iD1 Gi

where the second sum corresponds to all internal lines that connect the blocks G1 ; : : : ; Gk : For the diagram G to be convergent, it is necessary and sufficient that, for each possible decomposition of the diagram G into blocks G1 ; : : : ; Gk ; the following conditions be satisfied: !.G/ < 0 and

!.Gj / < 0; j D 1; 2; : : : ; k:

The last statement is, in fact, a corollary of the Bogoliubov–Parasiuk theorem formulated in what follows. In conclusion, we analyze possible types of Lagrangians and their relationship with the degrees of divergence of diagrams. To this end, we introduce the notion of the degree of a vertex, namely, !i D

1 X .rl C d  2/ C mi  d; 2

(29.30)

lint 2i

where the sum is carried out only over the internal incoming (or outgoing) lines of the vertex i: One can easily rewrite relation (29.29) in the form !.G/ D

n n X 1X X .rl C d  2/ C m  d n C d D !i C d: 2 iD1 lint 2i

iD1

The degree of a vertex !i is maximal in the case where all lines at the vertex i are internal. The maximal degree !imax can be determined by the equality !i D !imax 

1 X .rl C d  2/: 2 lext 2i

(29.31)

368

Chapter 29

Then !.G/ D

n X

On Renormalization in Perturbation Theory

!imax C d 

iD1

1X .rl C d  2/: 2

(29.32)

lext

According to the definition of !imax ; it is clear that it depends only on the type of interaction. If !imax 6 0; then, for any n; one has !.G/ 6 d: Such theories are called renormalizable. If !imax > 0; then, for any integer N; there exists a diagram of order n D n.N / for which !.G/ > N: Such theories are called nonrenormalizable. We consider these problems in what follows.

29.3

Elimination of Divergences by the Method of Bogoliubov–Parasiuk R-Operation

In this chapter, we consider mathematical problems related to renormalization theory. As early as the 1940s, Tomonaga, Schwinger, Dyson, Salam, and other scientists realized that quantum field theory (in fact, perturbation theory) can be “rescued” only by the introduction of a procedure for the elimination of divergences arising in higher orders of perturbation theory. In 1952, Bogoliubov (see [20]) noted that these contributions contain products of singular casual functions, which should be considered as linear functionals on the corresponding spaces of test functions. In the classical theory of generalized functions, the product of two generalized functions is not defined (see, e.g., [207], Section 5.9]). Thus, an idea was proposed to connect a renormalization procedure with the definition of a new functional that is the product of singular generalized functions. According to relations (29.17) and (29.21), the divergent (as M ! 1) terms have supports on the surfaces on which the arguments of these functions coincide. This immediately enables one to define a limit functional on a special set M of test functions of a certain class K that vanish on these surfaces together .n/ with their partial derivatives up to a certain order. Indeed, if N .x1 ; : : : ; xN I M / is a functional of the form (29.17) or (29.21) regularized by the Pauli–Villars operation that corresponds to the Feynman nth-order diagram with N external lines and a test function ' 2 M is such that '.x1 ; : : : ; xN / D 0

for

xi D xj ;

i ¤ j;

i; j D 1; : : : ; N;

then there exists the limit .n/

.n/

.N ; '/ D lim .N .: : : I M /; '/ M !1 Z .n/ dx1    dxN N .x1 ; : : : ; xN I M /'.x1 ; : : : ; xN /; D lim M !1

which defines the limit functional  on M : Then, according to the well-known Hahn– Banach theorem, the functional  can be (not uniquely) extended to the entire space of functions K :

369

Section 29.3 Method of Bogoliubov–Parasiuk R-Operation

The procedure of subtraction of divergent terms can be considered as a constructive operation of extension of linear functionals defined by contributions of Feynman diagrams. Based on the works [24, 149] by Bogoliubov and Parasiuk (see also [25]), we briefly describe this procedure. Assume that the contribution of the (one-particle irreducible) Feynman diagram G with n vertices, L internal lines, and N external lines is proportional to the product of the causal functions c .xi  xj / .i; j D 1; 2; : : : ; n/: .n/

N .x1 ; : : : ; xN I y1 ; : : : ; yn / D

LCN Y

#

#

# .xi.l/  xj.l/ /;

(29.33)

lD1

where # .x  y/ D 1i c .x  y/ for each internal line and # .x  y/ D ı.x  y/ for each external line. The form of the causal function c .xi  xj / depends on the # specific form of the Lagrangian. Furthermore, xi D xi if the vertex is external, # # # xi D yi if the vertex is internal, and xi.l/ D xi.l 0 / if both lines l and l 0 .l ¤ l 0 / enter into the vertex i: Remark 29.3. To obtain the contribution of the diagram G to the analytic expression .n/ for the coefficient function FN .x1 ; : : : ; xN /; it is necessary to integrate .n/

N .x1 ; : : : ; xN I y1 ; : : : ; yn / with respect to the variables of each internal vertex. To define the product of distributions (29.33), it is necessary to replace the causal function c .x/ in (29.33) by a regularized (e.g., by the Pauli–Villars method) function c .x/ defined, in the general case, by analogy with (29.2) or (29.23). M If G is a primitively-divergent diagram of the form (29.3) or (29.18), then the Roperation is, in fact, defined by relations (29.14) or (29.20), i.e., the operator R.G/ acts according to the rule .2/ .2/ .2/  4 .kI M /  e  4 .0; M / R.G/e  4 .kI M / WD e

for diagram (29.3) or .2/ .2/ .2/  2 .kI M /  e  2 .0; M /  R.G/e  2 .kI M / WD e

.2/ @e  2 .kI M / jkD0 k  @k 

 2 .kI M / 1 @2 e jkD0 k  k  2Š @k  @k  .2/



for diagram (29.18). To write the operator R.G/ in abstract form, we define an operator M.G/ that associates the regularized (e.g., by the Pauli–Villars method) contribution .n/

0.n/

FG .k1 ; : : : ; kN I M / D ı.†ki /FG .k1 ; : : : ; kN I M /

370

Chapter 29

On Renormalization in Perturbation Theory

with the sum of the first terms of its Maclaurin expansion up to terms of order !.G/ inclusive (if !.G/ > 0). Recall that !.G/ is the degree of divergence of the diagram G: Thus, .n/

M.G/FG .k1 ; : : : ; kN I M / ´ 0.n/ ı.†ki /¹FG .k1 ; : : : ; kN I M /º!.G/ if !.G/ > 0; D 0 if !.G/ < 0;

(29.34)

where, for brevity, the braces with subscript !.G/ denote the multiple Maclaurin series up to terms of order !.G/: For diagrams (29.3) and (29.18), we have R.G/ D 1  M.G/: As noted above (see Remark 29.1), the subtraction procedure is ambiguous. For this reason, one fixes it by the choice of a point of expansion. To take this ambiguity into account, it is convenient to introduce an operator of finite renormalization P .G/ that .n/ acts on the contribution FG .k1 ; : : : ; kN I M / similarly to the operator M.G/ but, instead of the expansion of FG0 into a Maclaurin series, it transforms the Feynman amplitude FG0 into an arbitrary polynomial P!.G/ of degree !.G/; i.e., .n/

P .G/FG .k1 ; : : : ; kN I M / ´ ı.†ki /P!.G/ .k1 ; : : : ; kN / if !.G/ > 0; D 0 if !.G/ < 0:

(29.35)

The coefficients of the polynomial P!.G/ are chosen depending on the model with regard for the normalization of Feynman amplitudes (see Remark 29.1). Thus, for diagrams (29.3) and (29.18), the operator R.G/ has the following final form: R.G/ D 1  M.G/ C P .G/: (29.36) Operation (29.36) can be reduced to the subtraction of the Taylor series at an arbitrary point of expansion. Now consider the case where the diagram G is divergent but not primitively-divergent, i.e., it is a diagram of order n > 2 with divergent subblocks. Furthermore, the degree of divergence of the entire diagram can be negative. Intuitively, it is clear that, for the elimination of divergences of this diagram, it is necessary to eliminate divergences first in each subblock of the diagram G; then in each union of these subblocks, and finally in the union of all subblocks, i.e., in the diagram G itself. To construct the operator R.G/; we consider the decomposition of n vertices of the diagram G into arbitrary nonempty disjoint subsets of vertices G D G1      Gm ;

m 6 n:

Section 29.3 Method of Bogoliubov–Parasiuk R-Operation

371

On the Feynman amplitude corresponding to the block Gi ; we define operators .Gi / by the recurrence relations .Gi / D 1 and

X

.Gi / D M.Gi /

if

jGi j D 1

.Gi1 /    .Gik / C P .Gi /;

(29.37)

(29.38)

Gi1 Gik DGi

where the sum is taken over all decompositions of the set Gi into nonempty disjoint subsets. In (29.37), jGi j is the number of vertices in the subgraph Gi : Then the Roperation is defined by the operator X .Gi1 /    .Gim /; (29.39) R.G/ D 1 C Gi1 Gim

and, according to the Bogoliubov–Parasiuk theorem, the limit lim R.G/FG .I M / WD FGR ./

M !1

exists as a generalized function and is a linear continuous functional (generalized function) over the space of Schwartz test functions S .R4jGj / (see, e.g., [207], Chapter 8). The proof of this theorem, which is rather complicated, is beyond the scope of this book; it can be found in [149] and [25]. However, the methods used in the proof of this theorem are not optimal. The method of mathematical induction used in the proof is rather awkward, and the proof itself has many unclear moments. For this reason, the Bogoliubov–Parasiuk R-operation has not been well understood. In 1966, using a new technique, Hepp [89] substantially simplified and completed the proof of the theorem. Although this proof is also not simple (it is given in detail in [90]), the R-operation has become very popular since then and was called the Bogoliubov–Parasiuk–Hepp R-operation. Later, after the paper [228] by Zimmermann (see also [229]), who gave a solution to the recurrence equation (see also [227]) and reformulated the method, it was called the Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) R-operation. For details of the proof and the history of these problems, see [225, 226]. In conclusion, we present an example of application of the R-operation to a specific divergent diagram that is not primitively-divergent and make several remarks. Example 29.4. Consider the diagram shown in Fig. 29.2. It is easy to verify that the contribution of the entire diagram G D G123 is divergent, and, by relation (29.29), it has the degree of divergence !.G/ D 0: The block of the diagram that consists of vertices 2 and 3 and two internal lines also has the degree of divergence !.G23 / D 0: There exist only two nontrivial decompositions G1  G2  G3

and

G1  G23 ;

372

Chapter 29

On Renormalization in Perturbation Theory 2

1

3

Figure 29.2. Divergent diagram with divergent subblock.

where Gi ; i D 1; 2; 3; are simple vertices and G23 is a divergent subblock. According to (29.37) and (29.38), we have .Gi / D 1;

i D 1; 2; 3;

.G23 / D P .G23 /  M.G23 /; and .G/ D P .G/  M.G/Œ1 C .G23 /: Thus, the operator that corresponds to the complete subtraction (R-operation) has the form R.G/ D 1  M.G23 / C P .G23 / C P .G/  M.G/Œ1 C P .G23 /  M.G23 /: Without finite renormalization (i.e., for P .G/ D P .G23 / D 0), we obtain the relation R.G/ D .1  M.G//.1  M.G23 //: This means that it is necessary, first, to perform subtraction in the divergent block G23 and then to apply an analogous operation to the contribution obtained after the operation (1  M.G23 /). Remark 29.5. To define the action of the operator R.G/ in the coordinate space, it suffices to write the action of the operator M.i / on contribution (29.33), where i coincides either with the diagram G itself or with its subblock. This action can be defined by the relation .n/

M.i /N .x1 ; : : : ; xN I y1 ; : : : ; yN I M / 8 Q # # # 0; j1 jni D l…L.i / : 0 if !.i / < 0;

(29.40)

373

Section 29.3 Method of Bogoliubov–Parasiuk R-Operation #

#

where i is the subblock that combines the vertices ¹xj1 ; : : : ; xjn º connected by lines i l 2 L.i /; and g!.i / is the inverse Fourier transform of the polynomial M.i /Fi .kj1 ; : : : ; kjni I M / (as a generalized function)[see (29.34)] if !.i / > 0: The operator P .i / is defined by analogy with regard for (29.35). Remark 29.6. Relation (29.39) is not convenient for applications and for the proof of the main theorem (Bogoliubov–Parasiuk theorem). In [227] and, later, in [228], it was shown that the recurrence equations (29.37) and (29.38) can be solved and the R-operation can be represented in the form : R.G/ D :: .1  M.1 / C P .1 //.1  M.2 / C P .2 // :    .1  M.m / C P .m // :: ;

(29.41)

where 1  G and 2 ; : : : ; m are all divergent subgraphs of the graph G: Following : : Zav’yalov [225], we call the sign ::       :: the “three-point” product, in which the factors are arranged according to the following rules: (1) if i and j partially overlap, i.e., i \j ¤ ¿ but neither j nor j completely lies in the other, then :: : : Q.i1 /    Q.i /    Q.j /    Q.ik / ::  0; where Q.i / is one of the operators M.i / and P .i /; (2) if the product contains i such that Q.i / D P .i / and j such that j i ; then this product also becomes equal to zero; (3) if j i ; then the operator Q.j / is located to the right of Q.i / in the product, i.e., it is necessary to act first by the operator Q.j / and then by the operator Q.i /: If i \ j D ¿; then the order of factors is of no importance. To take rules (1)–(3) into account, it is necessary to perform multiplication in all pairs indicated in (1) and (2). Then, e.g., for i and j indicated in (1), it is necessary to apply the single operator .1  M.i /  M.j // to the contribution of the diagram G instead of the successive application of the operators .1  M.i //.1  M.j // : : because ::    M. /M. /    ::  0: i

j

Relation (29.41) can also be simplified if one redefines the operator M by displacing the center of expansion of Feynman amplitudes in powers of momenta, i.e., by using the Taylor series instead of the Maclaurin one. Assume that the operator T .i / replaces the Feynman amplitude of the subgraph i by a polynomial of degree !.i / in variables .u  u0 /; where u are certain scalar combinations of external momenta of the subgraph i and u0 are certain values of u

374

Chapter 29

On Renormalization in Perturbation Theory

specially chosen using physical reasoning. Then relation (29.41) can be rewritten in the form : : G.G/ D :: .1  T .1 //       .1  T .m // :: : (29.42) However, it is worth noting that operation (29.41) has more arbitrary constants (coefficients of the polynomials P .i /) than operation (29.42). Indeed, the displacement of the point of subtraction in (29.42) can always be taken into account by a proper choice of operators of finite renormalization P .i /: The converse statement is not true.

29.4

R-Operation and Counterterms of a Lagrangian

The Bogoliubov–Parasiuk R-operation eliminates the mathematical difficulties related to divergences in each order of perturbation theory for the scattering matrix. However, at first sight, this operation looks like a certain additional “recipe” for the quantum theory of interacting fields whose physical motivation is not quite clear. In this section, we show (though formally from the mathematical point of view) that divergences can be eliminated by the introduction of a (possibly even infinite) sequence of quasilocal operators (counterterms) into a Lagrangian. Definition 29.7. A quasilocal operator is defined as an operator-valued functional (operator-valued generalized function) of the form X 1 n K˛1 ;:::;˛n .x1 ; : : : ; xn / W uk1I˛ .x1 /    uknI˛ .xn / W ; ƒn .x1 ; : : : ; xn / D n 1 ˛1 ;:::;˛n

(29.43) where uiI˛i .xj / is a free field of the i th type (i D 1; n), ki are integer numbers, and the coefficient functions K˛1 ;:::;˛n .x1 ; : : : ; xn / have the form K˛1 ;:::;˛n .x1 ; : : : ; xn / D P˛1 ;:::;˛n .      ;

@ ;       / ı.x1  x2 /    ı.x1  xn /: @x (29.44) .˛/

@ Here, P˛1 ;:::;˛n .      ; @x ;       / is a polynomial in all possible derivatives @=@xj ; j D 1; 2; : : : ; n; ˛ D 0; 1; 2; 3; with constant coefficients. Also note that if the field ui .x/ is a spinor field or a complex field, then it appears  in Equation (29.43) in the form of combinations K ;˛;˛0 ; N ˛ ˛0 or K ;˛;˛0 ; u˛ u˛0 : Furthermore, if one of ki is equal to zero, then the function K˛;:::;˛ does not have the corresponding index ˛i :

Definition 29.8. We say that quasilocal operators ƒn .x1 ; : : : ; xn /; n D 1; 2; : : : ; are of the same type if they differ from each other only by their coefficient functions, i.e.,

Section 29.4

R-Operation and Counterterms of a Lagrangian

375

the number and the type of the operator fields under the sign of normal product on the right-hand side of Equation (29.43) are the same for all quasilocal operators of a given type. In relation (29.43), the type of a quasilocal operator is determined by the types of the fields ui and their powers ki in the product. In the general case, a quasilocal operator is a sum of quasilocal operators of different types. It is easy to see that the integration of a quasilocal operator ƒ.x1 ; : : : ; xn / with respect to x2 ; : : : ; xn yields an ordinary local operator whose analogs appear in the definition of Lagrangians (5.54) and (6.114). Now let us analyze the form of the S-operator in each order of perturbation theory from the viewpoint of indeterminacy of the T -product in the definition of Sn (see (26.36) and (26.37)): Sn .x1 ; : : : ; xn / D i n T .LI .x1 /    LI .xn //:

(29.45)

In Chapter 26 (see Remark 26.2), we have established that this relation is ambiguous if some arguments x1 ; : : : ; xn coincide, e.g., xi1 D xi2 for i1 ¤ i2 or xi1 D xi2 D xi3 ; etc. Thus, instead of Equation (29.45) in the definition of S operator, we can consider the relation Sn .x1 ; : : : ; xn / D i n T .LI .x1 /    LI .xn // C Sn .x1 ; : : : ; xn /;

(29.46)

where Sn are defined in terms of quasilocal operators ƒ2 .x1 ; x2 /; : : : ; ƒn .x1 ; : : : ; xn / of the form (29.43). These operators can be chosen so that Equation (29.46), similarly to Equation (29.45), satisfies necessary conditions for relativistic invariance, unitarity, causality, etc. To verify this statement, we refer the reader to the book [26] (Chapter 18) by Bogoliubov and Shirkov for the axiomatic approach to the construction of the S-matrix in powers of interaction. In connection with representation (29.46), the natural question arises of whether it is possible to choose operators Sn .x1 ; : : : ; xn / so that the divergences that appear due to the use of Equation (29.45) are eliminated. A positive answer to this question follows from the analysis of these divergences and the structure of the R-operation considered above. We begin with the case n D 2: The expression i 2 T .LI .x1 /LI .x2 // contains contributions of all primitively-divergent diagrams of the second order. The application of the Bogoliubov–Parasiuk R-operation to these diagrams is equivalent to the subtraction of quasilocal operators from them. These operators can be defined with the use of the operator M.G/ (see relation (29.40)). Thus, it is necessary to set S2 .x1 ; x2 / D i ƒ2 .x1 ; x2 /; where ƒ2 .x1 ; x2 / is the sum of all quasilocal operators each of which corresponds to the expression (29.47) M.Gi / .2/ .x1 ; x2 / D g!.Gi / .x1 ; x2 /;

376

Chapter 29

On Renormalization in Perturbation Theory

Gi ; i D 1; 2; : : : ; are all primitively-divergent diagrams in the expression for i 2 T .LI .x1 /LI .x2 //; and !.Gi / are their degrees of divergence. The function g!.Gi / is the coefficient function of the corresponding quasilocal operator. This function is constructed according to the “recipe” described in Remark 29.5. For example, for interaction (25.3), according to Equations (29.16) and (29.21), the quasilocal operator ƒ2 .x1 ; x2 / has the form .2/

.4/

ƒ2 .x1 ; x2 / D ƒ2 .x1 ; x2 / C ƒ2 .x1 ; x2 /; .2/

.2/

ƒ2 .x1 ; x2 / D i 722 a4 ı.x1  x2 / W ' 2 .x1 /' 2 .x2 / W ; .4/

.2/

.2/

ƒ2 .x1 ; x2 / D i 962 Œa2 ı.x1  x2 / C b2 x1 ı.x1  x2 / W '.x1 /'.x2 / W ; (29.48) .2/

.2/

.2/

where the coefficients a4 ; a2 ; and b2 are the limit values of the constants .2/ .2/ .2/ a4 .M /; a2 .M /; and b2 .M / as M ! 1: Thus, these coefficients are infinite constants. Therefore, the formal expression for S2 .x1 ; x2 / determined with the use of Equation (29.46) should be understood as follows: First, one considers regularized (e.g., by the Pauli–Villars method) Feynman propagators, then the divergent terms that are cancelled with the corresponding counterterms are selected, and, finally, the intermediate regularization is removed. The numerical factors 72 and 96 appear due to the reduction of the operator S2 .x1 ; x2 / to the normal form, i.e., the application of the Wick theorem for the T -product. Thus, by the proper introduction of a quasilocal operator into the expression for S2 .x1 ; x2 / (see (29.46)), one can eliminate divergences in Equation (29.45). Now consider (29.45) for n D 3: All Feynman third-order diagrams can be obtained from second-order diagrams by connecting vertices in various ways, namely, the disconnected union of vertices, which corresponds to the expression W LI .x1 /LI .x2 /LI .x3 / W ; the connection of a vertex with the use of a single internal line, which corresponds to the pairing of one of operators in LI .x3 / with some operator in T .LI .x1 /LI .x2 //; etc. It is clear from this construction that the diagrams formed by the connection of vertices to primitively-divergent second-order diagrams remain divergent because they have a divergent subblock. According to the structure of the R-operation, to eliminate the divergence corresponding to this subblock it is necessary to subtract the contribution of this diagram in which the contribution associated with the divergent subblock is replaced by an expression of the form (29.47) from the contribution of the entire original diagram. In fact, this implies that the structure of counterterms corresponding to this subtraction has the form i 2 T .LI .x1 /ƒ2 .x2 ; x3 // C i 2 T .LI .x2 /ƒ2 .x1 ; x3 // C i 2 T .LI .x3 /ƒ2 .x1 ; x2 //:

(29.49)

Section 29.4

377

R-Operation and Counterterms of a Lagrangian

x1

x2

x3

Figure 29.3. Third-order diagram with second-order divergent subblock. x2 x1

x3

x2

x3

x1

(a)

(b)

Figure 29.4. Diagrams with two divergent subblocks.

Counterterms (29.49) eliminate the divergent contributions of the diagrams from the expression for T .L .x1 /L .x2 /L .x3 // that have only one divergent second-order subblock. An example of this third-order diagram is given in Fig. 29.3 for interaction (25.3). The corresponding counterterm is contained in the expression .2/

T .ƒ2 .x1 ; x2 / W ' 4 .x3 / W/: However, in the case of diagrams presented in Fig. 29.4 (a) and Fig. 29.4 (b), for the elimination of divergences in only one block it suffices to use counterterms (29.49) .4/ with ƒ2 .xi ; xj /; i < j; i; j D 1; 2; 3; instead of ƒ2 .xi ; xj /: The result of this subtraction can be represented in the form of diagrams given in Fig. 29.5, where the bold vertex corresponds to the contribution obtained by the application of the operation 1  M.G12 / to the contribution of diagram (29.3). However, the obtained .3/ .3/ contributions .1  M.G12 //4 and .1  M.G12 //2 are again divergent. For the complete elimination of divergences, it is necessary to use the operation 1M.G123 /; which is equivalent to the introduction of the quasilocal counterterm .4/

.2/

ƒ3 .x1 ; x2 ; x3 / D ƒ3 .x1 ; x2 ; x3 / C ƒ3 .x1 ; x2 ; x3 /; .4/

(29.50) .3/

.2/

where ƒ3 is constructed on the basis of the contribution .M.G12 //4 ; and ƒ3 .3/ is constructed on the basis of the contribution .M.G12 //2 : If, in addition to the obtained divergences, new divergences emerge in the third order, i.e., if the contribution  .3/ .x1 ; x2 ; x3 / of the third-order diagram G; as a whole, has the degree of divergence !.G/ > 0; then it is necessary to subtract the expression M.G/ .3/ .x1 ; x2 ; x3 /; which must be included into the counterterm ƒ3 .x1 ; x2 ; x3 /:

378

Chapter 29

On Renormalization in Perturbation Theory

x3

x1

x1

x3

x2

x2

(a)

(b)

Figure 29.5. Diagrams of Fig. 29.4 after one subtraction.

Thus, in the third order of perturbation theory, the counterterm S3 .x1 ; x2 ; x3 / has the structure S3 .x1 ; x2 ; x3 / D iT ¹LI .x1 /ƒ2 .x2 ; x3 / C LI .x2 /ƒ2 .x1 ; x3 / C LI .x3 /ƒ2 .x1 ; x2 /º C i ƒ3 .x1 ; x2 ; x3 /: By analogy, one can eliminate divergences in all orders of perturbation theory. Thus, the Bogoliubov–Parasiuk R-operation is equivalent to the formal definition of operator (29.45) by Equation (29.46), i.e., the introduction of a sequence of quasilocal operators ƒn .x1 ; : : : ; xn /: The operator Sn has the following general form: Sn .x1 ; : : : ; xn / D

n X ik kŠ

kD1

X

0 X

m1 ;:::;mk ;mi >1 m1 CCmk Dn

2Pn

T .ƒm1 .x.1/ ; : : : ; x.m1 / //

   ƒmk .x.m1 CCmk1 C1/ ; : : : ; x.n/ /;

(29.51)

where ƒ1 .x/ WD LI .x/ and the sum †0 over  2 Pn means the sum over all possible permutations of indices ¹1; : : : ; nº between the groups of m1 ; m2 ; : : : ; mk indices. This sum has nŠ=m1 Š    mk Š terms. At first sight, the situation is the same as in the case of application of the additional process of R-operation, i.e., for the complete determination of the S -matrix, it is not sufficient to define an interaction Lagrangian; it is also necessary to define an infinite sequence of quasilocal operators ƒn : However, it turns out that the combinatorial structure of (29.51) enables one to rewrite the expression for the complete S-operator in the form of an ordinary T product with a more complicated Lagrangian that contains the entire sequence of quasilocal operators ¹ƒn º: In other words, the following combinatorial lemma is true: Lemma 29.9. The S-operator S.g/ D 1 C

Z 1 X 1 Sn .x1 ; : : : ; xn /g.x1 /    g.xn /dx1    dxn nŠ

nD1

Section 29.5 Renormalizable and Nonrenormalizable Theories

379

with coefficient operators (29.51) can be represented as an ordinary T -product, namely,  R  S.g/ D T e i LI .xIg/dx Z 1 X im D T .LI .x1 I g/    LI .xm I g//dx1    dxm ; mŠ mD0

where the new Lagrangian LI .xI g/ is defined by the infinite series LI .xI g/ D

Z 1 X 1 ƒm .x; x1 ; : : : ; xm1 /g.x/g.x1 /    g.xm1 /dx1    dxm1 : (29.52) mŠ

mD1

Proof. For the proof of the lemma, see Problem 29.8. Lemma 29.9 enables one to define the S-operator in the standard way by using relation (26.39) with more complicated Lagrangian (29.52) containing a sequence of quasilocal operators.

29.5

Classification of Interactions: Renormalizable and Nonrenormalizable Theories

We now analyze a relationship between the type of interaction (which determines the form of vertices in Feynman diagrams), the degrees of divergence of diagrams, and the type of counterterms (see Definition 29.2) that should be introduced into Lagrangian (29.52) to eliminate all divergences in the S-matrix. At the end of Section 29.2, the theories whose interaction Lagrangians lead to Feynman diagrams with a maximum degree of a vertex !imax 6 0 have been called renormalizable (see (29.30) and (29.31)). To substantiate this name, we consider relation (29.32). Taking into account that !imax 6 0 and considering the worst case where all rl  0; for any diagram G with N external lines we get 1 !.G/ 6 N C d  Nd: 2 For the real 4-dimensional space-time .d D 4/; we have !.G/ 6 4  N: This inequality implies that only diagrams G with N 6 4 external lines can have the degree !.G/ > 0: Thus, according to Definition 29.2, only a finite number of types of counterterms ƒn .x1 ; : : : ; xn / corresponding to different values of N is possible.

380

Chapter 29

On Renormalization in Perturbation Theory

The counterterms corresponding to divergent diagrams with N external lines have the form / ƒ.N n .x1 ; : : : ; xn / X K˛.n/;N .x1 ; : : : ; xn / W u1I˛i1 .xi1 /    uN I˛iN .xiN / W; D i ;:::;˛i ˛i1 ;:::;˛iN

1

N

where K .n/;N n and  is the coupling constant. Taking into account the form of the coefficient functions (29.44), we can perform the integration with respect to dx1 ; : : : ; dxn1 in (29.52) by “transferring” the action of derivatives to the operators of the field (for g.x/  1; xn D x) and the formal summation of the coefficient functions with respect to the powers n for each type of counterterms. As a result, we obtain a new Lagrangian, which differs from the “bare” Lagrangian L .x/ by a finite number of infinite counterterms: X X # # / K˛.N W u1I˛1 .x/    uN I˛N .x/ W ; LI .x; 1/ D LI .x/ C 1 ;:::;˛N N >0 ˛1 ;:::;˛N

#

where uiI˛ .x/ coincides either with the field uiI˛ .x/ or with the derivative of this field with respect to @ : In what follows, we show that the constants K .N / can be associated with renormalization constants of the main physical parameters of the corresponding model (i.e., mass, charge (or coupling constant), etc.). For this reason, theories with a finite number of counterterms are called renormalizable. In the case where !imax > 1; it is always possible to construct a Feynman diagram with !.G/ > 0 for an arbitrarily large number of external lines N: This implies that the number of types of counterterms that must be included in the definition of Lagrangian (29.52) is infinite. For this reason, these theories are called nonrenormalizable. For an analysis of specific model examples of local interactions, see [26], Chapter 28. One should also note the idea of association of interactions of the second kind (i.e., nonrenormalizable interactions) with the nonlocal character of this type of interactions (see [26], Section 28.3).

29.6

Relationship between Counterterms and the Renormalization of Main Constants of the Theory

In the previous chapter,we have noted that the number of counterterms in renormalizable theories is finite. After the formal summation of each type of counterterms in all orders of perturbation theory, we obtain an additional Lagrangian L of the same structure as the original total Lagrangian L .x/ but with coefficients dependent on parameters of intermediate regularization (masses Mi in the case of Pauli–Villars regularization). As Mi ! 1; these coefficients diverge and compensate for divergences of the perturbation theory constructed on the basis of the Lagrangian LI .x/; which

381

Section 29.6 Renormalization of Main Constants of the Theory

is a perturbation of the Lagrangian L0 .x/ .L .x/ D L0 .x/ C LI .x//: It turns out that, for most renormalizable theories, this process of elimination of divergences is equivalent to the renormalization of the initial, or “bare,” constants (mass, coupling constant, etc.). This fact was first noted by Dyson [43]. He also developed a scheme for the construction of renormalizable Green functions and matrix elements of the S-operator. We illustrate this scheme for interaction (25.3). We write the total Lagrangian that describes the self-interaction of a real scalar field in the form L .x/  L .'.x/I m2 I / D

1 W '.x/Π m2 '.x/ W  W ' 4 .x/ W : 2

(29.53)

For now, we do not associate the field '.x/ and parameters m and  with actual particles and regard them as certain variables. Furthermore, in Equation (29.53), the term corresponding to the free Lagrangian has a form different from the standard one (5.54). For convenience, we transfer the derivative @ (which is possible because L .x/ is in the expression for the action) and use the definition of d’Alembert operator (5.5). The analysis of divergent diagrams carried out in Section 29.1 shows that, in addition to vacuum diagrams, diagrams with two and four external lines are divergent. It is clear that, in higher orders of perturbation theory, though diagrams with six (and more) external lines have the degree of divergence !.G/ < 0; they can contain divergent subblocks. Taking relations (29.16) and (29.21) into account, one can easily obtain the following form of counterterms that must be included in Lagrangian (29.53) in the second order of perturbation theory (see also relation (29.48)): .2/

.2/

L .2/ .x/ D 962 b2 W '.x/'.x/ W 962 a2 W '.x/2 W .2/

 722 a4 W '.x/4 W :

(29.54)

Counterterm (29.54) corresponds to the first terms of the Maclaurin expansion (up .2/ .2/ to order !.G/) of the regularized functions e  4 .kI M / and e  2 .kI M /, which are analytic expressions of diagrams (29.3) and (29.18) with an ordinary propagator replaced by the regularized expression (29.7). Taking into account Remark 29.1, one can replace the Maclaurin expansion by the Taylor expansion at some proper point of expansion. According to physical arguments, this is a point at which the external momenta lie on the mass surface k 2 D .k 0 /2  k2 D m2r ; where mr is the mass of an actual observable particle. In the case where mr D 0; it is reasonable to take a spacelike point with k 2 D 2 < 0: Setting m D mr and  D r (r is an observable coupling constant), we rewrite the counterterm in the form (in what follows, the index “r” is omitted) 1 .2/ .2/ 1 .2/ .2/ L .2/ .x/ D K2 B2 W '.x/Œ  m2 '.x/ W  2 K2 A2 W ' 2 .x/ W 2 2 .2/ .2/  K4 A4 W ' 4 .x/ W : (29.55)

382

Chapter 29

On Renormalization in Perturbation Theory

Counterterm (29.55) corresponds to the quasilocal operator ƒ2 .x1 ; x2 / and is the .2/ .4/ sum of quasilocal operators of two types ƒ2 and ƒ2 : In the third order of perturbation theory, the counterterm ƒ3 .x1 ; x2 ; x3 / in the interaction Lagrangian (29.52) is expressed in terms of operator (29.50), which has the same structure as countert.3/ .3/ .3/ .3/ .3/ erm (29.55) and differs only by the coefficients K2 ; K4 ; B2 ; A2 ; and A4 : Formally summing up the obtained expressions in all orders with respect to ; we obtain the following final form of the counterterm for Lagrangian (29.53): 1 1 L .x/ D .Z'  1/ W '.x/Œ  m2 '.x/ W  Z' ım2 W ' 2 .x/ W 2 2 4  .Z  1/ W ' .x/ W; where the constants Z' ; Z ; and ım2 are represented in the form of formal series as follows: Z4 D 1 C Z' ım2 D

1 X

1 X

.n/

.n/

n K2 B2 ;

Z D 1 C

nD2

1 X

.n/

.n/

n1 K4 A4 ;

(29.56)

nD2

.n/ .n/ n K2 A2 :

(29.57)

nD2 .n/

.n/

.n/

.n/

.n/

Remark 29.10. The constants B2 D B2 .M /; A2 D A2 .M /; and A4 D .n/ A4 .M / tend to infinity as M ! 1: However, even for M < 1; series (29.56) and (29.57) are not absolutely convergent for any  because the combinatorial factors .n/ .n/ K2 and Kn increase with n as .nŠ/2 (Wick theorem). Moreover, the summation of divergent series for B2 D B2 .M /; A2 D A2 .M /; and A4 D A4 .M / may give finite expressions even for M ! 1: For example, as M ! 1; the left-hand side of the series e

 M 2

1 X ./n M 2n ; D nŠ

 > 0;

nD0

tends to zero, whereas its right-hand side formally tends to infinity. Therefore, the expressions for all renormalization constants must be considered within the framework of perturbation theory exclusively as formal series in : Thus, analyzing the divergent diagrams of perturbation theory in model (29.53) and selecting all divergent parts, we construct the Lagrangian Lr .x/ D L .'.x/I m2 I / C L .x/:

(29.58)

Then the perturbation theory constructed for the interaction Lagrangian LIr .x/ D  W ' 4 .x/ W CL .x/ is free of ultraviolet divergences.

(29.59)

Section 29.6 Renormalization of Main Constants of the Theory

383

We now show that the introduction of the counterterm L .x/ can be reduced to the renormalization of main parameters of the theory (field, mass, and coupling constant). To this end, we rewrite Lagrangian (29.58) in the form 1 1 Lr .x/ D W '.x/Z' Œ  m2 '.x/ W  ım2 Z' W ' 2 .x/ W Z W ' 4 .x/ W : 2 2 (29.60) It is easy to see that this Lagrangian has the same form as (29.53). Let L 0 .x/ WD L .' 0 .x/I m20 I 0 / denote the Lagrangian corresponding to certain “bare” values of the field ' 0 .x/; mass m0 ; and coupling constant 0 : We see that L 0 .x/ D Lr .x/ if we set 1=2 0

'r .x/ D '.x/ D Z4

' .x/;

(29.61)

r D  D Z 1 Z42 0 ;

(29.62)

m2r

(29.63)

2

Dm D

m20

2

 ım :

The perturbation theory constructed on the basis of Lagrangian (29.59) does not have ultraviolet divergences. This means that the perturbation theory constructed on the basis of the Lagrangian L 0 .x/ also does not have divergences if the “bare” values of the field, mass, and coupling constant in the Lagrangian are replaced by the corresponding ones renormalized according to rules (29.61)–(29.63). This leads to the following practical question: How to construct a new perturbation theory for which analytic expressions corresponding to specific matrix elements (or Green functions) are expressed in terms of renormalized quantities? To formulate these rules, we first note a simple result that follows directly from the form of the Lagrangians L 0 .x/ (see (29.53) for  D 0 ) and Lr .x/ [see (29.60)]. A formal perturbation theory in terms of the contributions of Feynman diagrams for the theory with Lagrangian (29.60), which, according to the examples presented above, should not have divergences, can be obtained from the perturbation theory constructed in the theory with Lagrangian L 0 .x/ by replacing the propagator 1i D c .x  yI m20 / D i.  m20 /1 .x; y/ by the propagator iZ'1 .  m20 /1 .x; y/; m20 D m2 C ım2 , the vertex contribution .i 0 / by the contribution .iZ /, and, in the case of diagrams with external lines on the mass surface (for matrix elements of the S -matrix), 1=2 the factor ' 0 .x/ by '.x/ D Z' ' 0 .x/: We write the indicated replacements in the momentum space: Z'1 1 ! ; .2/4 i.m20  p 2  i "/ .2/4 i.m20  p 2  i "/ 0 ! Z D Z'2 0 ;

'Q 0 .p/ ! 'Q 0 .p/ D Z'1=2 :

(29.64)

384

Chapter 29

On Renormalization in Perturbation Theory

Analyzing the expansion of the total Green function G2 .x; y/  G20 .x  y/ (see e 2 .p 2 I m2 ; 0 /) in a series in 0 ; e 2 .p/  G (28.44)) (or, in the momentum space, G 0 one can easily verify that replacement (29.64) is equivalent to the replacement of the contribution of each diagram by the same contribution multiplied by the factor Z'1 ; in which  must again be expressed in terms of the nonnormalized constant according to (29.62), i.e., it is necessary to perform the substitution e 2 .p 2 I m20 ; 0 / WD G e r2 .p 2 I m2 ; /: e 2 .p 2 I m20 ; 0 / ! Z'1 G G

(29.65)

Remark 29.11. By analogy, we obtain a rule of renormalization of the coefficient e 2 .pI m2 ; 0 /; namely, function F 0 e 2 .p 2 I m20 ; 0 / ! Z' F e 2 .p 2 I m20 ; 0 / WD F e r2 .p 2 I m2 ; /; F

(29.66)

which agrees with the rule of renormalization of the main quantities (29.61)– (29.63). Indeed, for a matrix element, it is necessary to consider the operator e 2 .p 2 I m2 ; 0 /' 0 .p/: Then ' 0 .p/F 0 e 2 .p 2 I m20 ; 0 /' 0 .p/ D 'r .p/F e r2 .p 2 I m2 ; /'r .p/: ' 0 .p/F

(29.67)

To establish the relationship of a regularized vertex part (amputated oneparticle irreducible Green function of four variables (see Section 5.6)) e  r4 with e  4 , we define it, for convenience, in the form e  4 .p1 ; p2 ; p1 C p2  p3 / D

1 X

.n/

W4 .p1 ; p2 ; p1 C p2  p3 /;

(29.68)

nD1 .n/

where W4 is the sum of all strongly connected (one-particle irreducible) nth-order diagrams with four external lines after the separation of the ı-function ı.p1 C p2  p3  p4 /; which corresponds to the law of conservation of 4-momentum. If we now consider the perturbation theory for the total Green functions with N external lines GN .x1 ; : : : ; xN / or the coefficient functions of the S -matrix FN .x1 ; : : : ; xN /; then it is easy to see that the sum of contributions of diagrams in all orders can be reduced to the sum of contributions of so-called skeleton diagrams in which each internal line is associated with the total Green function G2 .x1 ; x2 / and each vertex is associated with the factor .0 /4 .x1 ; x2 ; x3 ; x4 ;m20 ; 0 /: If these functions are replaced by the renormalized functions G2r and 4r ; then these diagrams must be convergent. It is also easy to verify that the skeleton diagrams corresponding to GN with N > 5 in which the total Green function G2 is replaced by the free Green function G20 and the total vertex 4 is replaced by unity do not have divergences. Thus, the behavior of the renormalized functions G2r and 2r must be at least no worse than  r4 ; by the behavior of free functions. To determine the relationship between e  4 and e analogy with (29.65) and (29.66) we set y e  4 .   I m20 ; 0 /;  r4 .   I m2 ; / WD Z x Z4 e

where the exponents x and y are unknown.

Section 29.6 Renormalization of Main Constants of the Theory

385

Now consider an arbitrary skeleton (strongly connected) diagram with n vertices  4 .   I m20 ; 0 /; L internal each of which is associated with the contribution .0 /e e 2 .I m2 ; 0 /; and N lines each of which is associated with the total Green function G 0 external lines .N > 5/ each of which is associated with the external factor 'Q 0 .p/: The contribution of this diagram can be written in the following symbolic form: .n/

WN .p1 ;    / Z

n



 n e 2 .   I m20 ; 0 / L 'Q 0 ./ N d M K; e D 0  4 .   I m20 ; 0 / G where d M K .M D 4.L.n1/// means integration with respect to the internal mo 4; menta that remain after the elimination of the .n1/th ı-function. We express 0 ; e e 2 ; and 'Q 0 in terms of the corresponding renormalized quantities (29.61), (29.65), G and (29.68), taking into account that, in the theory W ' 4 W; exactly four lines intersect at every vertex and the order of a diagram is related to the numbers of internal lines L and external lines N as follows: 1 1 n D L C N: 2 4 As a result, we obtain n.1x/

Wn.n/ .p1 ;    / D Z

nyCL N 2n

2 Z' Z

r n r  e 2 .   I m2 ; / L .'.// e  4 .   I m2 ; / G  n Q N d M K:

.n/

Thus, for WN to be completely expressed in renormalized quantities, one should set x D 1 and y D 0: Then relation (29.68) takes the form e  4 .   I m20 ; 0 /:  r4 .   I m2 ; / D Z e

(29.69)

To complete the process of renormalization, it is necessary to evaluate the indetermie 2 and Z  e e r and nate forms Z'1  G  4 ; i.e., to construct the renormalized quantities G 2 e  r4 : In Section 28.6, we have established the relationship between the 2-point Green function G2 and its strongly connected part, i.e., the self-energy part 2 .x; y/ D 1P .x; y/ (see relations (28.47) and (28.49)). Using (28.47) and (28.48), we get i e 2 .p 2 / D G

m20

i : e 2/  C †.p p2

We define the renormalized self-energy part by the relation 

er .p 2 / D Z' †.p e 2 /  .p 2  m2 /† e00 .m2 / ; e 2 /  †.m †

(29.70)

(29.71)

386

Chapter 29

On Renormalization in Perturbation Theory

where 2e e00 .m2 / D @ †.p/ jp2 Dm2 : † @p  @p 

In the second order with respect to ; this relation coincides with the definition of D 1i †r;.2/ by relation (29.21) because, in this case, Z' must be taken in the

R;.2/ 2

.0/

zero order with respect to ; i.e., Z' D 1 (see (29.56)). er;.n/ n > 3; it is necessary to use the In higher orders, for the determination of † Bogoliubov–Parasiuk R-operation and prove that it leads to relation (29.71) after the summation of all orders of perturbation theory. This is a nontrivial problem. First of all, we show that the renormalized self-energy part thus chosen agrees with (29.65). e er .p/ into the rightIndeed, substituting †.p/ expressed in terms of the function † hand side of (29.69) and taking m20 D m2 C ım2 ;

e 2 /; ım2 D †.m

e00 .m2 /; Z'1 D 1  †

we obtain equality (29.65) with G2r .p 2 ; m2 ; 2 / D

m2



i : er .p 2 / C†

p2

(29.72)

The total vertex part can be represented in the form e e4 .p; k; p C k  q/;  4 .p; k; p C k  q/ D 1 C ƒ e4 is the sum of contributions of strongly connected diagrams in all orders where ƒ n > 2: We define a regularized function ƒr4 by the relation 

e4 .m2 / ; er4 .p; k; p C k  q/ D Z ƒ e4 .p; k; p C k  q/  ƒ ƒ where the point of expansion can be chosen, e.g., for p 2 D m2 and k D q D 0; i.e., e4 .p; k; p C k  q/j p2 Dm2 : e4 .m2 / D ƒ ƒ kDqD0

As in the previous case, relation (29.72) coincides with the definition of the regularR;.2/ ized function 4 in the second order of perturbation theory by relations (29.16) and (29.17). Taking e4 .m2 /; Z 1 D 1 C ƒ we get e er4 .p; k; p C k  q/:  r4 .p; k; p C k  q/ D 1 C ƒ For the final construction of a renormalized theory, it is necessary to prove that relations (29.70) and (29.72) define the completely renormalized self-energy and vertex parts and to construct a perturbation theory free of divergences. This is realized in Part VII for a model of quantum electrodynamics (see also [176]). For models (28.63) or (25.3), this can be done by analogy.

Section 29.7 Equivalent Types of Renormalizations

29.7

387

Equivalent Types of Renormalizations

After Hepp’s work [89], where the author, in fact, reproduced the Parasiuk– Bogoliubov proof with certain modifications and correction of some technical inaccuracies, new renormalization schemes appeared. Among them, first of all, one should mention the analytic Speer renormalization (see [189]) and the renormalization based on the introduction of an arbitrary dimension (see [201]). One should also mention the renormalization with respect to lines (Slavnov) [188] and the original idea (though not further developed) of Petryna and the author of this monograph on the connection of perturbation theory with projection-iterative methods for the solution of equations of quantum field theory [159] (it may be called the operator renormalization by projection-iterative method). In this chapter, we describe only the main ideas related to the approaches indicated above (for details, see the original works). The main idea of the Speer analytic renormalization is to replace the ordinary intermediate Pauli–Villars regularization (described in detail in Section 29.1) by a Feynman propagator of the form (analytic regularization) e c .p/ D D

.m2

Z.p/ ;  p 2  i "/1C

where  2 C; i.e.,  is a complex parameter. Then the Feynman amplitude F .G/ that corresponds to the diagram G becomes an analytic function for sufficiently large Re ; and, furthermore, its domain of analyticity depends on the degree of divergence of the diagram G: The renormalization of amplitude and the removal of regularization . ! 0/ reduce to the subtraction of poles (terms of the Laurent expansion with negative exponents). It turns out that this process is equivalent to the R-operation (see, e.g., [225]). The idea of renormalization based on the introduction of an arbitrary dimension is to replace the 4-dimensional space-time by a d -dimensional one. It turns out that Feynman amplitudes can be calculated for any d D 4  ı; and they are analytic functions of the complex parameter ı: As in the case of analytic renormalization, the determination of a renormalized amplitude reduces to the elimination of poles in the complex plane of the variable ı: This renormalization is also equivalent to the R-operation (see [225]). The renormalization based on the introduction of an arbitrary dimension is a fairly simple and efficient method and is extensively used. To illustrate its action, we use it for the calculation of integral (29.5) corresponding to the contribution of diagram (29.3). Assume that the integration in (29.5) is carried out over the d -dimensional Minkowski space. Using the relation 1 D ab

Z 0

1

d Œa.1  / C b 2

388

Chapter 29

On Renormalization in Perturbation Theory

where a D m2  q 2  i " and b D m2  .q  k/2  i "; and performing the translation q  k ! q; we obtain .2/ e  4 .k/ D

1 .2/4

Z

1

Z dq

0

Md

1 ; Œm2  q 2  .1  /k 2  i "2

(29.73)

where dq D dq 0 dq 1    dq s and d D sC1: We rotate the contour of integration in the complex plane q 0 by 90ı (see Section 34.1) and perform the substitution q 0 D i q d : Then Z 1Z i 1 .2/ e  4 .k/ D dq 2 ; (29.74) 4 2 .2/ 0 Œm C q  .1  /k 2 2 Rd

where dq D dq 1    dq d and the integration is taken over the Euclidean space Rd : .2/ It is worth noting that, in the case where the amplitude e  4 .k/ is determined for physical values of external momenta, i.e., on the mass surface k 2 D m2 ; the integral in (29.74) is not singular. The integral is taken according to the known formula Z dq .˛  d=2/ ; (29.75) D  d=2 x d=2˛ 2 ˛ .q C x/ .˛/ Rd

where  is the Euler gamma function. We can analytically extend the right-hand side of (29.75) to the complex plane with respect to the variable d and treat equality (29.75) as the definition of the integral on the left-hand side of (29.75) for an arbitrary (even complex) d: Setting ˛ D 2; we get .2/ e  4 .k/ D

i

16

.2  4d=2

d / 2

Z

1

0

d

d Œm2  k 2 .1  / 2 2 :

(29.76)

Thus, for d D 4; we have .0/ D 1; which corresponds to the divergence of integral (29.5). This divergence must be eliminated by the subtraction procedure. To this .2/ end, we consider e  4 .k/ as a function of the complex variable d (or ı D 4  d ) and expand it in a Laurent series in the domain of its analyticity. As a result, we obtain .2/ e  4 .k/ D

i i C const  8 2 ı 16 2

Z

1 0

d lnŒm2  k 2 .1  / C O.ı/;

(29.77)

where O.ı/ contains all terms of the expansion with ı n ; n > 1: Also note that, in the calculation of the coefficients a1 and a0 of the Laurent expansion, we have used the asymptotic representation of the function .ı=2/ for small ı, namely, 2 ı . /   C; 2 ı

Section 29.7 Equivalent Types of Renormalizations

389

where C is the Euler constant, and the following formula well known from analysis: ax  1 D ln a: x!0 x lim

It readily follows from (29.77) that the renormalized amplitude defined by (29.14) coincides with expression (29.15) calculated with the use of the Pauli–Villars regularization. Remark 29.12. In the calculation of Feynman amplitudes of models containing fermions, it is necessary to formulate rules for the calculation of the trace of the product of -matrices and integrals over a d -dimensional Euclidean space. Some of these rules are presented below: Tr 1 D d;  

Tr.  / D dg  ; Z

Tr.       / D d.g  g   g  g  C g  g  g  /; Z ı  dpp  p  f .p 2 / D dpp 2 f .p 2 /; d d d R R Z 1 x ˛1 dx D a˛ˇ B.˛; ˇ  ˛/ .x C a/ˇ 0 .˛/.ˇ  ˛/ D a˛ˇ : .ˇ/

In Slavnov’s work [188], the renormalized amplitude is recursively constructed using the amplitude corresponding to a diagram the number of whose external lines is lesser by one. This scheme makes the combinatorial structure of renormalization very simple. The renormalization with respect to lines is rather general, and the previous two renormalizations (analytic and with respect to dimension) are its special cases. In conclusion, we briefly describe the main idea of [159]. According to it, the equation for the S-matrix, i.e., the equation for coefficient functions (27.20), can be regarded as an ill-posed problem of mathematical physics. Indeed, Equation (27.20) written in a Euclidean domain (the Euclidean approach to quantum field theory is considered in detail in Part VI) has the form F D AF C F .0/ : The operator A can be defined in a certain Banach space B: However, no matter how this space is chosen, we have F .0/ … D.A/; where D.A/ is the everywhere dense set of vectors .D.A/  B/ that is the domain of definition of the operator A: Thus, this equation cannot be solved by the method of successive approximations because AF .0/ D 1: Using the theory of projection-iterative methods, we can propose the following scheme:

390

Chapter 29

On Renormalization in Perturbation Theory

Let Ar be an operator that is a certain extension of the operator A; i.e., Af D Ar f

if

f 2 D.A/

and, furthermore, D.A/ D.Ar / and

F .0/ 2 D.Ar /:

It was shown in [159] that the operator Ar can be constructed so that Anr F .0/ 2 D.Ar / and the formal iterative solution of the new equation F D Ar F C F .0/ is meaningful and equivalent to the perturbation theory for the coefficient functions of the S-matrix after the application of the Bogoliubov-Parasiuk R-operation. As a whole, this series is obviously divergent.

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory The functional integration was developed by Daniell [31] in 1919 and later by Wiener in a series of studies in connection with problems of Brownian motion (see, e.g., [218]). Wiener developed a rigorous method for assigning a probability to a random path of a particle. A functional integral is also known as a path integral (or a Feynman path integral). Feynman developed another notion of functional integral, a path integral, which is useful for the determination of the quantum properties of systems. In the Feynman path integral, the classical notion of the unique path of a particle is replaced by a continual sum of classical paths, each weighted differently according to its classical properties. All these integrals are path integrals of the Gaussian type. At the level of physical intuition, the path integral can be regarded as the limit of an n-fold integral in Rn as n ! 1: Indeed, considering the points x0 ; x1 ; : : : ; xn .xi 2 R1 / as values of a certain continuous function x.t /; t 2 Œ0; T ; i.e., x0 D x.0/; x1 D x.t1 /; : : : ; xn D x.tn /; 0 < t1 < t2 < t3 <    < tn < T; and connecting the points .x1 ; x2 /; .x2 ; x3 /; : : : ; .xn1 ; xn / by lines in the plane .x; t /; we obtain a piecewise-linear curve that tends to x.t / as n ! 1: For a certain continuous function 'n .x.// D F .x1 ; : : : ; xn /; the n-fold integral Z Z    F .x1 ; : : : ; xn /dx1    dxn R1

R1

can be considered as a single integral with respect to a variable that runs over all possible piecewise-linear curves originating in the point x0 2 X D R1 of the half plane TX: It tends (under the assumption that the limit exists) to the integral Z Y '.x.t // dx.t /; (30.1) 

t2Œ0;T

where is the collection of all possible continuous functions (paths) x./ on Œ0; T  that start at the point x0 : In fact, notation (30.1) is formal. The rigorous mathematical meaning can be given to the weak limit w  lim p.x0 ; x1 I t1 /p.x1 ; x2 I t2  t1 / n!1

   p.xn1 ; xn I tn  tn1 /dx1    dxn D dx0 .x.t //; where p.x; yI  / is a certain specially constructed function and x0 ./ is called the Wiener measure (for the corresponding p.x; yI  /); for details, see, e.g., [170], Chapter X.11. In the late 1940s, based on this approach, Feynman [53, 56] developed

392

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

a new formulation of quantum electrodynamics and the diagram technique. As an example, in Section 26.4, we have applied this technique to a scalar field by using operator methods. The Feynman technique was called the path-integral quantization. In contrast to the canonical quantization, this method has certain advantages in the gauge field theory. For the first time, it was used in the gauge field theory by Faddeev and Popov in [47] and de-Witt in [34, 35, 36]. We consider this problem in detail in Part VII. The aim of the present chapter is to give a brief introduction to the formal technique of path integration and the application of this technique to the quantization of gauge non-Abelian fields. Unfortunately, at present, a rigorous mathematical substantiation of Feynman integrals is absent. Rigorous representations of Green functions can be obtained only in several cases of interaction of scalar fields in a Euclidean two-dimensional space-time domain (see Part VI).

30.1

Notion of Path Integration and Main Formulas

We begin with the following known formula of the theory of Gaussian integrals in the space Rn : Z 1 1 1 e  2 .y;C y/C.z;y/ dy D .2/n=2 .det C /1=2 e 2 .z;C z/ ; (30.2) Rn

where C is a real strictly positive-definite n  n matrix. This matrix is called the covariance matrix, or the covariance operator, of the Gaussian probability measure ./ defined by the relation .dy/ D

.det C 1 /1=2  1 .y;C 1 y/ e 2 dy: .2/n=2

(30.3)

In Equations (30.2) and (30.3), .u; v/ denotes the scalar product of vectors u; v 2 Rn . It is easy to verify that ./ is a probability measure. To this end, we integrate (30.3) and use (30.2) for z D 0: As a result, we get Z .dy/ D 1: Rn

The matrix elements of C are the second moments of the measure ./; i.e., if yi and yj are components of the vector y 2 Rn ; then Z yi yj d.y/ D Cij : Rn

393

Section 30.1 Notion of Path Integration and Main Formulas

To generalize relation (30.2) to the case where yj is a function, i.e., the index j takes a continuum of values, we rewrite det C in a different form. In the case where the matrix C is diagonal, we easily obtain ln det C D Tr ln C:

(30.4)

It is easy to verify that this equality holds for all (nondiagonal) nonsingular .det C ¤ 0/ n  n matrices. In the continuous case where y is a function, e.g., y 2 L2 .Rd /; C is an integral operator with kernel C.x; x 0 /; x; x 0 2 Rd : Z dx 0 C.x; x 0 /y.x 0 /: (30.5) .Cy/.x/ D Rd

Then the operations ln and Tr are standard and, in many cases, they can be rigorously defined, whereas the notion of det for this operator should be understood in the sense of relation (30.4). The formal inverse C 1 is also an operator of the form (30.5) with kernel C 1 .x; x 0 / such that Z dx 00 C.x; x 00 /C 1 .x 00 ; x 0 / D ı.x  x 0 /; (30.6) Rd

Q where ı.x  x 0 / D diD1 ı.x .i/  x 0.i/ / is the Dirac ı-function in Rd : In this case, we rewrite relation (30.2) as follows: Z Dye De

1 2 Tr

 12

R Rd

ln C

dx 0 y.x/C 1 .x;x 0 /y.x 0 /C

Rd 1 2

e

R

dx R Rd

dx

R

dx z.x/y.x/

Rd

R

dx 0 z.x/C.x;x 0 /z.x 0 /

Rd

;

where (formally)

Y

Dy D

x2Rd

dy.x/ : .2/d=2

(30.7)

(30.8)

Relations (30.2) and (30.7) can be generalized to the case of complex functions z.x/ and y.x/ as follows: Z N 1 z/C.z;y/ N N dzd ze N .z;C D .2/n .det C /e .y;Cy/ ; (30.9) Cn

where

Z

Z dz d zN .   /  2 Cn

Rn

Z d.Re z/ Rn

d.Im z/.   /

(30.10)

394

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

and Z DzD zN e De

1 2 Tr

 12

ln C

1 2

e

R

dx 0 nz.x/nC 1 .x;x 0 /z.x/C

Rd

R Rd

R

dxu.x/z.x/

Rd

R

dx

dx 0 u.x/C.x;x 0 /u.x 0 /

Rd

:

(30.11)

In Part VI, we show that relations (30.7) and (30.11) can be rigorously substantiated in the case of a real scalar field if it is considered in a Euclidean domain, i.e., for imaginary times. However, in the context of this presentation, we use the aforementioned technique as a convenient computation procedure leading to the correct final result. This statement is based on quantum electrodynamics as a theory whose conclusions agree with experimental data. The path integrals (30.7) and (30.11) are generalizations of the multiple Gaussian integrals (30.2). At present, for arbitrary measures, there do not exist constructive methods for the definition of path integrals on the basis of their internal structure (for abstract definitions of measures on infinite tensor products, see, e.g., [13], Chapter 2). However, for the needs of quantum theory (i.e., perturbation theory), the study of properties of Gaussian path integrals is sufficient. Therefore, in conclusion, we give several main formulas of Gaussian calculus, which are used in what follows. To this end, we (formally) rewrite relation (30.7) in the form of an integral over the Gaussian measure: Z R R R 1 0 0 0 (30.12) dC .y/e dx z.x/y.x/ D e 2 dx dx z.x/C.x;x /z.x / ; where 1

1

dC .y/ D e  2 Tr ln C e  2

R

dx

R

dx 0 y.x/C 1 .x;x 0 /y.x 0 /

Dy:

(30.13)

Note that the relation for the Gaussian measure (30.13) is formal even in the case where the integral in (30.12) is rigorously defined. Relation (30.12) is the main formula for the determination of moments of the Gaussian measure. 1. Formula for moments Z y.x1 /    y.xn /dC .y/ D ² D

P .i/n

R R 1 ın 0 0 0 e 2 dx dx z.x/C.x;x /z.x / jz0 ız.x1 /    ız.xn /

0; n D 2k  1; k 2 N; C.xi1 ; xi2 /    C.xin1 ;xin /; n D 2kI

(30.14)

here, the summation is carried out over all decompositions of indices ¹1; 2; : : : ; nº into different nonordered pairs ¹ik ; ikC1 º such that all terms of the sum are different. The number of these terms is equal to Nn D

.2k/Š D .2k  1/ŠŠ; kŠ2k

n D 2k:

395

Section 30.1 Notion of Path Integration and Main Formulas

It is easy to verify that relation (30.14) coincides with the relation for the free n D 2k-point Green function (23.48) with C.x; y/ D 1i D c .x  y/; which is obtained with the use of the Wick theorem (Theorem 23.7). In this connection, in the theory of Gaussian integrals, relation (30.14) is sometimes called the Wick theorem. 2. Formula of integration by parts Let F .y/ be a functional over a certain space of functions on which a path integral with respect to the Gaussian measure dC is defined. Then the following relation is true: Z Z Z ıF : (30.15) y.x/F .y/dC .y/ D dC .y/ dx 0 C.x; x 0 / ıy.x/ This R relation can easily be verified (Problem 30.1) for a functional of the form F .y/ D e i z.x/y.x/dx : This allows one to easily generalize relation (30.15) to functionals F .y/ of the polynomial form (Problem 30.2) and to extend it by continuity to a broader class of functionals F .y/: For the rigorous proof of relation (30.15), see, e.g., [74]. 3. Formula for a multiple Gaussian integral Relation (30.12) can be generalized to the case where y.x/ is an n-component vector function: y.x/ WD .y1 .x/; : : : ; yn .x//: In this case, the covariation operator C and its inverse C 1 are defined by the matrix kernels Cij .x; x 0 / and Cij1 .x; x 0 /; respectively, and relation (30.10) takes the form n Z X j D1

0 00 0 dx 0 Cij .x; x 0 /Cj1 k .x ; x / D ıik ı.x  x /:

Rd

Equation (30.12) takes the form Z

n R P

dC .y/ej D1

dxzj .x/yi .x/

De

1 2

n R P

dx

i;j D1

R

dx 0 zi .x/Cij .x;x 0 /zj .x 0 /

;

where dC .y/ D e

 12 Tr ln C

e

 12

n R P i;j D1

dx

R

1 n dx 0 yi .x/Cij yj .x 0 / Y

j D1

and Tr A D

n Z X j D1

dx Ajj .x; x/:

Dyj

(30.16)

396

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

Formally, Equation (30.16) can easily be obtained from relations of the form (30.2). To this end, we use the limit transition (as in (30.1)) on the left-hand side of (30.16) Q to pass from integrals with respect to Dyj  x .2/d=2 dyj .x/ to integrals with d mj

respect to .2/ 2 dyj;1    dyj Imj ; j D 1; : : : ; n; and, upon redefinition of variables, to integrals with respect to .2/N=2 dy .1/    dy .N / ; N D m1 C    C mn I then we use relation (30.2), return to the previous notation, and pass again to the limit as mj ! 1; j D 1; 2; : : : ; n: 4. Definition of functional ı-function By analogy with the definition of ordinary Dirac ı-function as a linear functional Q over a certain space of test functions, we define a functional ı-function as follows: Let F .y/ be a functional (over a certain space of functions y.x/; x 2 Rd ) from a certain space of “test functionals” (we do not define here the notion of the space of test functionals and only use this term by analogy with the space of test functions in the definition of ordinary ı-function). Then a functional ı-function is defined as a functional given by the relation Z Q  y 0 /Dy 0 D F .y/: (30.17) F .y/ı.y In view of the formal notation (30.8), we must obviously assume that Y Q  y0/ D ı.y.x/  y 0 .x//; ı.y x2Rd

where ı is the ordinary Dirac ı-function. To give a rigorous mathematical meaning to relation (30.17), we consider a functional F .'/ of the form 1

F .yI j / D e  2 ˛

R

dx

R

dx 0 y.x/C 1 .x;x 0 /y.x 0 /Ciˇ

R

dx y.x/j.x/

;

(30.18)

where ˛; ˇ 2 R1 and j.x/ is a certain smooth function, and represent the functional ıQ as follows: Z R Qı.y  y 0 / D De i .y.x/y 0 .x//.x/dx : (30.19) In this case, relation (30.17) can easily be verified by direct calculation (Problem 30.3) with the use of relations (30.12) and (30.13). Differentiating the obtained equality with respect to ˇ and properly choosing the parameters ˛ and ˇ and function j.x/; we extend equality (30.17) to a fairly broad class of functionals F .y/: Representation (30.19) can also be generalized to more complicated arguments as follows: Z R Qı.f .I y/  y 0 / D De i Œf .xIy/y 0 .x/ .x/dx ; (30.20)

Section 30.1 Notion of Path Integration and Main Formulas

397

where f .xI y/ D f0 .x/ C '.x/ C fQ.xI y/ and the functional fQ.xI y/ is defined by the series fQ.xI y/ D

Z 1 X gn Cn .xI x1 ; : : : ; xn /y.x1 /    y.xn /dx1    dxn : nŠ

nD1

Moreover, the equation

f .xI y/  y 0 .x/ D 0

(30.21)

(30.22)

has a unique solution y.y Q 0 /; which can be represented in the form of a series in g: We obtain the following relation of the form (30.17): Z

where

Q Q .I y/  y 0 / det k1 C ı f kDy D F .y/; Q F .y/ı.f ıy

(30.23)

´ " #μ ı fQ ı fQ det k1 C k D exp Tr ln 1 C ıy ıy Z ° ı fQ.x/ jx 0 Dx D exp dx ıy.x 0 / Z ± ı fQ.x/ ı fQ.x1 / 1 dxdx1 jxDx2 C    : C 2 ıy.x1 / ıy.x2 /

For the proof of relation (30.23), see [48, 49], Chapter II, Section 5. 5. Change of variables According to the formula for moments (30.14), the Gaussian integrals can easily be calculated for functionals of polynomial form by using the generating functional (30.12). We represent this functional in the form of the Fourier transform of the Gaussian measure: Z R I./ D dC ./e i .x/.x/dx Z R R R 1 0 1 0  12 Tr ln C D e  2 dx dx .x/C .x;x /.x/Ci .x/.x/dx : (30.24) e Consider the following change of variable: .x/ D f0 .x/ C y.x/ C fQ.xI y/;

(30.25)

398

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

where the functional fQ is defined by (30.1). Then Z R R 0 1 0 0 Q I./ D dC .y/e  dx dx Œf0 .x/Cf .xIy/ C .x;x /y.x / R ı fQ i dxŒf0 .x/Cy.x/CfQ.xIy/ .x/ e det 1 C : ıy

(30.26)

It is easy to verify Equation (30.26) by determining the Fourier images of the functionals I./ defined by Equations (30.24) and (30.26): Z R e (30.27) I . / D D I./e i .x/ .x/dx : Calculating the integral in (30.27) with I./ defined by (30.24), one should use relations (30.19) and (30.17). Calculating the integral in (30.27) with I./ defined by (30.26), one should use relations (30.1) and (30.23). Remark 30.1. Equation (30.26) can be rigorously substantiated only for a fairly narrow class of functionals of the form (30.1). For example, in the case where fQ  0; substitution (30.25) corresponds to the displacement (translation) of the Gaussian variable, whereas, for Z Z g2 C2 .xI x1 x2 /y.x1 /y.x2 /dx1 dx2 ; fQ.xI y/ D g C1 .xI x1 /y.x1 /dx1 C 2 it corresponds to a quadratic perturbation of the Gaussian measure. Under certain restrictions on f0 ; C1 ; and C2 ; these substitutions can be made absolutely rigorously (see, e.g., [74], Chapter 9). However, within the framework of perturbation theory, which lies in the foundation of all calculations of contemporary quantum field theory, the developed formalism is assumed to be mathematically substantiated. 6. Wick renormalization In the course of quantization of free fields, we have introduced the notion of normal product and formulated the Wick theorem, which gives rules for the multiplication of polynomials of field operators. These rules enable one to eliminate infinite terms that appear in the determination of a product of operator-valued generalized functions. It is obvious that, for the quantization of fields with the use of path integrals, the formalism of Wick calculation should be generalized to Gaussian fields over which the integration is carried out. In the Gaussian analysis, the formalism of Wick calculation is connected with the orthogonalization of polynomials P./; which are elements of the space L2 .S 0 ; dC /; where S 0 is the space of linear continuous functionals (over the space of test functions S ) to which the Gaussian fields .x/ belong (see, e.g., [13], Chapter 2, Section 2.).

Section 30.1 Notion of Path Integration and Main Formulas

399

In the case of a scalar free quantum field '.x/; the definition of normal (Wick) product is connected with the location of the creation ' C .x/ and annihilation '  .x/ operators in the corresponding product. The Gaussian fields .x/ 2 S 0 are not connected with this decomposition. For this reason, the operation of Wick multiplication is defined as an operation in the space L2 .S 0 ; dC / as follows: W .f1 /.f2 / W WD .f1 /.f2 /  .f1 ; Cf2 /; where

Z .f / D .f1 ; Cf2 / D

Z

f .x/.x/dx; f 2 S ; Z dx dyf1 .x/C.x; y/f2 .y/:

The construction of the Wick product of three and more fields is the procedure of orthogonalization of these monomials in the space L2 .S 0 ; dC /: Here, we do not describe this procedure and only note that it is completely defined by the Wick theorem (see Section 23.4) if the pairing of two Gaussian fields .x/ and .y/ is understood as their covariation C.x; y/: Thus, to define the operation of Wick multiplication W .x1 /    .xn / W; it is necessary to write the Wick theorem for all products .x1 /.x2 /; .x1 /.x2 /.x3 /; : : : ; .x1 /    .xn / and then solve the obtained relations with respect to the Wick products W .x1 /.x2 / W; W .x1 /.x2 /.x3 / W; : : : ; W .x1 /    .xn / W; i.e., express them in terms of ordinary products and products of the corresponding covariations. It follows from the definition of this procedure that, for an arbitrary n and arbitrary fi 2 S ; i D 1; 2; : : : ; n; one has Z dC ./ W .f1 /    .fn / WD 0: (30.28) It follows from (30.28) that Z dC ./ W e .f / WD 1: Using (30.12), we get

Z

1

dC ./e .f / D e 2 .f;Cf / :

(30.29)

It is easy to verify that 1

W e .f / WD e  2 .f;Cf / e .f / :

(30.30)

This relation can also be obtained for operator fields with the use of (22.36). Using (30.29) and (30.30), we obtain the important relation (Problem 4) Z (30.31) dC ./ W e .f / W W e .g/ WD e .f;Cg/ :

400

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

Using this relation, we obtain the following analog of the formula for moments (30.18) for Wick products of Gaussian fields (Problem 5): Z dC ./ W

m Y j D1

.xj / WW

n Y

.yk / WD ımn

X

C.x1 ; y.1/ /    C.xn ; y.n/ /:

2Sn

kD1

(30.32) Conclusion 30.2. Thus, in the formalism of path integration, the Wick product has the same sense as in the operator formalism. The covariation coincides with the Green function of the corresponding quantized fields, and the integration with respect to the Gaussian measure coincides with vacuum averaging.

30.2

Formalism of Feynman Integrals (Path Integrals) in Quantum Mechanics

Consider a quantum mechanical system that consists of identical particles of mass m: Let (30.33) H D H.q; p/N D H.q1 ; p1 I : : : I qN ; pN /; where qj and pj are the coordinates and momenta of particles, be the classical Hamiltonian of this system. In Section 4.2, we have briefly described the transition to a quantum mechanical system with the use of the transition to operator quantities qOj and pOj that satisfy the commutation relations (4.31) with constant (4.36). It is convenient to consider these operators in the coordinate representation of functions from L2 .R3N ; .d r/N /; where the operators pOj have the form (4.35) and the operators qOj are defined as operators of multiplication by a variable of the coordinate rj : The operators pOj and qOj have a continuous spectrum, and, therefore, their eigenfunctions (see, e.g., [115], Section 5) do not belong to L2 .R3N ; .d r/N /: ‰q D ‰.q/N .r/N D

N Y

ı.rj  qj /;

j D1

‰p D ‰.p/N .r/N

N Y

(30.34)

i 1 „ pj rj : D e .2/3=2 j D1

b D H.Oq; p/ As noted in Section 4.2, the construction of the Hamilton operator H O in the general case is a nontrivial problem because the operators qO and pO do not commute. Here, we consider the most important case where the energy of interaction between particles is independent of momenta. Then bD H

N X pOj2 j D1

2m

C U.Oq/N ;

(30.35)

Section 30.2 Formalism of Feynman Integrals (Path Integrals) in Quantum Mechanics 401

where U./ is the operator of potential interaction energy, which depends only on the location of particles. Thus, the problem of the construction of the operator is trivial. Within the framework of the Schrödinger approach, the state of the system depends on time, and its evolution is determined by the equation i„

@‰.t / b ‰.t /: DH @t

Thus, if, at an initial time t0 ; the state of the system is given by a state vector ‰.t0 /; then the state of the system at an arbitrary time is determined by the relation b .t; t0 /‰.t0 /; ‰.t / D U where

b b .t; t0 / D e  „i .tt0 /H U

(30.36)

is the evolution operator. It is assumed that complete information on the state of the system can be obtained if one knows the probability of transition in time  D t  t0 from the state ‰q0 in which all coordinates of particles are exactly known at time t0 to the state ‰q at time t: According to the rules of quantum mechanics, this probability is determined by the relation t D j.‰q ; U.t; t0 /‰q0 /j2 : Pqq 0

Formally considering the evolution operator (30.36) as an integral operator and taking into account the explicit form of the functions ‰q (30.34), we establish that the amplitude   b ..q/ I .q / / b .t; t0 /‰q0 D e  „i .tt0 /H (30.37) ‰q ; U 0 N N is, in fact, the kernel of operator (30.36) The Feynman approach is based on the representation of kernel (30.37) in the form of the path integral Z i b DpDqe „ A t;t0 .p;q/ ; (30.38) .‰q t ; U .t; t0 /‰q0 / D q t q0

where

Zt    A t t0 .p; q/ D p. /  q. /  H.q. /; p. // d 

(30.39)

t0

is the function of classical action (see (4.8)), the measure of integration has the form DpDq D

N Y

Y

j D1

0 2Œt0 ;t

d pj . 0 / d qj . 0 / ; .2„/3=2 .2„/3=2

(30.40)

402

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

and q t q0 is the set of all possible trajectories of particles that start from the phase point q0 at time t0 and arrive at the point q t at time t . b defined Relation (30.38) is formal. Nevertheless, for the Hamilton operator H by (30.35), it is possible to prove it almost rigorously and, in the absence of interaction i .U  0/; to derive a rigorous formula for the kernel of the operator e  „ H0 despite the fact that, in this case, the path integral becomes an ordinary one. To prove relation (30.38), we use the following theorem: Theorem 30.3 ([170]; Theorem IX.29). Let f0 2 L1 .Rn /: If f0 2 L2 .Rn // or the Fourier transform fQ0 belongs to L1 .Rn //; then Z n=2 .f0 .i r/'/.x/ D .2/ fQ0 .x  y/'.y/dy (30.41) for all ' 2 L2 .Rn /; and the integral converges for all x if f0 2 L2 .Rn / and for almost all x if fQ0 2 L1 .Rn /: First, we use relation (30.41) in the absence of interaction, i.e., for the operator b0 ; e „ H i

 D t1  t0 ;

N X pOj2

b0 D H

j D1

where j D

@2 @xj2

C

@2 @yj2

C

@2 @zj2

2m

D

N X „2 j ; 2m

j D1

is the Laplace operator in L2 .R3 drj / and rj D

.xj ; yj ; zj /: 2 „

Since the function f0 ./ D e i˛ ; ˛ D 2m ; does not satisfy the conditions of Theorem 30.3, we consider the regularized function f" ./ D e i.˛i"/ ; 2

" > 0;

(30.42)

which tends pointwise to f0 ./ as " ! 0 and satisfies the conditions of the theorem. We apply Theorem 30.3 to function (30.42) with argument i r in R3N :   @ @ i r WD i ; : : : ; i : @x1 @zN Then, taking into account that the Fourier transform of the function f" ./ has the form 2

x fQ" .x/ D .2i ˛ C 2"/3N=2 e  4.i˛C"/ ;

" > 0;

x 2 R3N ;

b0 : we obtain the following final formula for the kernel of the operator e  „ H i

e

b ..r/ I .r0 / / D N N

 „i H 0



N P

i 2„ .rj rj / m  3N 2 j D1 e : 2 i „ m

0 2

(30.43)

Section 30.2 Formalism of Feynman Integrals (Path Integrals) in Quantum Mechanics 403

In the theory of nonrelativistic scattering, this function (for N D 1) is called a free propagator. For convenience, we introduce the following notation: .r/N D .r1 ; : : : ; rN / DW q; .r0 /N DW q0 :

(30.44)

We divide the time interval  D t  t0 into n equal parts: n D

t  t0  D ; n n

tj D tj 1 C n ;

j D 1; 2; : : : ; n:

Applying the Trotter theorem (see, e.g., [169], Theorem VIII.30) to the operator i b i b0 Cb U/ e „ H D e  „ .H ;

we get

b0 Cb U/ e  „ .H D s  lim i

n!1

(30.45)

 i n b0 e  „i n b U : e „ n H

(30.46)

Taking into account that U e „ n b .qI q0 / D e  „ n U.q/ ı.q  q0 / i 

i 

and using relations (30.43) and (30.46), we write the action of operator (30.45) on the function f .q/ as follows:  e

b  „i .tt0 /H

where





f .q0 / D lim

n!1

m 2 i „n

 3N Z 2 n

i

.d q/n e „ At;t0 .q/n f .qn /;

R3N n

"  #  n X m qi  qi1 2 A t;t0 .q/n D n  U.qi / : 2 n

(30.47) (30.48)

iD1

To derive (30.38) from (30.47), it is necessary to represent the coefficient of the integral in (30.47), together with the exponential of the first term in (30.48), in the form of a Gaussian integral, namely,  2 n P  3N i m qi qi1 2 n „  n m 2 n iD1 e i n i n h P p2 Z Y q q n i i pi  i i1  2m d pi „  n n iD1 D e 3N 2 .2„/ 3N n iD1



R

(here, as in (30.44), pi is a 3N -dimensional vector, i D 1; n), set qi D q.ti /;

pi D p.ti /;

404

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

and, taking into account the form of H.q; p/ (see (30.35)), formally pass to the limit as n ! 1: To give a physical interpretation of relations (30.38), (30.47), and (30.48), we consider the case of a single particle N D 1: If this particle moves along a piecewiselinear curve so that it is at the point with coordinate q0 at time t0 ; at the point q1 at time t1 ; etc., then .qi  qi1 /=n is the average velocity of the particle. As n ! 1;  it tends to q.ti /; and the sum in (30.72) tends to an integral corresponding to the action functional (see (4.1) and (4.4)): Zt1 A t;t0 .q/ D

d0

i hm  q. 0 /2  U.q. 0 // : 2

(30.49)

t0

Then n integrals in (30.47) can be regarded as one integral along all possible piecewise-linear curves and, in the limit as n ! 1; as integral (30.38) along all possible continuous paths of the particle from the point q0 at time t0 to the point q t at time t: The action functional (30.49) can also be rewritten in the form (30.39) (see (4.8)). Let

q t q0 denote the set of all these paths. Remark 30.4. The developed formalism of path integration enables one to write representations not only for matrix elements of the form (30.37) but also for matrix elements of partially ordered products of Heisenberg operators of arbitrary form: bt F .Oq; p/e bt : O i H F .Oq.t /; p.t O // D e i H The corresponding representation can be written as follows .t > t1 >    > tm > t0 / W

 ‰q t .t /; F1 .Oq.t1 /; p.t O 1 //    Fm .Oq.tm /; p.t O m //‰q0 .t0 / Z i D DpDqF1 .q.t1 /; p.t1 //    Fm .q.tm /; p.tm //e „ At t0 .p;q/ ; (30.50) q0 q t

where bti ‰ ; ‰qi .t / D e „ H qi i

i D 1; 2;

t1 D t0 ; q1 D q0 ; t2 D t; q2 D q t

(see, e.g., the proof in [212], Section 9.1). Remark 30.5. Let us explain why this formalism is called the quantization of classical mechanical systems with the use of path integrals. In the case of ordinary canonical first quantization, observables are regarded as Hermitian operators that are functions of canonical variables p and q acting in the space of states, and their averages are determined according to general rules. For example, for a dynamical quantity F; one has b ‰/ .‰; F .Oq; p/‰/ O .‰; F D : F D .‰; ‰/ .‰; ‰/

405

Section 30.3 Formalism of Feynman Integrals for Systems with Constraints

Within the framework of the Feynman approach, these averages are determined with the use of path integrals of the form (30.38) and (30.50) as follows: Z F D F .q./; p.//d.q; p/; (30.51) 0

where the measure of integration i

d.q; p/ D N 1 e „ A t t0 .q;p/ DpDq

(30.52)

is concentrated on a certain phase space of trajectories .0/ : In fact, the observable F .q; p/ of classical mechanics is again associated with a classical object, which, however, depends not on fixed values of coordinates and momenta of particles (i.e., on a discrete number of variables) but on all possible trajectories of these particles (i.e., on a continual number of variables), which (for a given integral (30.51)) start (at time t0 ) and end (at time t ) at a given phase point .q0 ; p0 /: Therefore, the classical location of the particle .q0 ; p0 / is in a certain “cloud” of its possible locations in the phase space, which corresponds to the notion of quantum object. Theoretically, these locations can be at any distance from .q0 ; p0 /: However, calculations show that the probabilities of trajectories with maximal distance l from q0 are proportional to expŒcl 2 ; i.e., they are negligible for large l (see, e.g., [68]). Representation (30.51), (30.52) is also convenient for passing to the classical limit .„ ! 0/: To this end, it is necessary to use formal expansions of the integrands in powers of „: In fact, on the rigorous mathematical level, this can be done only for a fairly narrow class of model physical systems.

30.3

Formalism of Feynman Integrals for Systems with Constraints

Prior to passing to Feynman integrals for a mechanical system with additional constraints on generalized coordinates and momenta, we clarify the core of the problem, using an ordinary multiple integral as an example. Let xO 1 ; : : : ; xO N be canonical variables and let Z IN D .dx/N e iA.x/N (30.53) RN

be an analog of the integral on the right-hand side of relation (30.38). As shown above, this definition of IN is correct and gives true results in the theory if the variables x1 ; : : : ; xN correspond to independent canonical variables. If the classical theory contains the additional equations 'i .x/N D 0;

i D 1; m;

(30.54)

406

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

and these constraints must be taken into account in the construction of the integral IN ; then (under the assumption that Equations (30.54) are solvable with respect to the variables x1 ; : : : ; xm ) integral (30.53) has the form Z e .dx/N e i A.xmC1 ;:::;xN / ; (30.55) IN D RN

where, under the solvability condition (implicit-function theorem), one must take into account that @'i m @.'1 ; : : : ; 'm / ¤ 0: (30.56)  det @x @.x1 ; : : : ; xm / j i;j D1 The action in the exponential in (30.55) has the form e mC1 ; : : : ; xN / D A.f1 ; : : : ; fm ; xmC1 ; : : : ; xN /; A.x

(30.57)

and the functions fi .xmC1 ; : : : ; xN / D xi ; i D 1; m; are solutions of system (30.54). Thus, in this case, we have IN D 1 because Z Z e IN D dxmC1    dxN e i A.xmC1 ;:::;xN / .dx/m  1 D C1: Rm

RN m

It is obvious that e IN D

Z

e dxmC1    dxN e i A.xmC1 ;:::;xN /

(30.58)

RN m

is an analog of integral (30.53). However, the integral e I N can again be rewritten in terms of all canonical variables. Constraints (30.54) can be taken into account by replacing the ordinary Lebesgue measure on RN m by a certain measure ..dx/N / on RN ; which can easily be obtained by introducing additional integrations in (30.58) with the use of the Dirac ı-functions: Z 0 0 0 e IN D dx10    dxm ; dxmC1    dxN e iA.x1 ;:::;xm ;xmC1 ;:::;xN /



RN m Y

ı.xi0  fi .xmC1 ; : : : ; xN //:

(30.59)

iD1 0 ; we pass to new variables x ; : : : ; x In the integral with respect to dx10    dxm 1 m according to the rules

xi0 D 'i .x1 ; : : : ; xm ; xmC1 ; : : : ; xN / C fi .xmC1 ; : : : ; xN /;

i D 1; m:

Section 30.3 Formalism of Feynman Integrals for Systems with Constraints

407

This can always be done because the Jacobian is not equal to zero: @'i m J D det ¤ 0: @xj i;j D1 Since the integrand contains the functions ı.'i /; which are not equal to zero only for 'i D 0; we have fi D xi for all i: Finally, we obtain e IN D

Z N iA.x/N

.dx/ e

m Y @'i m det ı.'i .x/N /: @xj i;j D1

(30.60)

iD1

RN

In Section 4.1.4, we have briefly formulated the canonical formalism of classical mechanics with additional constraints (4.22). It has been established that, under the additional constraints (4.17), a Hamiltonian system in the phase space  2n of variables .q; p/ can be reduced to an ordinary Hamiltonian system in the space e  2.nm/ of variables .q; Q p/ Q that are already independent. The space e  2.nm/ can be realized 2n as a subspace of  by imposing, in addition to (4.22), the condition j .q1 ; : : : ; qn ; p1 ; : : : ; pn / D 0;

j D 1; m;

(30.61)

with the Poisson brackets ¹j ; k º D 0

(30.62)

n ¤ 0: det k¹'j ; k ºkj;kD1

(30.63)

and Condition (30.62) enables one to choose j so that, e.g., the first m variables of generalized momenta coincide with j ; i.e., p D .1 ; : : : ; m ; pQ1 ; : : : ; pQnm /:

(30.64)

Then condition (30.63) takes the form @'j m m det k¹'j ; k ºkj;kD1 D det ¤ 0: @q k j;kD1

(30.65)

Using this procedure, one can easily solve the system of equations (4.22) with respect  2.nm/ in  2n by the equations to qk ; k D 1; m; define the surface e pk D 0;

qk D qk .qQ 1 ; : : : ; qQ nm I pQ1 ; : : : ; pQnm /;

(30.66)

and consider the variables .qQ 1 ; : : : ; qQ nm I pQ1 ; : : : ; pQnm / as independent canonical variables. Then the averages of the evolution operator (30.37) will have the form (30.38) in terms of the variables .q; Q p/: Q

408

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

However, in practice, it is not always possible to solve the system of equations (4.22) in explicit form. Furthermore, it is necessary to use this formalism for the construction of path integrals in gauge-invariant theories and to represent these integrals in the Lorentz covariant form. Here, we show that integral (30.38) written in terms of the variables .q; Q p/ Q can be rewritten again in terms of the old variables .q; p/ but with respect to a measure that takes into account constraints (4.22) and (30.61). This integral has the form Z Y i e „ A t t0 .q;q/ d.q. /; p. //; (30.67) .‰q ; U.t; t0 /‰q0 / D

2Œt;t0

qq0

where m d. / D .2„/mn det k¹'j ; k ºkj;kD1

m Y

ı.j /ı.'j /

j D1

n Y

dqk . /dpk . /:

kD1

(30.68) To prove this statement, we first rewrite measure (30.68) in terms of variables (30.64) with regard for (30.65) as follows: n m Y Y @'j m d . Q / D .2„/mn det ı.p /ı.' / dqk . /dpk . /: (30.69) j j @q k j;kD1 j D1

kD1

In integral (30.67) with respect to measure (30.69), we pass to the variables pk D pk0 ;

k D 1; m;

pj Cm D pQj ;

qk .qQ 1    ; qQ nm I pQ1 ; : : : ; pQnm / D

j D 1; n  m;

qk0 ;

@' m Then, with regard for the Jacobian, det @qj k

j;kD1

qj Cm D qQj : is eliminated (see (30.54)–

(30.60)), and measure (30.69) takes the form d . Q / D .2/mn

m Y

0 0 ı.pj0 /ı.qj0  qj .qI Q p//dq Q j dpj

j D1

nm Y

d qQ k d pQk :

(30.70)

kD1

Substituting (30.70) into (30.67), integrating over functions, and taking into account that

Qm

0 0 j D1 d pQj d qQj

with the use of ı-

Q p/; Q H.q; p/j'DD0 D H.q; we obtain relation (30.38). Thus, for the Hamilton system with additional constraints (4.17), the evolution operator of the quantum system preserves the form of integral (30.38)) with a measure that takes the constraints into account, i.e., it has the form (30.67).

409

Section 30.4 Path Integral Representation for Scalar Fields

30.4

Path Integral Representation for Scalar Fields

The formal generalization of the Hamiltonian system described in Section 30.2 consists of the replacement of a finite collection of canonical variables .q; p/N in (30.33) by a continuum of variables .'.x/; .x//; x 2 M: To directly use the formalism presented in Section 30.2, it is necessary to define commutation relations for the operators '.x/ and .x/: In Chapter 5, we have considered representations of commutation relations in the Fock space. For the construction of the path integral (29.38) in quantum mechanics, we use the so-called coordinate representation, in which the coordinate operator acts as the operator of multiplication, and the momentum operator is the differential operator. An analogous representation can be obtained in quantum field theory. For this purpose, it is necessary to consider the space of states whose elements are functionals ˆ.'/ of the values of fields '.0; x/: The equal-time commutation relations (17.20) for x 0 D y 0 D 0 hold if the operators '.0; x/ and .0; x/ are defined as follows: '.0; x/ˆ.'/ D '.x/ˆ.'/;

.0; x/ D i

ı ˆ.'/: ı'.x/

In this case, the representation of a matrix element of an evolution operator (of the form (30.37)) is constructed by analogy for the system of self-interacting scalar field with the Lagrangian L .x/ D

m2 1 W @ '.x/@ '.x/ W  W ' 2 .x/ W  W ' 4 .x/ W 2 2

(30.71)

and Hamiltonian H D H.; '/ Z  1 1 m2 D W  2 .0; x/ W C W jr'.0; x/j2 W C W ' 2 .0; x/ W C W ' 4 .0; x/ W d x: 2 2 2 Then, using an analogy with a mechanical quantum system, we rewrite the path integral (30.38) as follows:  Z  A 0 0 . ;/ .x20 x10 /H ˆ1 .'/ D D De x2 ;x1 ; (30.72) ˆ2 .'/; e where x20 > x10 ; 0

Zx2 2

Z dx 0

Ax 0 ;x 0 . ; / D 1

x10

R3

d xΠ.x/@0 .x/  H . .x/; .x//;

(30.73)

410

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

H . .x/; .x// is defined so that Z H. ; / D d xH . .x 0 ; x/; .x 0 ; x//jx 0 D0 ; and D D D

Y x2Œx10 ;x20 ˝R3

d .x/ d.x/ : .2/2 .2/2

(30.74)

(30.75)

In relations (30.72)–(30.75), the variable  corresponds to the field operator '; and the variable of integration is associated with the canonical momentum variable. In quantum field theory, the main object is the Green function. The representation of the Green function in the form of a path integral can easily be obtained using the general formalism described in Section 30.1. Within the framework of perturbation theory, which is used here for path integrals, to obtain the required representation it suffices to use an analogy between relation (30.14) and the Wick theorem (Theorem 23.7). Setting 1 C.x; y/ D D c .x  y/ i in relation (30.14) and taking (23.18) into account, we obtain Z dC ./.x/.y/ D . 0 ; T .'.x/'.y// 0 /; where the measure dC is formally defined by relation (30.13) with C 1 .x; x 0 / D i.g  @ @0  m2 /ı.x  x 0 /: Taking into account the Fourier transforms of the functions ı.x  x 0 / and 1i D c .x  (see (26.92) and (23.9)), one can easily determine the infinite constant Z Z dp ln.p 2  m2 /: Tr ln C D dx .2/4

x0/

However, this constant must not be present in results because it is, in fact, a constant normalization factor for the Gaussian integral, and, thus, it must vanish if the integral is calculated correctly. By analogy, one can easily establish that (Problem 30.6) Z dC ./.x1 /    .xN / D . 0 ; T .'.x1 /    '.xN // 0 /: (30.76) Then the N -point Green function defined by (28.8) in the interaction representation can be rewritten in the form of a path integral (for interaction (25.3)) as follows: Z R 4 1 dC ./.x1 /    .xN /e i W .x/Wdx ; GN .x/N D S0

411

Section 30.5 Path Integral Representation for Fermi Fields

where

Z S0 D

dC ./e i

R

W 4 .x/Wdx

:

Taking (30.12) and (30.13) into account, we rewrite this relation as follows: Z R 1 GN .x/N D 0 D .x1 /    .xN /e i L .x/dx ; (30.77) S0 where L .x/ coincides with (30.71) and S00 differs from S0 by the factor e 1=2Tr ln C cancelled by the corresponding factor in the numerator. Relation (30.77) is called the Lagrangian form of the continual representation of the Green function. Relation (30.77) can be rewritten in the Hamiltonian form as follows: Z R 1 GN .x/N D 00 D D .x1 /    .xN /e i dxΠ.x/@0 .x/H . .x/;.x// : S0 (30.78) To verify the equivalence of relations (30.77) and (30.78), it is necessary to perform the translation-type change of variables [see (30.25)] .x/ D

0

.x/ C @0 .x/

(30.79)

in (30.78) and to carry out integration with respect to the measure D 0 : Relation (30.78) agrees with representation (30.72) written on the basis of analogy with a mechanical quantum system. Remark 30.6. Expanding the exponential in relation (30.77) in a series in  and integrating each term of this series with the use of relation (30.14), we obtain the same perturbation series for the Green function GN : Remark 30.7. Using relations (30.9) and (30.10), one can easily verify that the formalism developed above can be generalized to the case of complex scalar fields and to the case of self-interaction of several scalar fields. An analogous statement is true for vector fields. However, the procedure of construction of path integrals for an electromagnetic field and vector gauge fields has its specific features due to additional conditions (constraints). Therefore, it must be based on an analogy with the corresponding mechanical system described in Section 30.3. In what follows, we show that the formalism of integration can also be developed for matter fields, i.e., Fermi fields.

30.5

Path Integral Representation for Fermi Fields

The construction of a path integral for Fermi fields is based on the notion of Grassmann variable a or many Grassmann variables a1 ; : : : ; an ; which are called generatrices of a finite-dimensional Grassmann algebra and satisfy the anticommutation

412

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

relations Œai ; ak C D 0;

i; j D 1; n:

To apply the method of Grassmann variables to the description of quantum Fermi systems, it is necessary to consider Grassmann algebras with involution, i.e., to consider  the conjugate variables ai along with variables ai : In this case, the complete system of commutation relations has the form 

Œai ; ak C D ıik ;





Œai ; ak C D Œai ; ak C D 0:

(30.80)

Using (30.80), we obtain the important relations 

ai2 D a2i D 0:

(30.81)

An example of a Grassmann variable a is the complex matrix   0 z aD ; jzj2 D z zN D 1: 0 0

(30.82)

On a Grassmann algebra, one can define a space of analytic functions and construct differential and integral calculus. For details, see, e.g., [14], Chapter 3, and [50]. Here, we present only the main relations necessary for the construction of ordinary integrals and their continual analogs with respect to Grassmann variables. To define integrals with respect to the variables a1 ; : : : ; an ; we introduce symbols da1 ; : : : ; dan so that they satisfy the anticommutation relations 





Œdai ; daj C D Œd ai ; d ak C D Œdai ; d aj C D 0; Œai ; daj C D

 Œdai ; aj C





D Œai ; d ak C D

 Œai ; d aj C

(30.83) D 0: 

We define single integrals with respect to the variables ai and ai by the relations Z Z Z Z    ai dai D ai d ai D 1; dai D d ai D 0: (30.84) Using (30.80)–(30.84), we define the n-fold integrals Z





Z

P.a; a/d ada WD





P.a1 ; : : : ; an I a1 ; : : : ; an /

1 Y



d ai dai

iDn

on the polynomials 

P.a; a/ D

X .s/ N n ;.s/n

sNi ;si 2¹0;1º;iD1;n

where C.Ns /n I.s/n 2 C:





C.Ns /n I.s/n .a1 /sN1    .an /sNn .a1 /s1    .an /sn ;

(30.85)

413

Section 30.5 Path Integral Representation for Fermi Fields

To determine the general form of a polynomial of degree N D .N 1 ;    ; N n ; N1 ; : : : ; Nn /; the summation over .Ns /n D sN1 ; : : : ; sNn and .s/n D .s1 ; : : : ; sn / should be carried out for sNj D 0; 1; : : : ; N j ; and sj D 0; 1; : : : ; Nj ; j D 1; : : : ; n: However, by virtue of (30.81), only the terms with sNj ; sj 2 ¹0; 1º “survive.” For the same reason, the expansion of every analytic function in a series is a polynomial of the form (30.85). Using properties (30.80)–(30.84), we can take the following integral, which is an analog of the Gaussian integral (30.9) (see Problem 30.7): Z   (30.86) In D e .a;Ba/ d ada D det B; where B is an n  n skew-Hermitian matrix. For this purpose, we need the following generalization of this relation, which is analogous to (30.11): R









 e .a;Ba/C.;a/C.a;/ d ada 1 D e .;B / ; R   a;Ba e d ada

(30.87)

where a new extended Grassmann algebra is constructed on generatrices a1 ; : : : ; an and 1 ; : : : ; n that satisfy the same relations (30.80)–(30.84). Relation (30.87) follows from (30.86) if we perform the substitution ai ! ai C .B 1 /ij j in (30.87). Further, on the formal level, relations (30.86) and (30.87) are generalized to the continual case, i.e., to the case where the generatrix of the Grassmann algebra is the Grassmann field .x/; x 2 M; and a continuous analog of the matrices B and B 1 is associated with the differential matrix operator that defines the left-hand side of the Dirac equation (6.7) and its Green function, i.e., B.x; y/ D i.i   @  m/x ı.x  y/; 1 B 1 .x; y/ D S c .x  y/: i The corresponding relation has the form R

D ND

De

R

dx

ei R

R

R dx N .x/.i  @ m/ .x/Ci dxΠN .x/.x/C.x/ N .x/ R R  D N D e i dx N .x/.i @ m/ .x/

1 c dy .x/ N i S .xy/.y/

where D ND

D

;

4 Y Y x2M4 ˛D1

(30.88)

d N ˛ .x/d

˛ .x/:

(30.89)

414

Chapter 30

Method of Functional (Path) Integrals in Quantized Field Theory

Using (30.88), we obtain a formula for the moments of the measure d WD e i

R

dx N .x/.i  @ m/ .x/

D ND

by analogy with (30.14). The Grassmann variational derivatives ı=ı.x/ and ı=ı .x/ N act on the corresponding functionals in the same way as the ordinary variational derivatives (28.24) but with regard for their anticommutation with N ; ; ; N and : On the other hand, the application of this formula to an arbitrary polynomial of the Grassmann variables N and is equivalent to the application of the Wick theorem and the calculation of vacuum averages of the same polynomial with respect to the quantum Dirac fields N .x/ and .x/:11 This analogy enables one to represent the Green function of Fermi and Bose fields in the form of a path integral. In conclusion, we write this representation for fields with Yukawa interaction Lagrangian (30.90) LI .x/ D g W N .x/ .x/'.x/ W : We have G2nIm ..x/n I .y/n I .z/m / D '.z1 /    '.z/m e ig

1 S0 R

Z

.x1 /    .xn / N .y1 /    N .yn /

W N .x/ .x/'.x/Wdx

d. N ; ; '/;

(30.91)

where d. N ; ; '/ D e i

R

N .x/.i  @ m0 / .x/dxCi

Z

and S0 D

d. N ; ; '/e ig

R

R

'.x/.20 /'.x/dx

N .x/ .x/'.x/dx

:

D N D D' (30.92) (30.93)

Remark 30.8. Relations (30.88) and (30.91) have been obtained by using an analogy between the Wick theorem and the rule of calculation of integrals of the form (30.87). However, they can also be obtained by using the standard formalism of canonical variables. In contrast to scalar fields, for which an analogy between representations in the canonical form (30.78) and covariation form (30.77) is established with the use of the change of variables (30.79) and integration with respect to the canonical momentum variables D 0 ; for fermions this procedure is not necessary because the canonical momenta pj  i .x/ (see (6.129)) are not connected with qPj  @0 .x/ and, by virtue of (4.8), one has X pj . /qPj . /  H0 .p. /; q. // D L  N .x/.i   @  m/ .x/ j 11 To

avoid misunderstanding, one should distinguish between the quantum fields Grassmann variables of integration N .x/; .x/:

.x/; N .x/ and the

Section 30.5 Path Integral Representation for Fermi Fields

415

and [see (30.40) and (30.89] DpDq D D N D : Therefore, integration with respect to momentum and coordinate variables coincides with integration with respect to N and ; which guarantees the equivalence of Hamiltonian and covariation representations.

Chapter 31

Problems to Part V

Problem 25.1.

Prove that a free field of the form (25.16) satisfies the Klein–Fock– Gordon equation (5.6).

Problem 25.2.

Show that the Hamiltonian of self-interaction of the real scalar field (25.10) can be rewritten in the form (25.21)–(25.22).

Problem 25.3.

Find estimates of the form (25.31) for the operators H3;1 ; H2;2 ; H1;2 ; and H0;4 :

Problem 26.1.

Prove that the evolution operator defined by (26.20) satisfies conditions (26.21)–(22.23) on normalized states of the Hilbert space H .

Problem 26.2.

Write conditions for the relativistic invariance of the coefficient functions of the S-matrix for the following interactions: (a) LI D  W ' 4 W; (b) LI D  W N ' W :

Problem 26.3.

Prove that the S-operator defined with the use of self-adjoint operators H and H0 by relations (26.31), (26.28), and (26.19) commutes with the operator H0 :

Problem 27.1.

Prove that the sequence of coefficient functions of the Wick ordered functional T .W ' 4 .x/ W W / has the form W a4 .x/ W f ; where f is the sequence of coefficient functions of the Wick ordered functional (see (27.1)).

Problem 27.2.

Derive Equation (27.24) using the formal definition of operator symbols a˙ .x/:

Problem 27.3.

Write an expression for the Wick ordered functional in the case of Yukawa interaction LI .x/ D  W N .x/ .x/'.x/ W and establish the rules of action of the “operators” ‰ ˙ .x/ and ‰ ˙ .x/ on a sequence of coefficient functions.

Problem 27.4.

Write equations of resolvent and evolution types for the Yukawa model LI .x/ D  W N .x/ .x/'.x/ W :

417

Chapter 31 Problems to Part V

Problem 28.1.

Derive Equation (28.11) from the representation (28.7) of the Green functions by using the generalized Wick theorem.

Problem 28.2.

Deduce an equation for the vertex strongly connected Green functions 3 .x1 ; x2 ; x3 / in the theory ' 3 and 4 .x1 ; : : : ; x4 / in the theory ' 4 :

Problem 29.1.

Using the Pauli–Villars regularization, select the divergent part of the vertex function 4 (for the theory ' 4 ) in the second order of perturbation theory.

Problem 29.2.

Solve Problem 1 for the self-energy particle 2 :

Problem 29.3.

Select a divergence and determine the character of the contribution of the self-energy second-order diagram in the model with interaction Lagrangian ' 3 .x/ by the Pauli–Villars method.

Problem 29.4.

Determine the degree of divergence of the diagram presented in Fig. 25.1. Prove that the contribution of this diagram is divergent.

Problem 29.5.

Determine the degrees of divergence of the following diagrams (in the space M4 ):

.2/

a

b

c

With what interaction Lagrangian are these diagrams associated? Which contributions are convergent and which are divergent in these diagrams? Problem 29.6.

Find the maximum index of a vertex in models with interaction Lagrangians '44 ; '36 ; and '46 : Which theories are renormalizable and which are nonrenormalizable (the subscript denotes the dimension of the Minkowski space Md )?

Problem 29.7.

Express the degree of divergence of a Feynman diagram that can be decomposed into k blocks G1 ; : : : ; Gk in terms of the degrees of divergence of blocks.

Problem 29.8.

Prove Lemma 29.9 (Section 29.4).

418

Chapter 31

Problems to Part V

Problem 29.9.

Analyze the form of counterterms and write the complete renormalized Lagrangian for the interaction '43 :

Problem 30.1.

Verify relation (30.15) for the functionals  Z F .y/ D exp i z.x/y.x/dx :

Problem 30.2.

Verify relation (30.15) for the functionals F .y/ D P .y/; where Z P .y/ D Pn .x1 ; : : : ; xn /y.x1 /    y.xn /:

Problem 30.3.

Verify relation (30.17) using representation (30.15).

Problem 30.4.

Prove relation (30.31).

Problem 30.5.

Prove relation (30.32).

Problem 30.6.

Prove relation (30.76).

Problem 30.7.

Prove (for n D 2; 4) that relation (30.86) holds for any nonsingular skew-Hermitian n  n matrix B:

Part VI

Axiomatic and Euclidean Field Theories

Part VI

Axiomatic and Euclidean Field Theories

421

The development of a new physical theory starts, as a rule, from the theoretical substantiation of the phenomena observed in the nature or in carefully prepared and purposeful experiments. This kind of “spontaneous” development of the theory always leads to the appearance of problems (mathematical, technical, etc.). Sometimes, these problems are “spontaneously” solved but, in numerous other cases, lead to controversial results. In these cases, it becomes necessary for the researchers to reconsider the theory on the level of its foundations (axioms) and from the viewpoint of completeness of the assumptions which, in the course of time, turn into axioms. The first attitude of scientists to the axiomatic theory is somewhat skeptical, especially if they prefer to compute something specific but not to prove the existence of objects that were not observed earlier. However, in the course of time, it often turns out that the development of the axiomatic approach and, especially, the discovery of nontrivial examples satisfying the formulated system of axioms lead to new theoretical (and sometimes even experimental) results or to the formation of a new theoretical approach, which can be used in the other fields of the theoretical physics. As an example of this type of development, we can mention the constructive quantum field theory (CQFT) formed in attempts to find a nontrivial model satisfying the system of Wightman’s axioms . The CQFT is now extensively used in the statistical mechanics, solid state theory, etc. An important role played by the axiomatic theory is explained by the fact that a system of axioms, first, formulated for the basic objects of a physical theory can be rewritten in the form of a new system of axioms for the other objects connected, clearly, with the original objects but having absolutely different nature and, as a result, the investigation of these new objects becomes much simpler. As an example of this approach, we can mention the system of axioms for Wightman or Schwinger functions (Euclidean Green functions). The Heisenberg approach [88] was the first attempt to go beyond the framework of traditional approaches to the construction of the quantum theory of elementary particles. He introduced the notion of S-matrix as the main mathematical object in terms of which all observable quantities should be expressed. First, the idea was that it suffices to formulate a certain (necessary and sufficient) number of axioms for the S-operator and introduce the notion of the field. However, this approach appeared to be too general. Later, Heisenberg abandoned this approach himself. The subsequent development of the axiomatic approach was based on the study of local quantities: fields, Hamiltonians, etc. In the present Part, we briefly describe the main axiomatic approaches and indicate the literature required for a more profound investigation of this circle of problems. However, the main aim of the part is to fluently (i.e., logically) pass to the Euclidean field theory, which is more rigorous from mathematical point of view and, at present, generally accepted despite the fact that its substantiation remains incomplete. In conclusion, we advise the reader to get acquainted with the introduction to the work by Jost [101] in which he presents an original point of view on the history of development of the quantum theory and its axiomatization. Despite the fact that these

422

Part VI Axiomatic and Euclidean Field Theories

thoughts were formulated almost 50 years ago, they remain actual in the contemporary stage of development of the quantum theory. We also advise the reader to read an interesting brief survey of the history of axiomatic approach presented in [23].

Chapter 32

Wightman Axiomatics

The Wightman axiomatics is, on the one hand, the theory most substantiated from the mathematical point of view but, on the other hand, least connected with the physical reality because it does not contain the notion of elementary particle and does not establish relationships with the experimentally observed quantities. Nevertheless, we present a brief survey of the Wightman axiomatic theory because the formal and axiomatic aspects of this theory make it possible to substantiate the transition to the Euclidean quantum field theory. As in the previous parts, we formulate the axioms for real scalar fields. This restriction is not essential from the viewpoint of the general principles. The complete description of the Wightman axiomatic theory can be found in in [191] (see also [101]).

32.1

Wightman Axioms for Real Scalar Fields

In Part IV, we have formulated the system of axioms that enables one to construct a quantum field theory based on classical ideas and generalization of classical and quantum nonrelativistic mechanics. The system of Wightman axioms is a part of postulates AQ1–AQ9 (Section 17.3). This is the minimum number of axioms that define a quantum field theory. A1. Existence of a relativistic-invariant state 1. The space of states is a separable Hilbert space H : Each physical state is associated with unit bundle ‰ WD ¹e i' ˆj ' 2 R; ˆ 2 H ; kˆkH D 1º: 2. The relativistic law of transformation of vectors in the space H is defined by a continuous unitary transformation of the proper orthochronous Poincaré group .a; ƒ/ ! U.a; ƒ/: 3. The operator for simple translation .ƒ  1/ U.a; 1/ D exp¹iP  a º; where P D .P  /3D0 is the unbounded self-adjoint operator of the energy-momentum 4-vector. Its spectrum consists of the isolated point p D .0; 0; 0; 0/ and points 2  2 0 lying in the cone V C m WD ¹p j p D p p > m ; p > 0º that lies in the cone of future (1.7) (spectral condition, see AQ5).

424

Chapter 32

Wightman Axiomatics

4. The eigenvalue p D .0; 0; 0; 0/; up to the phase factor e i' ; corresponds to the unique eigenvector ˆ0 invariant under the transformations U.a; ƒ/: U.a; ƒ/ˆ0 D ˆ0 ;

(32.1)

which is called a vacuum state. A2. Existence of a relativistic-invariant quantum field 1. Over the space of test functions f 2 S .M/; there exists a linear operator-valued functional ' such that '.f / D < f; ' > is a linear closed (in the case at hand, self-adjoint) operator in the space H with everywhere dense (see Section 22.2.1) domain of definition D.'/; furthermore, U.a; ƒ/D.'/ D.'/;

'.f /D.'/ D.'/;

ˆ0 2 D.'/

and the vacuum vector ˆ0 is a cyclic vector in H (see the definition in Section 22.2.3). 2. The operators of the field '.f / satisfy the following conditions of relativistic invariance: (32.2) U.a; ƒ/'.f /U 1 .a; ƒ/ˆ D '.fƒ;a /ˆ for any ˆ 2 D.'/: Here, fƒ:a .x/ D f .ƒ1 .x  a//: 3. The fields '.f / satisfy the condition of local commutation. Therefore, if supports of the test functions f and g are separated by a space-like interval f .x/g.y/ D 0 for the points x; y

for which .x  y/2 < 0;

then Œ'.f /; '.g/ ˆ D 0;

ˆ 2 D.'/:

The system of axioms A1–A2 is also called the Garding–Wightman axioms [220]. Axioms A1–A2 are, in fact, the definition of a quantum field theory. Therefore, constructing the space H and the operators with conditions A1–A2, one obtains a certain example of the theory, which can be regarded as a quantum field theory. However, this example can be trivial. For example, this is true for a scalar field of the form '.x/ D const  1; where 1 is the identity operator in the space H : Another example is the free field defined in Chapter 18 and Chapter 22. A free field is not trivial from mathematical point of view and, in fact, substantiates consistency of the system of axioms A1–A2. However, it is of no interest from the physical point of view because it is an idealization of an actual physical system. For a constructed field or a system of fields to be nontrivial both from mathematical and physical points of view, a certain criterion or additional assumptions are required. For example, the existence of

Section 32.2 Wightman Functions and Their Properties

425

asymptotic states of the scattering theory and their completeness can be used as this additional criterion. Often, this condition is written in the form H D Hin D Hout ;

(32.3)

where Hin and Hout are, respectively, the Hilbert spaces of states before and after scattering. For the verification of condition (32.3), a certain constructive method for calculation (or determination) of these states is necessary. There are several approaches to this problem. We refer the reader to [101], Part VI where this problem is considered in detail within the framework of the theory of Haag–Ruelle asymptotic fields.

32.2

Wightman Functions and Their Properties

First, we formally introduce the functions wn .x1 ; : : : ; xn / D .ˆ0 ; '.x1 /    '.xn /ˆ0 /;

(32.4)

where '.x/ is the Heisenberg scalar field and ˆ0 is the physical vacuum state. The existence of '.x/ as an operator-valued functional over the space of tempered test functions S .M/ and the vacuum vector ˆ0 2 H is a corollary of axioms A1 and A2. To attach a meaning to (32.4), it is necessary, first, to consider the multilinear functional wn .f1 ; : : : ; fn / WD .ˆ0 ; '.f1 /    '.fn /ˆ0 / of arguments fj 2 S .M/; j D 1; : : : ; n: By virtue of axioms A1, 4 and A2, 1, it is well-defined on S .M/˝n : According to the Schwartz theorem on kernel (see, e.g., [191], Chapter 2), there exists a unique tempered distribution over S .M/˝n : Relation (32.4) defines this function. We briefly formulate main properties of functions (32.4), which are corollaries of axioms A1–A2. W0. Moderate growth. Generalized functions (distributions) wn .x1 ; : : : ; xn / are elements of the space S 0 .M˝n /: W1. Covariance. wn .f1Iƒ;a ; : : : ; fnIƒ;a / D wn .f1 ; : : : ; fn /:

(32.5)

Recall that fiIƒ;a .x/ D fi .ƒ1 .x  a//: In terms of functions (32.4), relation (32.5) has the form wn .x1 ; : : : ; xn / D wn .ƒx1 C a; : : : ; ƒxn C a/:

(32.6)

426

Chapter 32

Wightman Axiomatics

W2. Spectral condition. For each function wn ; there exists a tempered distribution (see, e.g., [206]) Wn1 . 1 ; : : : ; n1 / that depends on the difference coordinates j D xj C1  xj ;

j D 1; 2; : : : ; n  1;

(32.7)

wn .x1 ; : : : ; xn / D Wn1 . 1 ; : : : ; n1 /;

(32.8)

such that furthermore, the Fourier transforms of these functions are connected by relations (see Problem 32.2) wQ n .p1 ; : : : ; pn / e n1 .p1 ; p1 C p2 ; : : : ; p1 C    C pn1 / (32.9) D .2/4 ı.p1 C    C pn /W e n1 satisfy the condition and the functions W e n1 .q1 ; : : : ; qn1 / D 0 W if at least one qi does not belong to the spectrum of the energy-momentum operator. W3. Hermiticity. wn .x1 ; : : : ; xn / D wn .xn ; xn1 ; : : : ; x1 /:

(32.10)

W4. Locality. wn .x1 ; : : : ; xk1 ; xk ; xkC1 ; xkC2 ; : : : ; xn / D wn .x1 ; : : : ; xk1 ; xkC1 ; xk ; xkC2 ; : : : ; xn /;

(32.11)

if .xk  xkC1 /2 < 0: W5. Positive definiteness. For any finite sequence fO D ¹fj ºjND1 of test functions fj 2 S .M4j /; the inequality w. O fOI fO/ WD

N X j;kD0

wj Ck .f j ˝ fk / D

1 Z X

Z 

fj .x1 ; : : : ; xj /

j;kD0

 wj Ck .xj ; xj 1 ; : : : ; x1 ; y1 ; : : : ; yk /  fk .y1 ; : : : ; yk /dx1    dxj dy1    dyk > 0 is true.

(32.12)

427

Section 32.3 Reconstruction Theorem

W6. Weak cluster property. If a 2 M4 is a space-like vector, i.e., a2 < 0; then w  lim wn .x1 ; : : : ; xj ; xj C1 C a; : : : ; xn C a/ !1

D wj .x1 ; : : : ; xj /wnj .xj C1 ; : : : ; xn /:

(32.13)

Relations (32.6)–(27.13) should be understood as operations with generalized functions(distributions) in the space S 0 .M4n /: Here, only properties of Wightman functions for real scalar fields are presented. The general case is considered in [191], Chapter 3. The proof of properties W1–W6 is also given there (for the proof in the case of a scalar field, see Problems 32.1–32.4).

32.3

Reconstruction Theorem

Properties W1–W6 are corollaries of axioms A1–A2. In this chapter, we show that the converse statement is also true: the existence of a sequence of generalized functions of moderate growth that possess properties W1–W6 guarantees the existence of a space of states and a local Heisenberg field that satisfy axioms A1–A2. This is a corollary of a so-called theorem of reconstruction. Theorem 32.1 ([191], Theorems 3–7). Let ¹wn º1 nD1 be a sequence of generalized functions of moderate growth, i.e., functionals over a sequence of spaces of tempered test functions ¹S .M4n /º1 nD1 ) that satisfy conditions W1–W6. Then there exist the separable Hilbert space H ; the continuous unitary representation U.a; ƒ/ of the proper orthochronous Poincaré group in H ; the unique vacuum vector ˆ0 (or the unique bundle ‰0 D e i' ˆ0 ) invariant under U.a; ƒ/; and the self-adjoint operator '.f /; f 2 S .M4 /; such that wn .x1 ; : : : ; xn / D .ˆ0 ; '.x1 /    '.xn /ˆn /: Note that “uniqueness” means uniqueness up to unitary equivalence. For the complete proof of this theorem, see [191], Chapters 3 and 4. We only call the reader attention to some moments. For the construction of the space of states H ; first, the linear space E of infinite sequences of complex functions fO D .f0 ; f1 ; : : : ; fk ; : : :/; where f0 2 C and fk 2 S .M4k /; is used. Then, the scalar product is constructed in E: For any fO; gO 2 E; .fO; g/ O D w. O fOI g/ O D

1 X j;kD0

wj Ck .f j ˝ gk /;

428

Chapter 32

Wightman Axiomatics

where w. O fOI g/ O is defined by relation (32.12) with gk instead of fk and N D 1: The property W5 guarantees nonnegativity of the norm kfOk2 D .fO; fO/: It is necessary to supplement the space H with sequences fO for which kfOk < 1 and to supplement E with this norm. The supplement of the linear space E leads to the complete separable space H : The linear unitary transformation U.a; ƒ/ is defined by the relation U.a; ƒ/fO D fOƒ;a WD .f0 ; f1Iƒ;a ; : : : ; fkIƒ;a ; : : :/; (32.14) fkIƒ;a D fk .ƒ1 .x1  a/; : : : ; ƒ1 .xk  a//: The vacuum vector can be defined by the sequence ˆ0 D .1; 0; : : :/:

(32.15)

Relations (32.14) and (32.15) guarantee the condition of invariance (32.1). Remark 32.2. At first sight, it seems that the vacuum vector (32.15) coincides with vacuum vector 0 for a free scalar field (see relation (22.30)). However, the vector

0 is a vector of the Fock space F and the vector ˆ0 2 H : For a unitary operator that establishes the unitary equivalence of the spaces H and F ; the unitary image of the vector ˆ0 in the Fock space has another form and is a vector with infinite number of nonzero components. For each test function h 2 S .M/; we define the linear operator .'.h/fO/0 D 0; .'.h/fO/k D h ˝ fOk1 ;

k > 1:

It is also easy to verify (see Problem 32.5) that the set ° ± D0 WD fO 2 H j 9 n > 0; is such that fk  0 for k > n

(32.16)

(32.17)

is everywhere dense in H and D.'.h//  D0 ;

'.h/D0 D0 :

In addition (Problem 32.6), for any real h 2 S .M/; the operator '.h/ is symmetric in H ; i.e., for any fO; gO 2 D0 ; the equality .'.h/fO; g/ O D .fO; '.h/g/ O (32.18) is true. It is also easy to verify (Problem 32.7) the transformation properties of the field '.h/: (32.19) U.a; ƒ/'.h/U.a; ƒ/1 D '.hƒ;a /: The uniqueness of the vacuum state ˆ0 follows from property W6 (see the proof in [191], Chapters 3 and 4). Finally, for the proof of the spectral axiom A1, 3, see [101], Part III, Chapter 4.

Chapter 33

Other Axiomatic Approaches

33.1

Haag–Ruelle Scattering Theory (HRST)

In the case of quantization of the free fields, the notion of particles associated with these fields appears quite naturally. Therefore, the quantum theory of free fields can be regarded as the theory of noninteracting elementary particles. Unfortunately, at present, this is not clear for the Heisenberg interacting field. A similar problem is encountered in the axiomatic approach. Intuitively, it is clear that, as x 0 ! ˙1; the Heisenberg field '.x/ must act upon the vectors of states as a free field (however, this does not mean that the asymptotic field coincides with the operator of free field, i.e., the field without interaction) and the action of '.x/ on the vacuum must give a single-particle state corresponding to given quantum numbers (spin, charge, mass, etc.). Within the framework of the axiomatic Wightman theory, partial answers to these questions are given by the HRST. For the detailed presentation of this theory, see, e.g., [101], Chapter VI, and [23] Chapter 4, Section 1. In the present chapter, we give only the main theorems and make some conclusions which enable us to understand the general picture of construction of the axiomatic fields and states which can be associated with the actual interacting particles. Axiom A2 determines the existence of an operator-valued functional ' over the space of test functions S .M/; i.e., for any f 2 S .M/; of the linear operator '.f / with everywhere dense domain of definition D.'/  H : To construct time asymptotics (as x 0 ! ˙1) of the field operators, it is necessary to define the operator Z (33.1) '.x 0 ; g/ D '.x/g.x/d x; g 2 S .R3 /; in the space H . Unfortunately, it is not a corollary of axiom A2, 1. Thus, for the construction of an asymptotic field, we first construct constructed a certain auxiliary field A such that its Fourier transform e A is connected with Fourier transform 'Q of the field ' by the relation e Q (33.2) A.p/ D h.p 2 /'.p/; where h 2 C 1 .R/ and satisfies the conditions h.m2 / D 1;

h.p 2 / D 0 for jp 2  m2 j > m2 :

(33.3)

430

Chapter 33

Other Axiomatic Approaches

The field thus defined is smooth with respect to the variable x 0 ; i.e., “smoothing” it with respect to the space variable, i.e., representing in the form Z 0 A.x ; g/ D A.x/g.x/d x; g 2 S .R3 /; (33.4) after elementary calculations (Problem 33.1), we establish that A.x 0 ; g/ is a linear operator in the space H with domain of definition D.'/: Indeed, the new field A.x 0 ; g/ is easily expressed in terms of the field ' as follows: Z 0 A.x ; g/ D f .y 0  x 0 ; y/'.y/dy; (33.5) where f .y/ D

1 .2/4

Z

gQ .p/h.p 2 /e ipy dp:

(33.6)

It follows from the construction of the function h and gQ 2 S .R3 / that gQ h 2 S .M/: Thus, the function f .y 0  x 0 ; y/ 2 S .M/ for every fixed x 0 : It follows from axiom A2,1 that A.x 0 ; g/ is a well-defined operator in H because it coincides with '.f / for the special choice of f defined by relation (33.6). Recall that free fields also have this specific feature (smoothness with respect to 0 x ). Using relation (33.5), we easily verify that the field A.x 0 ; g/ is nonlocal with respect to the variable x 0 : This field satisfies all axioms just as the field ' except for the condition of locality. Now we introduce the notion of a smooth solution of the Klein–Fock–Gordon equation as a solution of Equation (5.6) that satisfies the conditions f .0; x/ 2 S .R3 /

and

fx0 0 .0; x/ 

@f .x/ j 0 2 S .R3 /: @x 0 x D0

Using the field A constructed above, we construct the operator Z $ Af .x 0 / D i f .x/@0 A.x/d x:

(33.7)

(33.8)

If f is a smooth solutions of the Klein–Fock–Gordon equation, then Af .x 0 / is the operator defined on the everywhere dense domain D.'/ (Problem 33.1). Now we formulate the main result of the Haag–Ruelle theory. Theorem 33.1. Let ˆ.x 0 / D ˆn .x 0 I .f /n / WD Af1 .x 0 /Af2 .x 0 /    Afn .x 0 /ˆ0 ;

431

Section 33.1 Haag–Ruelle Scattering Theory (HRST)

where f1 ; : : : ; fn are smooth solutions of the Klein–Fock–Gordon equation. Then in in WD ˆout there exist limit vectors ˆout n n .f1 ; : : : ; fn / and ˆn WDˆn .f1 ; : : : ; fn / such 0 that, as x ! ˙1 with respect to strong convergence in the space H ; lim

x 0 !1

kˆn .x 0 /  ˆin n k D 0;

lim

x 0 !C1

kˆn .x 0 /  ˆout n k D 0:

Within the framework of the proper orthochronous Lorentz group, these limits are independent of the coordinate system in which operators (33.1) are defined. Let Hin and Hout denote Hilbert spaces spanned by the systems of vectors 1 out 1 ¹ˆin n ºnD0 and ¹ˆn ºnD0 (i.e., Hin and Hout are closures with respect to norm conout in vergence of linear combinations of the vectors ˆin n and ˆn /: On the vectors ˆn and out ˆn ; we define the linear operations in ' in .f /ˆin n .f1 ; : : : ; fn / D ˆnC1 .f; f1 ; : : : ; fn /;

(33.9)

out ' out .f /ˆout n .f1 ; : : : ; fn / D ˆnC1 .f; f1 ; : : : ; fn /:

Theorem 33.2. Operators defined as linear expansions of operations (33.9) define two free real scalar fields ' in .x/ and ' out .x/ as operator-valued functionals of the form Z $ ' in .f / D i f .x/@0 ' in .x/d x; Z ' out .f / D i

$

f .x/@0 ' out .x/d x:

The fields ' in and ' out satisfy the relations U.a; ƒ/' in .x/U 1 .a; ƒ/ D ' in .ƒx C a/; U.a; ƒ/' out .x/U 1 .a; ƒ/ D ' out .ƒx C a/ and (see (21.28))

b in .x/‚ b D ' out .x/; ‚'

bDT bC bP b: ‚

(33.10)

To prove Theorems 28.1 and 28.2, Wightman axioms A1–A2 should be complemented with asymptotic completeness axiom Hin D H :

(33.11)

Equality (32.3) follows from (33.11) and (33.10) because, by virtue of (33.10), b in D Hout ; ‚H

b ‚H D H:

In addition, we make the following two additional assumptions on the spectrum of the energy-momentum operator and the single-particle Hilbert space H1 :

432

Chapter 33

Other Axiomatic Approaches

1. First assumption. In addition to the isolated point p D .0; 0; 0; 0/ corresponding to the vacuum vector, the spectrum of the energy-momentum operator contains the isolated point ¹pjp0 > 0; p 2 D m2 º corresponding to the single-particle state of a particle with mass (in the case of one field) and a continuous spectrum in the cone V C 2m : 2. Second assumption. The restriction of the representation U.a; ƒ/) to H1 is irreducible and stands for spin 0 and mass m: Axioms A1–A2, together with asymptotic completeness axiom (33.11) and these two assumptions, enable one to prove Theorems 33.1 and 33.2 and, hence, to interpret the axiomatic Wightman theory in terms of particles. In addition, equality (32.3) guarantees the existence of the unitary operator S; which is uniquely defined by the conditions ' out .x/ D S 1 ' in .x/S

(33.12)

Sˆ0 D ˆ0 ;

(33.13)

and where the S-operator is the scattering operator, which, on the formal level, has been studied in detail in previous parts. The condition of Lorentz-covariance of the fields ' in and ' out ; together with (33.12), leads to invariance of the Soperator under the transformations U.a; ƒ/: U.a; ƒ/SU 1 .a; ƒ/ D S: Remark 33.3. Probably, it is impossible to construct a mathematically rigorous HRST in a certain space of states. However, it can be considered from the point of view of an abstract scattering theory with some couple of Hilbert spaces. This approach was proposed by Kato in [108]. Later, it was developed in works of many mathematicians (see, e.g., [111] and references therein). It is also necessary to note the approach prosed by Faddeev in [46], which establishes the connection between HRST and perturbation theory. It was used in certain specific models (see, e.g., [3] and [180]).

33.2

Lehmann–Symanzik–Zimmermann Axiomatics

HRST briefly described in the previous chapter, in fact, establishes the existence of the scattering operator and enables one to interpret the axiomatic Wightman theory in terms of particles. However, for the completeness of the theory, it is necessary to establish a relationship with observable quantities. In this respect, the axiomatic LSZ

433

Section 33.2 Lehmann–Symanzik–Zimmermann Axiomatics

theory (see [121, 122]) better meets these requirements. This theory appeared almost simultaneously with the Wightman theory [219]. Analyzing the perturbation theory, Lehmann, Symanzik, and Zimmermann determined a reduction formula that connects matrix elements of the S -operator with vacuum averages of the T -product of Heisenberg fields (Green functions) in out in .ˆout n ; ˆm / D .ˆn .f1 ; : : : ; fn /; ˆm .g1 ; : : : ; gm // Z c D .i /mCn GmCn .x1 ; : : : ; xn ; y1 ; : : : ; ym /f1 .x1 /    fn .xn /

 g.y1 / : : : g.ym /dx1 : : : dym ;

(33.14)

out where ˆin in and ˆn are intuitively defined just as in Theorem 33.1 and the casual c Green function GmCn has the form 2 3 N Y c GN .x1 ; : : : ; xN / D 4 .xj  m2 /5 GN .x1 ; : : : ; xN /; j D1

GN .x1 ; : : : ; xN / D .ˆ0 ; T .'.x1 /    '.xN //ˆ0 /: The reductive relation (33.14) is a main relation for calculations of physical processes. For more visuality, we rewrite it in the momentum space by formally replacing the functions fi .xi / and gj .yj / in (33.14) by wave packets corresponding to free outparticles with momenta p1 ; : : : ; pn and free in-particles with momenta pnC1 ; : : : ; pN ; furthermore, these packets do not overlap pi .m2 C p2i /1=2 ¤ pj .m2 C pj2 /1=2

for i ¤ j:

Then we have in .ˆout n .p1 ; : : : ; pn /; ˆN n .pnC1 ; : : : ; pN //

p

N

D .i 2/

N Y

.pk2  m2 /

kD1

e N .p1 ; : : : ; pn ; pnC1 ; : : : ; pN /j 0 2 2 1=2 G p D.p Cm / ;iD1;:::;N : i

i

(33.15)

At first sight, each factor in the product, for pk0 D .p2k C m2 /1=2 ; is equal to zero. e N has the structure of the However, the Fourier transform of the Green functions G product ! N Y 1 e N .p1 ; : : : ; pN / D G G.p1 ; : : : ; pN /: .2/4 i.m2  pk2  i "/ kD1

This can be easily verified by formally considering a perturbation series for Green functions.

434

Chapter 33

Other Axiomatic Approaches

Thus, the corresponding zero factors cancel on the mass surface .p 2 D m2 /: According to relation (33.15), the scattering amplitude (the left-hand side of relation (33.15)) admits an extension outside the mass surface with the use of retarded Green functions in terms of which, the spectral properties, locality, and asymptotic completeness can be expressed. Green functions (for ordered time variables, they coincide with Wightman functions) also admit an analytic extension to the complex plane of their variables. Therefore, they are limit values of certain analytic functions whose properties can be studied based on general theorems of the theory of functions of many complex variables. For the rigorous substantiation of relation (33.15), the following additional assumptions as compared with Wightman theory are necessary for the LSZ axiomatics: ALSZ1. The asymptotic completeness H D Hin D Hout : ALSZ2. The asymptotic LSZ condition lim .ˆ; Af .t /‰/ D .ˆ; '

t!˙1

out in

.f /‰/;

where Af .t / is defined with the use of (33.2)–(33.8) and ˆ; ‰ 2 D.'/: ALSZ3. The existence of generalized retarded functions. These functions are expressed in terms of multiple commutators ri .x0 ; : : : ; xn / D

0 n1 X Y  j D0

0 0 .x.j /  x.j C1/ /

  ˆ0 ; Œ: : : Œ'.x.0/ /; '.x.1/ /; '.x.2/ / : : : ; '.x.n/ /ˆ0 ; where the sum over  is taken for all rearrangements of indices ¹0; 1; 2; : : : ; nº for which .0/ D i: The functions ri .x0 ; : : : ; xn / are generalized functions of moderate growth. In addition, within the framework of the LSZ axioms, a connection between asymptotic fields and the Heisenberg field '.x/ is established. If, for the field '; the current is defined by the relation j.x/ D .  m2 /'.x/; then so-called Yang–Feldman equations Z in '.x/ D ' .x/ C D ret .x  yI m/j.y/dy Z out D ' .x/ C D adv .x  yI m/j.y/dy

(33.16) (33.17)

are true. For the proof of relations (33.14), (33.16), and (33.17) see, e.g., [101, 23, 22].

Section 33.3 Bogoliubov–Medvedev–Polivanov (BMP) Axiomatic Approach

33.3

435

Bogoliubov–Medvedev–Polivanov (BMP) Axiomatic Approach

Probably, the BMP axiomatic approach is most close to the Heisenberg idea. The main object is the S-matrix and local quantities (fields, currents, etc.) are defined as variational derivatives of the S-operator that, first, is considered as a functional of classical additions to asymptotic fields. The approach is based on an operatorfunctional of the form XX Z FNI.˛/N .x/N S./ D N .˛/N

M4jNj

    .1/ .1/ .1/ .k/ .k/ .k/ .k/  W  .1/ C  .1/ .x1 /     .k/ .xNk / C  .k/ .xNk / W ; ˛1

˛1

˛N

˛N

k

(33.18)

k

where N D .N1 ; : : : ; Nk /; jNj D N1 C    C Nk ;   .1/ .1/ .2/ .2/ .k/ .k/ .x/N D x1 ; : : : ; xN1 I x1 ; : : : ; xN2 I : : : I x1 ; : : : ; xNk ; ˛ are quantum asymptotic fields in the given theory (k species of fields), and ˛ are classical additions (functional arguments of the S -operator). If the fields ˛ are Bose fields, then ˛ are ordinary functions, if ˛ are Fermi fields, then ˛ are Grassmann variables (see Section 30.5), i.e., they anticommute among themselves and with fields ˛ : It is clear that the S-operator is defined by the relation S D S./j0 : The operator-valued functional S./ is called the S -matrix extended outside the mass surface. For different versions of these extensions, see [22], Chapter 14. We also define variational derivatives of S with respect to the quantum fields ˛ by the formal relation ˇ ˇ ˇ ı N S./ ıN S ˇ WD : (33.19) ˇ .i1 / .i1 / .iN / .iN / .i1 / .i1 / .iN / .iN / ˇ ı .i1 / .x1 /    ı .iN / .xN / ı .i1 / .x1 /    ı .iN / .xN / ˇ ˛ ˛ ˛ ˛ 1

N

1

˛;

Note that, for the Fermi fields

0

the derivatives are anticommutative, i.e.,

ı2A ı

N

˛1 ı

˛2

D

ı2A ı ˛2 ı

: ˛1

Taking (33.19) into account, we define so-called radiation operators, which are main local objects within the framework of the BMP approach .i/

.x/N WD HN .x1 ; : : : ; xN / D H.˛/N N where S  D S 1 and i˛ are In-fields.

ıN S ıi˛11 .x1 /    ıi˛NN .xN /

S ;

(33.20)

436

Chapter 33

Other Axiomatic Approaches

Using interaction (25.3) as an example, we show how, with the use of operators (33.20), to determine an operator-valued distribution for a scalar field. To this end, we introduce the operator-valued distribution corresponding to the current j.x/ D i

ıS ı' out .x/

S  D iS 

ıS : ı' in .x/

(33.21)

Note that definition (33.21) is independent of the type of interaction (see, e.g., [10, Chapter 14]). If the interaction contains fields of particles of several species, then we obtain relations for current operators for particles of each species. To determine the field operator, we use one of the Yang–Feldman equations (33.16), (33.17): Z ıS out '.x/ D ' .x/ C i dyD adv .x  yI m/ out S ı' .y/ Z ıS in : (33.22) D ' .x/  i dyD ret .x  yI m/S  in ı' .y/ Using formal operations, the expression for the Heisenberg field can be represented in the form (see [22], Section 14.2) '.x/ D S  T .S' in .x// D T .' out .x/S /S  :

(33.23)

We briefly discuss main requirements (axioms) for the operator S and the radiation operators (33.20) within the framework of the BMP approach. S1. Existence of a space of states A space of asymptotic states is the equipped(see, e.g., [13]) Hilbert space HC

H H in which representations of the proper inhomogeneous Lorentz group and groups of external symmetries are realized. The triple .HC ; H ; H / must be considered as an extension of the Hilbert space H by generalized solutions (of the form of a plane wave) from H ; which are functionals over smooth functions HC H : The necessity of this extension is caused by the condition of asymptotic completeness, which we formulate together with postulate of spectrality. S2. Spectrality and asymptotic completeness There exists a unique invariant vacuum state ˆ0 and a collection of generalized physical states ˆn .kn / 2 H that correspond to positive masses and energies and form a complete system of vectors in H : Within the framework of the BMP approach, this postulate is used for the decomposition of a matrix element of the product of two operators A and B in the complete system of intermediate states .ˆ; AB‰/ D .ˆ; Aˆ0 /.ˆ0 ; B‰/ XZ C d kn .ˆ; Aˆn .kn //.ˆn .kn /; B‰/; n

Section 33.3 Bogoliubov–Medvedev–Polivanov (BMP) Axiomatic Approach

437

where kn is the summary momentum of the intermediate state n and n is the collection of all other quantum numbers. S3. Stability of vacuum and single-particle state Sˆ0 D ˆ0 ;

Sˆ1 D ˆ1 :

(33.24)

S4. Unitarity of the S -matrix SS  D S  S D 1:

(33.25)

Remark 33.4. Expansion (33.18) of the operator-valued functional S./ can be considered as the process of extension of the S-matrix outside the mass surface. This means that, in expansion (33.18), the asymptotic field, together with its classical term, ˛ .x/; which already does i.e., ˛ .x/ C ˛ .x/; can be considered as a separate field e not satisfy the free Klein–Fock–Gordon equation. Moreover, the process of taking the variational derivative with respect to S./ itself requires an extrapolation of the S-operator outside the mass surface. For details, see [22], Chapter 14. However, the process of extension of the S-operator outside the mass surface is not unique. This nonuniqueness can be eliminated by imposing on the functional S./ the same conditions as on the S-matrix. Assume that conditions (33.24) and (33.25) are also satisfied for the operator S./: In addition, for calculations within the framework of the BMP approach, relations that express the commutators of creation and annihilation operators of free asymptotic fields with S-operator are important. These commutators can be easily derived by using expansion (33.18) and relations of the form (26.74). It can be shown (see, e.g., [26], Section 45.3) that they are expressed in terms of variational derivatives of the S-operator with respect to asymptotic fields. In this connection, the condition of preservation of the form of these commutation relations for the extended S -matrix is important. Finally, last two requirements are connected with radiation operator introduced by relation (33.20). S5. Space of distributions Vacuum averages over the radiation operators (33.20) are tempered distributions, i.e., belong to the space S 0 .M4N /: It is worth noting that this mathematical axiom has a deep physical meaning. In fact, according to this axiom, the matrix elements of the S -operator are polynomially bounded in momentum variables. This condition is not consistent with perturbation theory for the S-matrix because, say, for nonrenormalizable interactions (see Part V), after the application of the R-operation, the Feynman amplitude contains polynomials in momentum variables of an arbitrarily large degree. Thus, this axiom restricts a class of interactions (see [23], Chapter 4, Section 3.5).

438

Chapter 33

Other Axiomatic Approaches

S6. Microcasuality of the S -operator If x . y (i.e., x 0 < y 0 and .x  y/2 < 0), then the following relation is true:   ıS ı  S D 0: (33.26) ı˛ .x/ ıˇ .y/ Relation (33.26) can be understood in the sense of definition of (33.19). More rigorously, we can write   ıS./  ı S ./ D 0 (33.27) ı˛ .x/ ıˇ .y/ for x . y: Axioms S1–S6 enable one to use the methods based not on the perturbation theory for the S -matrix and obtain important relations (the so-called dispersion relations) for the physical scattering amplitudes. These relations play an especially important role in the theory of strong interactions for which the perturbation theory is not correct even on the formal level. For the detailed description of the procedure of getting dispersion relations, see [26], Part IX.

Chapter 34

Euclidean Field Theory

The development of the Euclidean field theory (EFT) can be conventionally split into two periods. The first period is connected with the appearance of the Euclidean approach to the elimination of technical difficulties caused by the presence of divergencies in the Feynman diagrams. The second period is connected with the appearance of the notion of Euclidean field as a mathematical object that can be connected with the actual physical fields. In 1949, Dyson [43] proposed (for the first time) a formal procedure of analytic continuation of the Feynman amplitudes into the domain of imaginary energies p 0 D ip 4 ; p 4 2 R1 ; with an aim to avoid poles in the expressions for Feynman propagators N N Z.p 0 ; p/ Z.ip 4 ; p/ ! : 4 2 0 2 2 4 2 .2/ i.m  .p / C p  i "/ .2/ i.m C .p 4 /2 C p2 /

(34.1)

In [217], Wick developed this idea and proposed to rotate the contour by 90ı in the complex plane of the variable p 0 (see Section 34.1). Similar ideas were independently proposed by Schwinger [182] and Nakanishi [135]. A new stimulus for the development of these ideas was given by Schwinger [183] and Nakano [136] who proposed to formulate the field theory in terms of fields in Euclidean domains constructed as operator-valued functionals (by analogy with the ordinary fields). Later, Symanzik [195] associated these formal objects with generalized random processes and laid foundations of the future Euclidean axiomatics. In the early 1970s, Nelson [138, 139] developed Symanzik’s ideas and showed a way of returning to the Minkowski domain, i.e., a method for the reconstruction of a quantum field according to an Euclidean field satisfying certain conditions. In the mid-1970s, Osterwalder and Schrader [146] (with later corrections in [69, 147]) formulated a system of axioms for the Green functions in the Euclidean domain (Schwinger functions) and proved the possibility of their analytic continuation to Wightman functions. Thus, the equivalence of the Euclidean and Wightman approaches was proved on the level of axiomatic theory. At the same time, Petryna, Ivanov, and Rebenko developed an approach to the Euclidean theory of scattering matrix (for details, see [158]). Here, we briefly discuss the principal features of the EFT.

440

Chapter 34

34.1

Euclidean Field Theory

Analytic Continuation of Feynman Amplitudes

First, we consider the following example of the Feynman diagram of order 3 for the interaction ' 3 .x/: p2

q2 p1

:

q3 q1

(34.2)

p3

Using the Feynman rules for matrix elements of the S -matrix (26.93)–(22.95), we write the contribution of this diagram to the expression for coefficient functions (i.e., ignoring contributions from external lines) in the momentum space by integrating with respect to the variables q2 ; q3 .q1 D q/: .3/ e I 3 .p1 ; p2 ; p3 / D 3 ı.p1 C p3  p2 / Z dq 0 d q  : .m2  q 2  i "/Œm2  .q  p1 /2  i "Œm2  .q C p3 /2  i "

(34.3)

Integral (34.3) is convergent for large values of momenta (the degree of divergence of diagram (34.2) !.G/ D 2). We consider integral (34.3) as an integral with respect to the complex variable q 0 and the integrand as a function of the external complex variable q 0 and the external complex variables p10 and p20 not associating the .3/ contribution e I 3 with matrix element of the S-matrix and considering the variables .3/ I 3 is considered pi0 ; pi ; i D 1; 2; 3; as independent. Therefore, the amplitude e 2 2 outside the mass surfaces pj D m ; j D 1; 2; 3: The contour of integration in the complex plane q 0 goes along the line =q 0 D 0: It is easy to see that the integrand as a function of the complex variable q 0 is an analytic (holomorphic) function in q 0 in the entire complex plane except for poles defined by the functions p 0 D ˙ q2 C m2  i "; q1;2 q 0 q3;4 D p10 ˙ .q  p1 /2 C m2  i "; (34.4) q 0 q5;6 D p30 ˙ .q C p3 /2 C m2  i ": First, assume that the values of the variables p10 and p30 are small in comparison with corresponding radicals in relations (34.4). Then the poles lie in the second and fourth quadrants of the complex plane of q 0 and are at the distance " from the axis =q 0 D 0: The first and third quadrants are domains of analyticity. We can rotate the contour of integration counterclockwise by 90ı (Fig. 34.1).

441

Section 34.1 Analytic Continuation of Feynman Amplitudes Im k 0

90◦ Re k 0

Figure 34.1. Rotation of contour of integration.

Performing in integral (34.3) the change of variable q0 D i q4

(34.5)

and continuing the function e I 3 with respect to the variables pj0 to the imaginary axes pj0 D ipj4 ; we obtain the relation .3/

.3/ e I 3 .p1 ; p2 ; p3 / D 3 ı.p1 C p3  p2 / Z d 4q ;  .q 2 C m2 /Œ.q  p1 /2 C m2 Œ.q C p3 /2 C m2 

(34.6)

where, pj D .pj1 ; pj2 ; pj3 ; pj4 / 2 R4 ; q 2 D .q 1 /2 C    C .q 4 /2 ; d 4 q D dq 1    dq 4 : In addition, in (34.6), we performed the formal change ı.p10 C p30  p20 / ! i ı.p14 C p34  p24 /:

(34.7)

This procedure is equivalent to the analytic continuation of amplitude (34.6) to the complex domain with respect to the variables pj0 . Therefore, prior to performing this continuation, it is necessary to integrate a generalized function with a certain test analytic function, to rotate the contour of integration, and to perform change (34.5). If the values of p10 and p30 are not small, then the rotation of the contour in the plane k 0 and the analytic continuation pj0 ! ipj4 must be performed simultaneously to avoid the intersection of the contour with poles.

442

Chapter 34

Euclidean Field Theory

A similar procedure can be carried out in each order of perturbation theory. This can be easily proved by the method of mathematical induction expressing the sum of contributions of all n-order diagrams in terms of contributions of all .n  1/-order diagrams with the use of equations of the resolvent type (27.20) F .k/ D BF .k1/ ; .k/

where F .k/ D ¹FN º1 N D1 (see the proof in [158]). For divergent diagrams, first, it is necessary to introduce an intermediate regularization (e.g., by the Pauli–Villars method). Remark 34.1. By analogy with process described above, an analytic continuation of Feynman amplitudes can be performed in the coordinate space if the arguments of the .k/ amplitude FN .x1 ; : : : ; xN / satisfy the condition .xi  xj /2 ¤ 0;

i ¤ j;

i.e., do not lie on the surface of the light cone. In this case, an analytic continuation is performed with respect to the time variables to a domain of imaginary times xj0 ! ixj4 ;

xj4 2 R1 :

It is easy to verify that the analytic continuation of the free 2-point Green function 1 c i D .x  y/ to the Euclidean domain is the function Z ik.xy/ 1 4 e G0 .x  y/ D d k ; .2/4 k 2 C m2 k  .x  y/ D k 1 .x 1  y 1 / C    C k 4 .x 4  y 4 /;

(34.8)

d 4 k D d k 1    d k 4 ; k 2 D .k 1 /2 C    C .k 4 /2 :

34.2

Operators of Free Euclidean Fields

34.2.1 Real scalar field By virtue of postulate of local commutativity AQ6, the Bose operators of the scalar field '.x/ and '.y/ commute if .x  y/2 < 0:

(34.9)

In the Euclidean domain, i.e., for x 0 D ix 4 ; y 0 D iy 4 ; x 4 ; y 4 2 R1 ; relation (34.9) is automatically true. Thus, according to intuitive reasonings, analogs of the field

443

Section 34.2 Operators of Free Euclidean Fields

operators in the Euclidean domain commute for all values of x; y 2 R4 : Denote this field by aE .x/  a.x/: Thus, a.x/ and a.y/ must satisfy the condition Œa.x/; a.y/ D 0:

(34.10)

For a Euclidean analog of free fields, we have a.x/ D aC C a .x/; Œa .x/; aC .y/ D G0 .x  y/;

(34.11)

where G0 is defined by relation (34.8). Their Fourier transforms have the form 1 a .x/ D .2/2 ˙

Z

e ikx d 4k p a˙ .k/; 2 2 k Cm

(34.12)

Œa .k/; aC .p/ D ı.k  p/; where k  x D k 1 x 1 C    C k 4 x 4 and k 2 D .k 1 /2 C    C .k 4 /2 : The operators a˙ .k/ are operator-valued distributions, i.e., for each test function f 2 S .R4 /; the relations Z ˙ a .f / D d 4 kf .k/a˙ .k/ (34.13) define mutually adjoint operators in the Euclidean Fock space FBE with everywhere dense domain of definition, i.e., .a˙ .f // D a .f /;

D.a.f //  FBE :

(34.14)

The space FBE is constructed by analogy with Fock space (see Chapter 22 and, for details, [158] and a representation of the commutation relations (34.10)–(34.12) is constructed in the space FBE by analogy with representation of the operators of a free scalar field (see Section 22.2). We easily obtain E . E 0 ; a.x/a.y/ 0 / D G0 .x  y/;

(34.15)

E where E 0 is the vacuum vector in FB : This vector has the structure analogous to the structure of the vacuum vector 0 in the Fock space (22.30). In what follows, we show that the operators a.x/ can be constructed based on the definition of a free field at the time x 0 D 0 .'.0; x// and the free evolution (25.16). Other definition of a free Euclidean field is given by Symanzik in [195] and Nelson in [138, 139], who, in fact, developed the Symanzik ideas. They define a free Euclidean space as a generalized Gaussian process  indexed by functions f 2 L2 .R4 /: This means that, in the rigged Hilbert space H .HC L2 .R4 / H /, there exists

444

Chapter 34

Euclidean Field Theory

a Gaussian measure (see, e.g., [13], Part 2, Chapter 1, Sections 4–9) 0 such that, for f; g 2 L2 .R4 /; Z Z d0 ./.f /.g/ D .f; G0 g/ D f .x/G0 .x  y/f .y/ d 4 x d 4 y: However, this interpretation of Euclidean fields is convenient only for Bose fields. For Fermi fields, an ordinary operator approach is more clear.

34.2.2 Euclidean Fermi fields For fermion fields, the structure of Green functions (in particular, a 2-point function in the Minkowski space) is defined not only by dependence on the square of momentum p 2 D .p 0 /2  p2 but also by properties of the Dirac matrices   : In the Euclidean domain, the metric is defined by the Kronecker symbol ı ; ; D 1; 2; 3; 4: It is obvious that ı must be connected with certain new Hermitian matrices ˛ for which (34.16) ˛ ˛ C ˛ ˛ D 2ı : Schwinger was the first to propose a system of these matrices [183]. For a 4dimensional space, we choose   3 0 (34.17) ; ˛5 D 0 ˛k D 0 k ; k D 1; 2; 3; ˛4 D i 0 5 l3 ; l3 D 0 3 if the matrices  are taken in representation (6.10). In the study of fermion models in a 2-dimensional space-time, we can take the matrices ˛1 D 0 1 ;

˛2 D i 1 ;

˛3 D 0 :

(34.170 )

The 2-point Green function of fermion fields has the form Z 1 c pO C m 1 S .x  y/ D e ipx dp; 4 2 i .2/ i m  p2  i " where pO D p 0  0  p k  k and the other quantities are the same as in the scalar case. We define the Euclidean Green function by the relation 1 1 1 S E .x  y/ WD e  4  i˛4 S c .i.x 4  y 4 /; x  y/ 0 e  4  i˛4 : i

Taking into account properties (34.16), we finally obtain (Problem 34.3) Z ˛  p C ˛5 m 4 1 E e ip.xy/ d p; S .x  y/ D .2/4 p 2 C m2

(34.18)

(34.19)

445

Section 34.2 Operators of Free Euclidean Fields

where p D p  :

˛  p D ˛1 p1 C ˛2 p2 C ˛3 p3 C ˛4 p4 ;

To introduce operators of Euclidean Fermi fields, it is convenient to introduce a transformation of the Foldi–Wouthuysen type (see [181], Chapter 4, Section 6) p (34.20) e iS.k/ .˛  k C m˛5 /e iS.k/ D ˛5 k 2 C m2 ; where S.k/ D 

i jkj m ˛5 ˛  k arctan ; 2m jkj m

which yields

e iS.k/ ˛5 e iS.k/ ˛  k C m˛5 D : p k 2 C m2 k 2 C m2 Now we define operators of a free Euclidean Fermi field as follows: E ˛ .x/

D D

E;C .x/ C ˛E; .x/ ˛ 4 Z iS.k/ p˛ / 1 X 5 ˛ˇ ikx .e e p 4 2 C m2 .2/2 k ˇ D1

.bˇC .k/ C bˇ .k//d k;

N ˛E .x/ D N ˛E;C .x/  N ˛E; .x/  p  4 Z  . ˛5 e iS.k/ /ˇ ˛ 1 X C ikx  e b .k/  b .k/ d k: D p ˇ 4 ˇ .2/2 k 2 C m2

(34.21)

(34.22)

ˇ D1

The introduced operators satisfy the anticommutation relations   D ı˛ˇ ı.k  p/; b˛˙ .k/; b  ˇ C

h

E;˙ .x/; ˛

i N E; .y/ D S E .x  y/; ˇ C

 E ˛ .x/;





E ˇ

C

D

 N ˛E .x/; N E .y/ ˇ

D 2ı˛ˇ h

E ˛ .x/;

i

E ˇ .y/ C

D

h

1 .2/4

Z

C ik.xy/ e

p

k 2 C m2

i h N ˛E .x/; N E .y/ D ˇ C

d k;

E ˛ .x/;

i N E .y/ D 0: ˇ C

(34.23) For the first time, operators (34.21) and (34.22) were introduced in [167] and, in somewhat other form, in [145]. The representation of the commutation relations (34.23) in a Euclidean Fock space is defined by analogy with representation of free Fermi fields (see [158], Section 11.2).

446

34.3

Chapter 34

Euclidean Field Theory

Euclidean Green Functions of a Free Scalar Field

Representation of the theory in a Euclidean domain essentially extends the possibility of application of rigorous mathematical methods and enables one to go beyond the framework of perturbation theory. Of course, this does not mean that the conventional difficulties caused by volume and ultraviolet divergences noted in Chapter 25 can be overcome. However, introducing the corresponding “cuts,” i.e., introducing the function of interaction intensity (25.27), and bounding the upper limits of integration in momenta variables jpi j 6 ~; 0 < ~ < 1; we can rigorously determine expressions for Green functions of interacting fields. To verify this, we, first, construct the free Euclidean Green function and operators of a free Euclidean field starting from a free scalar field that satisfies the Klein–Fock–Gordon equation. First, for each real function f 2 S .R3 /; we define the operator Z '0 .0; f / D '0 .0; x/f .x/d x; (34.24) where '0 .0; x/ is the operator of the free scalar field (18.2) for x 0 D 0: We consider the free evolution of this operator corresponding to the Euclidean time x 0 D i t; t > 0 (see relation (25.16)) (34.25) 'O0 .t; f / D e tH0 '0 .0; f /e tH0 ; Z

where H0 D

d k k 0 '0C .k/'0 .k/;

k0 D

p k2 C m2

is the free Hamiltonian of a real scalar field (see relation (18.16) for a complex scalar field). It is easy to verify (Problem 34.6) that the action of operator (34.25) on the vector F 2 D0loc  FB [see (22.27)] is the same as the action of the operator Z 'O0 .t; f / D 'O0 .t; x/f .x/d x; (34.26) Z d k ikx k 0 t C 1 0 'O0 .t; x/ D e Œe '0 .k/ C e k t '0 .k/: (34.27) p 3=2 0 .2/ 2k Formally, the operator 'O0 .t; x/ coincides with relation (18.2) for x 0 D i t: For operators (34.26) and (34.27), we introduce the anti-time-ordered product as follows: ´ 'O0 .t1 ; x1 /'O0 .t2 ; x2 /; t1 6 t2 ;  T .'O0 .t1 ; x1 /'O0 .t2 ; x2 // D 'O0 .t2 ; x2 /'O0 .t1 ; x1 /; t2 6 t1 : We easily obtain (Problem 34.7) 

0 ; T  .'O0 .x14 ; x1 /'O0 .x24 ; x2 // 0 p Z Z 4 2 2 e ikxjx j k Cm 1 e ikx 1 4 d k d D k ; D p 16 3 .2/4 k 2 C m2 k2 C m2

(34.28)

447

Section 34.4 Euclidean Green Functions of Interacting Fields

where x WD x1  x2 and x 4 D x14  x24 : Comparing (34.15) with (34.28), we get E . 0 ; T  .'O0 .x1 /'O0 .x2 // 0 / D . E 0 ; a.x1 /a.x2 / 0 /

or, in terms of operators (34.26) and (34.27), 4 4 E . 0 ; T  .'O0 .x14 ; f1 /'O0 .x24 ; f2 // 0 / D . E 0 ; a.x1 ; f1 /a.x2 ; f / 0 /:

(34.29)

Using the Wick theorem for the T  -product, which is formulated analogously to the corresponding theorem for the T -product for the operators '0 .x/; we easily generalize relation (34.29) to the case of polynomials Pn .'0 / of degree n: . 0 ; T  .Pn1 .'O0 .x14 ; f1 //    Pnk .'O0 .xk4 ; fk // 0 / 4 4 E D . E 0 ; Pn1 .a.x1 ; f1 //    Pnk .a.xk ; fk // 0 /:

(34.30)

Finally, equality (34.30) is generalized to all bounded functions Fk ./ of the operators 'O0 or a: . 0 ; T  .F1 .'O0 .x14 ; f1 //    Fk .'O0 .xk4 ; fk // 0 / 4 4 E D . E 0 ; F1 .a.x1 ; f1 //    Fk .a.xk ; fk // 0 /:

(34.31)

This is a consequence of the result that operators of Bose fields form a ring [14] whose weak closure contains all bounded functions of these fields.

34.4

Euclidean Green Functions of Interacting Fields

To construct the Green function in a Euclidean domain not based on perturbation theory in the coupling constant ; we define the regularized interaction Hamiltonian HI~ .g/ by relations (25.10), (25.21), and (25.22) but with volume and ultraviolet cutoff Z ~ HI .g/ D d xg.x/ W '0~ .0; x/4 W Z D

d k1    R3

g.k1 C    C k4 /

Z R3

d k4 4

4 Q

.2/3=2

j D1

4 Q j D1

~ .kj /

W '0 .k1 /    '0 .k4 / W; q 2 4 2 kj C m (34.32)

where '0 .k/ D '0C .k/ C '0 .k/; ´ 1; jkj 6 ~  ı; ~ .k/ D 0; jkj > ~; and, for ~  ı 6 jkj 6 jkj; the function ~ 2 C 1 .R3 /:

(34.33) (34.34)

448

Chapter 34

Euclidean Field Theory

Then the regularized field corresponding to operator (34.27) has the form 'O0~ .t; x/ D 1 D .2/3=2

Z

i p ~ .k/e ixk h t pk2 Cm2 C t k2 Cm2  dkp p ' .k/ C e ' .k/ : e 4 2 k2 C m2 (34.35)

Finally, the total Hamiltonian has the form H ~ .g/ D H0 C HI~ .g/:

(34.36)

The operators H0 ; HI~ .g/; and H ~ .g/ are well-defined operators in the Fock space FB : In addition, the operator H ~ .g/ is essentially self-adjoint (see [72]) and bounded below (Problem 34.8) by virtue of identity (25.5), which already does not have infinite constants due to ultraviolet cut. The lower bound of the spectra of the operator H ~ .g/ is an isolated eigenvalue of multiplicity one (for the proof, see [73]). This means that there exists a unique vector ˆ0 D ˆ0 .~; g/ 2 FB such that H ~ .g/ˆ0 D E .~; g/ˆ0 ;

kˆ0 k D 1;

and E .~; g/ D inf¹spectrum H ~ .g/º: Therefore, instead of the operator H ~ .g/; it is more convenient to consider the operator b ~ .g/ D H ~ .g/  E .~; g/ H whose spectrum is nonnegative and contains ¹0º as an isolated point of the spectrum corresponding to the physical vacuum ˆ0 : Using the arguments in [82], we easily obtain the important limit b~ .g/ D P ; s  lim e t H 0

(34.37)

t!C1

where P0 is the projector onto the proper subspace corresponding to the eigenvalue E D 0: In addition, (34.38) .ˆ0 ; 0 / ¤ 0: With the use of the projector P0 ; the vector ˆ0 can be expressed in terms of the “bare” vacuum vector 0 (22.30), i.e., 1 ˆ0 D P0 0 ; c

c D kP0 0 k:

(34.39)

By analogy with (34.25), we consider the total evolution (see (25.11)) of operator (34.24) corresponding to the imaginary time x 0 D i t; t > 0; b .g/ ' .0; f /e t H b .g/ : 'O ~ .t; f / D e t H 0 ~

~

(34.40)

Section 34.4 Euclidean Green Functions of Interacting Fields

449

Then, by analogy with (34.28), it is quite natural to define the Green functions of fields (34.40) as follows: E .x1 ; : : : ; xN I ~; g/ WD .ˆ0 ; T  .'O ~ .x1 /    'O ~ .xN //ˆ0 /; GN

(34.41)

where 'O ~ .x/ D 'O ~ .x 1 ; x 2 ; x 3 ; x 4 / .x 4  t / are connected with operators (34.40) by relations (34.26). Then we express the right-hand side of (34.41) in terms of the operators of free Euclidean fields (34.11). In fact, this means the passage to the interaction picture in terms of Euclidean fields or representation of relation (28.1) in the form of (28.7) in the Euclidean space R4 : To this end, we expand the operator ~

~

e tH .g/ D e t.H0 C H .g// in series in  using the theorem on decomposition of semigroups in a perturbation series (see [92], Part XIII) e

tH~ .g/

N X

D s  lim

N !1

e

 n H0

Zt n

Zn1

d 1   

./

d n

nD0

0 0 ~ . n1  n /H0 HI .g/e    HI~ .g/e .t 1 /H0 Zt N

D s  lim

N !1

X

./n

nD0

d 1 0

Zn1

d n HI~ .n ; g/    HI~ .1 ; g/e tH0

 0

Zt N X ./n  T . d  HI~ .; g//n e tH0 D s  lim N !1 nŠ nD0

0

Zt N X ./n tH0  e D s  lim T . d  HI~ .  t; g//n ; N !1 nŠ nD0

0

where HI~ .; g/

De

 H0

(34.42)

Z HI~ .g/e H0

D

d x g.x/ W 'O0~ .; x/4 W :

(34.43)

Remark 34.2. In the case at hand, (34.42) is a formal expansion. For operations to be absolutely strict, it is necessary to perform an intermediate regularization of

450

Chapter 34

Euclidean Field Theory

the operator HI~ .g/ by replacing it by the operator HI~ .g; "/; " > 0; with bounded operators ~ 2 (34.44) '"~ .0; f / WD '0~ .0; f /e "'0 .0;f / ; " > 0; instead of fields '0~ .0; f /; f 2 S .R3 /: Then expansion (34.42) is convergent in a strong sense (s  lim) and the intermediate regularization must be eliminated ." ! 0/ E after the passage to Euclidean fields. in the final expression for the Green function GN For the procedure not to be very awkward, we omit these transformations (for the detailed proof, see [158], Section 16). We perform necessary transformations for the 2-point Green function .N D 2/: Let, for x14 D t1 < t2 D x24 and fk 2 S .R3 /; k D 1; 2; Z Z G2E .t1 ; f1 I t2 ; f2 / D dx1 dx2 G2E .x1 ; x2 /f1 .x1 /f2 .x2 /: R3

R3

Using (34.41) and (34.40), we get G2E .t1 ; f1 I t2 ; f2 / D .ˆ0 ; b~ .g/ ' .0; f /e .t2 t1 /H b~ .g/ ' .0; f /e t2 H b~ .g/ ˆ / e t1 H 0 1 0 2 0 1 ~ . 0 ; e .tCt1 /H .g/ D s  lim t!C1 S0 .t / ~

~

 '0 .0; f1 /e .t2 t1 /H .g/ '0 .0; f2 /e .tt2 /H .g/ 0 / D lim

N1 ;N 2 ;N3 X

lim

t!C1 N1 ;N2 ;N3 !1

 e tH0 T  .

n1 ;n2 ;n3 D0

./n1 Cn2 Cn3 1 . 0 ; n1 Šn2 Šn3 Š S0 .t /

tCt Z 1

d HI~ .  t; g//n1 'O0 .t1 ; f1 / 0



tZ 2 t1

d HI~ .

T .

n2



tt Z 2

d HI~ . C t2 ; g//n3 0 /

C t1 ; g// 'O0 .t2 ; f2 /T .

0

0 tR1

tR2

~

 d HI . ;g/  d HI~ . ;g/ 1 D lim . 0 ; T  Œe t 'O0 .t1 ; f1 /e t1 t!1 S0 .t / 

Rt

d HI~ . ;g/

 'O0 .t2 ; f2 /e  0 / 1 ~ . E ; a.t1 ; f1 /a.t2 ; f2 /e  HI .g;a/ E D lim 0 0 /; t!1 S0 .t / t2

(34.45)

451

Section 34.4 Euclidean Green Functions of Interacting Fields

where Z

Zt HI~ .g; a/

D

d 4 x g.x/ W a~ .x/4 W;

d x g.x/ W a.; x/ WD

d t

Z 4

R3

R4

g.x/ D Œt;t .x 4 /g.x/;

Œt;t . / D

~

´ 1;  2 Œt; t ;

(34.46)

0;  62 Œt; t ; ~

 HI .g;a/ E

0 /; S0 .t / D . 0 ; e 2tH .g/ 0 / D . E 0 ;e

and 1 a .x/ D .2/2

Z

~

R4

~ .k/e ixk a.k/ d 4 k: p k2 C m

We used the relation ~

.ˆ0 ; .: : :/ˆ0 / D lim

t!C1

~

.e tH .g/ 0 ; .: : :/e tH .g/ 0 / ~

. 0 ; e 2tH .g/ 0 /

;

which is a corollary of (34.37)–(34.39), in the second equality (34.45), expansion (34.42) in the third, after the change of variables, rewrote the sums in the form of exponents in the fourth, and used relation (34.31) in the fifth. Recall that the operator HI~ .; g/ is assumed to be approximated by a sequence of bounded operators HI~ .; g; "/ (see Remark 34.2). Similar transformations can be performed for the N -point Green function. Without the passage to the limit as t ! C1; the N -point Green function in a Euclidean domain of variables has the form E .x1 ; : : : ; xN I ~; g/ GN

D

R 1  g.x/Wa~ .x/4 Wd 4 x . E

E 0 ; a.x1 /    a.xN /e 0 /; S0 .g; ~/

(34.47)

S0 .g; ~/  S0 .t /: Thus, in fact, we proved that the time cut can be eliminated in a weak sense in relation (34.47) for the Euclidean N -point Green function, i.e., E E .x1 ; : : : ; xN I ~; g/ D GN .x1 ; : : : ; xN I ~; g/; w  lim GN t!C1

(34.48)

where the function of inclusion of interaction g is defined by relation (34.48). For the first time, this result has been obtained in [75] for the model (25.3) and in [168] for the Yukawa model. It is easy to obtain similar relations for Green functions of fermion fields (Problem 34.9).

452

Chapter 34

Euclidean Field Theory

Remark 34.3. For the Green functions corresponding to the Bose fields (34.25), (34.26), the transition to the limit as ~ ! 1 can be performed under the condition that the dimension of the space-time d D s C 1 D 2 (see, e.g., [158] and the references therein).

Chapter 35

Euclidean Axiomatics

The transition to imaginary times described in the previous chapters of the present part proves to be a very efficient method for the investigation of model systems in the relativistic quantum field theory. This procedure enables us to apply rigorous mathematical methods (in fact, for the first time) and go beyond the scope of the perturbation theory. However, this becomes possible only for regularized (with ultraviolet cuts) theories or theories in which the space-time dimension d 6 3: Thus, it is not clear that we indeed deal with the Euclidean field theory but not with the Euclidean statement of the relativistic theory of quantized fields. In order to justify the approach outlined above, it is necessary to have a scheme (on the level of an abstract theorem), which would enable us to clearly formulate the conditions for returning to a pseudo-Euclidean domain, i.e., conditions guaranteeing the equivalence of the Euclidean and relativistic field theories. We formulate this equivalence in the language of the Osterwalder–Schrader axioms (see [69, 147] for Schwinger functions [Euclidean Green functions)] and the theorem of reconstruction guaranteeing the existence of the Wightman functions (see Chapter 32) satisfying properties W1–W6. Thus, by virtue of Theorem 27.1, there exists a space of states and a local Heisenberg field satisfying axioms A1–A2. Prior to the formulation of the Osterwalder–Schrader axioms, we briefly discuss the analytic properties of the Wightman functions defined in Section 30.2.

35.1

Analytic Continuation of Generalized Wightman Functions

We introduce new definitions of domains in the space ntimes

MC

˝n

‚ …„ ƒ D MC ˝ ˝MC

(see Section 1.4). We call the domain TnC WD ¹.z1 ; : : : ; zn / 2 .MC/˝n j Im .zj C1  zj / 2 VC ; j D 1; n  1º

(35.1)

a tube of future and the domain Tn a tube of past (see (1.7)). In the same way as in (1.7), Tn˙ denote closed sets. We also define an extended tube of future by the relation [ ƒTnC : (35.2) Tnext WD ƒ2LC .C/

454

Chapter 35

Euclidean Axiomatics

It follows from definition (35.1) that TnC does not contain any real point .x1 ; : : : ; xn / 2 .M/˝n : However, the extended tube Tnext contains these points, which are called Jost points due to the theorem proved by Jost in [99]. Theorem 35.1. . The point .x1 ; : : : ; xn / 2 M˝n \ Tnext if and only if the convex cone C.x1 ; : : : ; xn / generated by the points x1 ; : : : ; xn contains only space-like points. Remark 35.2. The convex cone C.x1 ; : : : ; xn / is defined as a set of points 1 x1 C n P 2 x2 C    C n xn ; where k > 0 but k > 0 (for the proof, see [101], Part IV, kD1

Chapter 5). Denote the set of all Jost points by J: We introduce the definition of a symmetrized extended tube of future [ ext eext T TnI ; (35.3) n WD 2Sn ext TnI

is the extended tube of future (35.2) with respect to the variables where .x.1/ ; : : : ; x.n/ / for any rearrangement  2 Sn of the variables x1 ; : : : ; xn : The main problem is to establish maximum domains of analyticity of Wightman functions in the space .MC/˝n : We perform it for three steps. 1. Analytic continuation to the tube of future TnC : The following theorem guarantees this continuation (see [101], Part IV, Chapter 2): Theorem 35.3. The generalized Wightman functions wn .x1 ; : : : ; xn / are limit values of the analytic functions wn .z1 ; : : : ; zn / holomorphic in the tube of future TnC : The Wightman functions are invariant under proper transformations of orthotropous nonhomogeneous Lorentz transformations. The theorem is a corollary of several general theorems of the theory of Laplace transformations for functions of many variable and spectral condition W2 (see Section 30.2). 2. Analytic continuation to the extended tube of future Tnext : This analytic continuation is based on the known Bargman–Hall–Wightman theorem published for the first time in [86]. Theorem 35.4. If the function f .z1 ; : : : ; zn / is holomorphic (analytic) in the tube of future TnC (35.1) and invariant under nonhomogeneous proper orthochronous Lorentz transformations, then it admits an analytic continuation to the extended tube Tnext (see (35.2)) and is invariant under L .C/ [see (1.22) and (1.23)]. The proof (see [191], Chapter 3, [23], Chapter 5) is based on condition W1 (Section 30.2).

Section 35.2 Euclidean Green Functions. Osterwalder–Schrader Axioms

455

eext 3. Analytic continuation to the symmetrized extended tube of future T n (see (35.3)). Restricting the analytic function wn .z1 ; : : : ; zn / by the Jost points .x1 ; : : : ; xn /; we obtain (35.4) wn .x1 ; : : : ; xn / D wn .x.1/ ; : : : ; x.n/ / WD wn .x1 ; : : : ; xn / for any  2 Sn : Equality (35.4) is true by virtue of Theorem 35.1 and condition W4 (see relation (32.11)). By virtue of Theorem 35.4, the analytic extension of wn ext : This to the tube TnI ; in fact, defines the analytic extension of wn to the tube TnI ext e implies that equality (35.4) is analytically extended to the domain T n : The following theorem is true [101], Part IV, Chapter 5: Theorem 35.5. There exists analytic functions wn .z1 ; : : : ; zn / holomorphic in the doeext main T n and symmetric with respect to the variables z1 ; : : : ; zn and their limit values in the domain M˝n restore all generalized Wightman functions wn .x1 ; : : : ; xn /:

35.2

Euclidean Green Functions. Osterwalder–Schrader Axioms

In Sections 34.1–34.4, we have introduced the notion of a Euclidean domain that can be defined as a domain of the complex Minkowski space MC D MC 4 (see Section 1.4). This domain can be defined by the relation E 4 D E WD ¹z 2 MCjz 0 D i t; t 2 R; z 2 R3 º: We denote the corresponding domain in the space MC ˝n by E ˝n : We also define the off-diagonal Euclidean domain En by the relation En WD ¹.z1 ; : : : ; zn / 2 E ˝n jzi ¤ zj for i ¤ j º: The following (almost obvious) lemma is true: eext Lemma 35.6. The domain En lies in the symmetrized extended tube of future T n : To prove the lemma, it suffices to show that, for any Euclidean point z D .z1 ; : : : ; zn / 2 En ; there exist at least one Jost point x D .x1 ; : : : ; xn / 2 J and transform ƒ 2 LC .C/ such that (Problem 35.1) z D ƒx .zj D ƒxj ; j D 1; n/: Then the statement of the lemma follows from definitions (35.2) and (35.3) and the fact that Jost points lie in Tnext (Theorem 35.1). Lemma 35.6 enables one to determine Euclidean Green functions.

456

Chapter 35

Euclidean Axiomatics

Definition 35.7. We call the restriction of the analytic function wn .z1 ; : : : ; zn / holomorphic in Tnext by the domain En an n-point Euclidean Green function or a Schwinger function GnE .x1 ; : : : ; xn / D wn .x1 ; ix14 ; : : : ; xn ; ixn4 /;

(35.5)

where xj D .xj1 ; xj2 ; xj3 ; xj4 / 2 R4 : It follows from Definition 35.7 that both the generalized Wightman functions wn .x1 ; : : : ; xn /; xj 2 M4 ; and the Schwinger functions GnE .x1 ; : : : ; xn / xi 2 R4 ; are limit values of one analytic function wn .z1 ; : : : ; zn / for .z1 ; : : : ; zn /  .x1 ; : : : ; xn / 2 M4n and .z1 ; : : : ; zn / 2 En ; respectively. The Schwinger functions thus constructed satisfy the following conditions: E0. Moderate growth The distributions GnE .x1 ; : : : ; xn / .G0 D 1/ are elements of the space 0 S 0 .R4n /: The space of test functions is defined by the relation 0

S .R4n / WD ¹f 2 S .R4n / j f .k/ .x1 ; : : : ; xn / D 0; for xi D xj ; i ¤ j I k  0º;

where f .k/ is the derivative of the function f of order k: E1. Euclidean invariance GnE .x1 ; : : : ; xn / D GnE .Rx1 C a; : : : ; Rxn C a/;

R 2 SO4 ;

(35.6)

where SO4 is the group of Euclidean rotations in the space R4 : E2. Positive definiteness Let SC .R4n / be a space of test functions from S .R4n / whose support, together with derivatives, lies in the domain ¹.x1 ; : : : ; xn / 2 R4n j0 < x14 < x24 <    < xn4 º: For f 2 SC .R4n /; we define the operation  as follows: .f /.x1 ; : : : ; xn / D f . x1 ; : : : ;  xn /;  xj D .xj1 ; xj2 ; xj3 ; xj4 /: Then, for f0 2 C and fk 2 SC .R4k /; k D 1; 2; : : : , X E GnCm .fn ˝ fm / > 0: n;m>0

(35.7)

457

Section 35.3 Reconstruction of the Wightman Theory

E3. Symmetry GnE .x1 ; : : : ; xn / D GnE .x.1/ ; : : : ; x.n/ /;

 2 Sn :

E4. Cluster property For arbitrary n; m 2 N; f 2 SC .R4n /; g 2 SC .R4n /; and a D .a; 0/ 2 R4 ; E E lim ŒGnCm .fn ˝ gmI a /  GnE .f /Gm .g/ D 0;

!1

(35.8)

where gmI a .x1 ; : : : ; xm / D gm .x1 C a; : : : ; xm C a/: Properties E0–E4 can be proved based on Wightman axioms W0–W6 (Problems 35.2–35.6). By analogy with (32.8), we can define functions that depend on the difference of arguments j D xj C1  xj : GnE .x1 ; : : : ; xn / D Sn1 . 1 ; : : : ; n1 /:

(35.9)

These functions can be represented in the form of the Laplace transforms of the functions Wn1 defined by relation (32.9) Z Pn 4 4 e n .q1 ; : : : ; qn /d 4n q: (35.10) Sn . 1 ; : : : ; n / D e  kD1 .k qk Ci k qk / W A collection of properties E0–E4 is called the Osterwalder–Schrader axioms. For the first time, they were proposed in [147]. The following problem is of interest: whether conditions E0–E4 are sufficient for the restoration of Wightman functions that satisfy conditions W0–W6. The positive answer to this question enables one, on the basis of Euclidean functions, to restore the relativistic field theory in the Minkowski space. We consider this problem in the next chapter.

35.3

Reconstruction of the Wightman Theory

Starting from Garding–Wightman axioms A1–A2 (Section 32.1) that define the relativistic quantum field theory, we, first, passed to other axiomatics (the Wightman axiomatics) and then to the Osterwalder–Schrader axioms for Euclidean Green functions, which are most convenient objects from mathematical point of view. The results of calculations of various physical characteristics obtained on the basis of perturbation theory and performed with the use of the Euclidean theory again can be rewritten in a physical domain using analytic extension. It is desirable to have this return from a Euclidean domain to a physical one not only for perturbation theory. To this end, we need a theorem on equivalence between axioms E0–E4 and axioms A1–A2. Therefore, it is necessary to show that the chain .A1–A2/ ) .W0–W6/ ) .E0–E4/

(35.11)

458

Chapter 35

Euclidean Axiomatics

is also true in the opposite direction. We make the first step in this direction using Theorem 27.1, which enables us, on the basis of Wightman functions that satisfy conditions W0–W6, to restore a field and a space of states of a system that satisfy axioms A1–A2. In 1973, Osterwalder–Schrader proposed a theorem on reconstruction of Wightman functions on the basis of Schwinger functions. Later, Schrader and Simon found a mistake in the proof of a lemma. According to this mistake, condition E0 is too weak to determine the corresponding Wightman functions with the use of the Laplace transform. Therefore, condition E0 is sufficient to guarantee the existence of the Laplace transform with respect to each complex variable separately but not with respect to a collection of all variables. Later, in [147], axiom E0 was reformulated in other form, which requires additional spaces of test and generalized functions. For 4n 4n j x 4 > 0 for all j D 1; : : : ; nº; the function ' 2 S .R4n C /; where RC WD ¹x 2 R j we define the Laplace transform 1 '.p Q 1 ; : : : ; pn / D .2/4n

Z



e

n P j D1

.pj4 xj4 Cipj xj /

'.x1 ; : : : ; xn /d 4n x;

where pj4 > 0; j D 1; n; and SL.R4n C / is the Schwartz space with an ordinary system of seminorms. Let the following condition be satisfied: E00 : Improved moderate growth The distributions GnE 2 0 S 0 .R4n / and the generalized functions (35.9) Sn1 2 0 S 0 .R4.n1/ /: Using this improvement, we formulate the following theorem: Theorem on reconstruction (E00 –E4) , (W0–W6). Conditions E00 ; E1–E4 for Euclidean Green functions are equivalent to Wightman axioms W0–W6 for generalized Wightman functions. Omitting the detailed proof, we dwell on several its main moments. First of all, note that, with the use of condition E00 , based on the functions GnE ; we determine a 4.n1/ certain generalized function wQ n1 .p1 ; : : : ; pn1 / from the space SL 0 .RC / such that GnE .x1 ; : : : ; xn / D Sn1 .x2  x1 ; : : : ; xn  xn1 / Z Pn1 4 4 4 D wQ n1 .p1 ; : : : ; pn1 /e  j D1 Œpj .xj C1 xj /ipj xj d 4.n1/ p:

(35.12)

Using this relation, we can define the generalized Wightman functions as follows: Z wn .x1 ; : : : ; xn / D

wQ n1 .p1 ; : : : ; pn1 /e

i

n1 P j D1

pj .xj C1 xj /

d 4.n1/ p:

(35.13)

Section 35.3 Reconstruction of the Wightman Theory

459

A convenient expression for the restoration of the Laplace transform is also proposed in [230]. Then property W0 is a corollary of definition (35.13) (Problem 35.7). Invariance W1 follows from (35.12) and (35.13) (Problem 35.8). Since the support wQ n lies in the space R4n Q n V nC ; which guarantees W2 C and it is Lorentz-invariant, sup w (Problem 35.9). Condition W3 is guaranteed by property E3 and relations (35.12) and (35.13) (Problem 35.10). Property of locality W4 is a corollary of symmetry E3 and the Jost theorem [101], Part IV (Problem 35.10). Positive definiteness W5 follows from the similar condition E2 (Problem 35.11). Finally, property W6 is a corollary of E4. Thus, chain (35.11) is true in the opposite direction, i.e., .E00 –E4/ ) .W0–W6/ ) .A1–A2/:

(35.14)

Remark 35.8. Here, we also mention the work by Zinoviev [230], where another version of the axiom of spectrality was proposed, which enabled the author to establish equivalence (35.14). Remark 35.9. The system of axioms E00 –E4 can be modified for the other fields. The case of arbitrary spinor fields is studied similarly (see [147], Remark 6).

Chapter 36

Problems to Part VI

Problem 32.1.

Establish the relativistic law of transformation for the Wightman functions (32.6).

Problem 32.2.

Prove that the Fourier transforms of the functions wn and Wn1 [see definitions (32.7) and (32.8)] are connected by relation (32.9).

Problem 32.3.

Prove conditions W3–W5 [relations (32.6)–(27.12)] for the Wightman functions of the scalar field.

Problem 32.4.

Prove the property of decomposition into bundles (32.13).

Problem 32.5.

Prove that the set D0 of all finite sequences fO D .f0 ; f1 ; : : : ; fN ; 0; : : :/, N 2 N [ ¹0º; is everywhere dense in H :

Problem 32.6.

Prove that, for h 2 S .M4 / and Im h D 0; the operator '.h/ is a symmetric operator in the space H ; i.e., satisfies equality (32.18).

Problem 32.7.

Check property (32.19).

Problem 33.1.

Show that the operator defined by relations (33.2)–(33.4) is an operator in the space H with the domain of definition D.'/ if '.f /; f 2 S .M/; is defined on D  H :

Problem 33.2.

Prove Theorem 33.1.

Problem 33.3.

Prove Theorem 33.2.

Problem 33.4.

Substantiate the expression for the current (33.21).

Problem 33.5.

Prove that the Heisenberg field can be represented in the form of the formal expression (33.23).

Problem 34.1.

Construct the Euclidean Fock space.

Problem 34.2.

Prove the symmetry properties of the “smoothed” Euclidean operators a.f / for a scalar field.

Problem 34.3.

Show that the analytic continuation of the fermion Green function 1 c i S .x  y/ and transformation (34.18) lead to relation (34.19).

461

Chapter 36 Problems to Part VI

Problem 34.4.

Prove relation (34.20).

Problem 34.5.

Prove that operators (34.21) and (34.22) satisfy the commutation relations (34.23).

Problem 34.6.

Show that operator (34.25) acts in the set D0loc ; in exactly the same way as operator (34.26), (34.27).

Problem 34.7.

Prove relation (34.28).

Problem 34.8.

Prove that the regularized operator of energy H ~ [see (22.24) and (34.32)–(34.36)] is bounded from below by the constant 9. 0 ; '0~ .0; x/2 0 /2 :

Problem 34.9.

E in terms of Represent the expression for the Green function Gm;2n the operators of Euclidean fields for the Yukawa interaction Lint D N ':

Problem 35.1.

Show that, for any z D .i s; z/; s; z k 2 R; k D 1; 2; 3; one can find the Jost point x D .0; x 1 ; x 2 ; x 3 /; x k 2 R; such that, e.g., for ƒ 2 LC .C/ of the form ƒ00 D ƒ11 D cos ';

ƒ22 D ƒ33 D 1;

ƒ21 D ƒ12 D sin ';

ƒ31 D ƒ13 D ƒ41 D ƒ41 D ƒ32 D ƒ32 D ƒ43 D ƒ34 D 0; we have z D ƒx: Find the vector x: Problem 35.2.

Prove that the functions GnE defined by relation (35.5) are generalized tempered functions.

Problem 35.3.

Prove the invariance (35.6) of the functions GnE :

Problem 35.4.

Prove relation (35.7).

Problem 35.5.

Prove the symmetry properties of the functions GnE :

Problem 35.6.

Prove the cluster property (35.8) for the functions GnE :

Problem 35.7.

Prove that the Wightman functions defined by relation (35.13) satisfy condition W0.

Problem 35.8.

Prove that the Wightman functions defined by relation (35.13) satisfy condition W1.

Problem 35.9.

Prove that the Wightman functions defined by relation (35.13) satisfy condition W2.

462

Chapter 36

Problems to Part VI

Problem 35.10.

Prove that the Wightman functions defined by relation (35.13) satisfy condition W3.

Problem 35.11.

Prove that the Wightman functions defined by relation (35.13) satisfy condition W4.

Part VII

Quantum Theory of Gauge Fields

Chapter 37

Quantum Electrodynamics (QED)

Following the history of construction of quantum interaction between charged particles (electrons, positrons, protons, etc.), one should begin with the classic Maxwell equations (8.2). Taking into account the explicit form of the current 4-vector J  .x/ D . .x/; J.x// (see (6.115)) and rewriting these equations in terms of the 4-potential A .x/ with the use of (8.4) and (8.5), we obtain A .x/ C @ .@ A .x// D e N .x/ .x/:

(37.1)

This equation should be considered together with Equation (6.91), which describes the motion of a charged particle in the field determined by the 4-vector potential A .x/ and the conjugate equation (6.92) (or Equation (6.93), which describes the motion of an antiparticle in the field A .x/). In Part III, we have considered this system of equations based on the requirement of local gauge invariance of the Dirac equations and Lagrangian (11.6). This requirement, which was reduced to the main principle of the theory, leads to Lagrangian (11.9) and the corresponding system of equations (11.11). Note that system (11.11) can have infinitely many solutions, including the solution B .x/ D A .x/, i.e., a solution corresponding to an electromagnetic field. We rewrite the corresponding Lagrangian in the form L .x/ D L .x/ C LA .x/ C LI .x/;

(37.2)

where L and LA are defined by (6.114) and (8.18), respectively, and LI .x/ describes the interaction of these fields: A.x/ .x/: LI .x/ D A .x/J  .x/ D e N .x/b

(37.3)

We now pass to the quantum case, i.e., to the quantization of the fields .x/ and A .x/: Note that system (11.11) should be considered as a relation between the operator-valued generalized functions .x/ and A .x/ in which one should perform the first regularization related to the definition of the product of fields at the same point: .i   @  m/ .x/ D e  W A .x/ .x/ W; i @ N .x/  C m N .x/ D e W N .x/  A .x/ W; A .x/ C @ .@ A .x// D J .x/ D e W N .x/ .x/ W :

(37.4)

466

Chapter 37

Quantum Electrodynamics (QED)

The definition of normal product for interacting fields should be considered as the regularization defined in Section 25.1 (see relations (25.5) and (25.6)). Similarly to the case of a real scalar field with interaction Lagrangian (25.3) (see Equation (25.4)), this regularization does not eliminate the mathematical difficulties related to the definition of the energy operator (complete Hamiltonian). Moreover, not only problems become more complicated in the technical aspect but, at the formal level, their number even increases due to the gauge invariance of system (37.4). We begin with the quantization of an electromagnetic field and the definition of the Hamiltonian of a system of interacting fields.

37.1

Quantization of Interacting Electromagnetic Fields

In this chapter, we briefly describe several approaches to the quantization of an electromagnetic field and problems connected with each approach. At present, there is no method that solves all problems or, at least, is formally consistent with the condition of gauge invariance and main axioms of quantum theory. As in the case of a free field, we begin with a generalization of the Gupta–Bleuler formalism to the case of an interacting field.

37.1.1 Gupta–Bleuler Formalism for Interacting Electromagnetic Fields Recall that the main idea lies in the replacement of the Lorentz condition (8.8), which is not consistent with the commutation relations (20.25), by the weaker condition (20.33). It is obvious that, in the case of an interacting field, the Lorentz condition should not be imposed as well because this contradicts the possibility of continuous transition to a free field with elimination of interaction. On the other hand, condition (20.33) can easily be realized because the operator of a free electromagnetic field is decomposed into two terms: the creation operator AC  and the annihilation operator . In the case of its interaction with other fields, A A  depends on the current J  generated by matter fields and satisfies the equation A .x/  @ @ A .x/ D J .x/:

(37.5)

In this case, an artificial “trick” is used. Namely, one considers the so-called pseudoMaxwell equation (37.6) A .x/  @ @ A .x/ D J .x/ for  ¤ 1: If the Lorentz condition (8.8) is satisfied, then Equations (37.5) and (37.6) are equivalent. We introduce the auxiliary operator L.x/ D @ A .x/:

(37.7)

Section 37.1 Quantization of Interacting Electromagnetic Fields

467

Assume that the current is preserved, i.e., the continuity equation @ J .x/ D 0 is satisfied. Acting by the operator @ on Equation (37.6) and performing summation with respect to , we obtain .1  /L.x/ D 0: This means that the field L.x/ satisfies the free wave equation for  ¤ 1 and L.x/ can be rewritten in the form (8.15): L.x/ D LC .x/ C L .x/: In the same way as for a free electromagnetic field, we can define the following set (see (20.36)) in the space of states H : P WD ¹ˆ 2 H jL .x/ˆ  0º: On this set, Equations (37.5) and (37.6) are equivalent in the weak sense, i.e., as the mean values, because, for ˆ, ‰ 2 P, we have .ˆ; L.x/‰/  0: It is clear that the construction of the space of states H and the set D is a more complicated problem because it is necessary to construct the linear span of everywhere dense vectors in the space H with the use of polynomials in the operators A , , and N : For a more detailed analysis of this construction and problems connected with its realization, see [133] (also [190], Chapters 6 and 12).

37.1.2 Quantization of Interacting Electromagnetic Fields in the Coulomb Gauge In Section 20.2.3, we have, in fact, quantized an electromagnetic field by the canonical method under condition (8.54) for the canonical variables of the field Ai .x/, i D 1; 2; 3, which fixes a gauge (Coulomb gauge). In contrast to a free electromagnetic field, the operator A0 .x/ cannot be set identically equal to zero because it is connected with the distribution of charges in the system of interacting (via an electromagnetic potential) particles of matter (i.e., electrons and positrons). This connection is defined by relation (8.56), in which '.x/ can be set equal to zero. Therefore, the generalized momentum, which, according to (8.21), is a variable canonically conjugate to the generalized coordinate Ai .x/, has the form ! @Ai .x/ @A0 .x/ i D @0 Ai .x/  @i A0 .x/:  (37.8) A .x/ D  @x0 @xi

468

Chapter 37

Quantum Electrodynamics (QED)

This somewhat changes the form of the second constraint (20.44), namely '2 .A; A /  @i Ai .x/ C A0 .x/ D @i Ai .x/  J0 .x/ D 0;

(37.9)

because, by virtue of (37.8), (8.54), and (8.55), we have @i Ai .x/ D J0 .x/: The charge density J0 .x/ is a function of the matter fields N and and does not depend on the canonical variables Ai .x/ and Ai .x/: This implies that relations (20.45)– (20.47) do not change, and the canonical commutation relations for the fields Ai .x/ and momenta are defined by (20.48). Since Ai .x/ depends on A0 .x/, which is expressed in terms of the fields and N , the canonical variables of the electromagnetic field Ai .x/ do not commute with the matter fields. The next step in the construction of quantized fields is the construction of the complete Hamiltonian. According to the general principle of canonical formalism (4.8), we get j

H .x/ DW  .x/@0 .x/ W C W A .x/@0 Aj .x/ W  W L .x/ W;

(37.10)

where  .x/, Ai .x/, and L .x/ are defined by (6.129), (37.8), and (11.9), respectively. Taking into account relations (8.55) and (8.56) with '  0 and the relation for the current operator (6.115), we represent the energy operator as follows: Z 0 (37.11) H D P D d xH .x/ D H0 C HI ; where H0 D H0; C H0;A ; Z  H0; D d x W .x/.i ˛  r C ˇm/ .x/ W; Z 1 H0;A D d xŒW .@0 A/2 W C W .r  A.x//2 W 2

(37.12) (37.13) (37.14)

are the energy operators of free fields and HI D HI;A C HI;A0 ; Z HI;A D d x W J  A W; Z Z W J 0 .x/J 0 .y/ W 1 HI;A0 D HCoul D dx dy ; 2 4jx  yj J  .x/ D e W N .x/  .x/ W :

(37.15) (37.16) (37.17) (37.18)

Section 37.1 Quantization of Interacting Electromagnetic Fields

469

In the same way as for the scalar field (see relation (25.11)), the fields .x/, N .x/, and A .x/ that satisfy Equation (37.4) are constructed with the use of the formal relations 0 0 (37.19) Fi .x/ D e ix H Fi .0; x/e ix H ; i D 1; 2; 3; where F1 D , F2 D N , and F3 D A (Problem 37.2) and the fields Fi .0; x/ satisfy the commutation relations (19.1) and (20.48). Remark 37.1. General mathematical problems arising in the substantiation of relation (37.19) are analogous to those for field (25.11) (see Section 25.2). However, technical problems are much more complicated, first of all, due to more singular integrals of the form (25.32). For the detailed analysis of these problems, see [158]. In addition, analogous problems arise in the construction of the scattering matrix. In conclusion, we consider the problem of construction of a photon propagator associated with internal lines of Feynman diagrams constructed according to the rules formulated in Section 37.2.

37.1.3 Photon Propagator and Gauge Conditions In Section 26.4, for the self-interaction of a scalar field, we have established that a propagator associated with internal line of a Feynman diagram is the Green function of the differential operator that defines the left-hand side of the classical equation for a free field. In Chapter 8, we have shown that, for an electromagnetic field, this equation can have different forms depending on the choice of gauge. For example, Equation (8.6) is gauge-invariant and Equation (8.9) is equivalent to Equation (8.6) under the Lorentz condition (8.8). First, we try to determine the Green function of the operator corresponding to the gauge-invariant equation 8.6. Then the corresponding A .x/ must satisfy the equation Green function G A A .x/  @ @ G .x/ D g ı.x/: @˛ @˛ G

We seek a solution in the form of the Fourier transform Z 1 A eA G .x/ D d k e ikx G  .k/: .2/4 eA Thus, the corresponding equation for G  .k/ has the form eA .ı k 2  k k  /G  .k/ D g :

(37.20)

We rewrite Equation (37.20) in the matrix form e 0 .k/G A .k/ D G; …

(37.21)

470

Chapter 37

Quantum Electrodynamics (QED)

where the matrix G is defined by the matrix tensor g [see (1.2)] and the matrix element of the matrix … D ….k/ has the form e 0 D k 2 ı  k g k : … It is easy to establish (Problem 37.3) that e 0 D 0: det … e 0 .k/ does not have an inverse matrix and, hence, This implies that the matrix … Equations (37.20), (37.21) cannot be solved for the function G A .k/: To overcome this difficulty, a gauge should be fixed. The most convenient realization of this problem is to add the gauge-noninvariant term LL .x/ D 

1 1 L.x/2 D  .@ A .x//2 ; 2˛ 2˛

(37.22)

where the parameter ˛ is arbitrary and can be assumed tending to infinity after transformations, to the Lagrangian LA .x/: Since the theory is gauge-invariant, the physical results must not depend on ˛: Therefore, we can retain a finite value of the parameter ˛ and set its value depending on the problem. For this reason, the first two equations of motion in (37.4) do not change for the new Lagrangian L 0 .x/ D L .x/ C LL .x/ and the third equation takes the form A .x/ C

˛1 @ .@ A .x// D J .x/: ˛

In this case, the Green function satisfies the equation A @ˇ @ˇ G .x/ C

˛1 A @ @ G .x/ D g ı.x/ ˛

eA and, for the Fourier transform G  .k/, we have  ı k 2 

 ˛1 eA k k  G  .k/ D g : ˛

(37.23)

It is easy to verify that for k 6 0 and ˛ ¤ 0, 1, the determinant of the matrix of the left-hand side of the equation is not equal to zero and a solution of Equation (37.23) is the matrix with matrix elements  1 k k A c e e (37.24) G  .k/ WD D  .k/ D  2 g C 2 .˛  1/ : k k

471

Section 37.2 S -Matrix in QED

Repeating all reasonings in Section 23.1, we obtain the following expression for the casual Green function of an electromagnetic field: c D .x

1  y/ D  .2/4

  e ik.xy/ k k dk 2 g C 2 .˛  1/ ; k C i" k

Z

(37.25)

where ˛ is an arbitrary number and the physical observable quantities do not depend on it. Therefore, the conditions imposed on ˛ can be forgotten. The quantity ˛ D 1 is associated with Feynman propagator ig D0c .x  y/ (see (23.15)), which is the Green function of the d’Alembert operator. Very often, in transformations, a so-called Landau gauge .˛ D 0/ is used. In the theory of Yang–Mills gauge fields, the axial gauge is used. Main types of gauges are described in what follows.

37.2

S -Matrix in QED

37.2.1 Perturbation Theory. Feynman Diagrams The formal expression for the S-operator (26.36)–(26.39) in the case of QED has the form Z (37.26) S D T expŒi e W N .x/  .x/A .x/ W dx: M

Expanding the exponent in relation (37.26) in a series and using the Wick theorem for the T -product (see Section 23.4.2), we can rewrite the expansion coefficients Sn in the form of the sum of normal products of the fields N , , and A for all possible pairings. Using this representation, we can immediately formulate correspondence rules between analytic expressions and their graphic representations (Feynman diagrams). It is obvious that, in the case of QED, internal lines are lines of two types. The first type corresponds to pairing of two operators of an electromagnetic field (see (23.49)). We denote the corresponding pairing (as a result of pairing) in a Feynman diagram by a wavy line c .x  y/ iD

”

.; x/

. ; y/ :

(37.27)

The second type corresponds to pairing of two operators of an electron-positron field. We denote this pairing by a solid line with given direction 1 c S .x  y/ i

”

x

y:

(37.28)

The operator A .x/ is associated with a wavy external line that starts (or ends) at the point x x A .x/ ” (37.29)

472 and the operator

Chapter 37

Quantum Electrodynamics (QED)

.x/ is associated with a solid line directed to the point x .x/

”

x

:

(37.30)

The operator N .y/ is associated with a solid line directed from the point y N .y/

”

y

:

(37.31)

Finally, a vertex at which all three lines (37.29)–(37.31) encounter is associated with factor e  or, together with (37.29)–(37.31), e W N .x/  .x/A .x/ W

”

:

(37.32)

x

Thus, for the representation of an analytic expression by the corresponding Feynman diagram, it is necessary to use rules (37.27)–(37.32) starting from a fermion line leaving the diagram and moving in the opposite direction. By analogy with case with interaction (25.3) the matrix elements of the S -operator between the initial ˆs and final ˆf states are determined (see Section 26.4.3). In this case, the correspondence rules between matrix elements in the momentum space and the corresponding lines in Feynman diagrams are the following: (1) the motion of the electron with momentum p in the initial state is associated with factor p I (37.33) .2/3=2 v ; .p/ ” (2) the motion of the positron with momentum p and spin in the initial state is associated with factor .2/3=2 vN ; .p/

”

p

I

(37.34)

(3) the motion of the electron with momentum p and spin in the final state is associated with factor .2/3=2 vN ;C .p/

”

p

I

(37.35)

(4) the motion of the positron with momentum p and spin in the final state is associated with factor .2/3=2 v ;C .p/

”

p

I

(37.36)

473

Section 37.2 S -Matrix in QED

 and momentum k in the initial or (5) the motion of the photon with polarization e final state is associated with factor

k  e p

.2/3=2 2k 0

”

I

(37.37)

k

(6) the motion of a virtual electron from the point y to the point x is associated with factor Z p, e − m C pO 1 dp ” I (37.38) .2/4 i m2  p 2  i " (7) the motion of a virtual positron from the point x to the point y is associated with factor Z p, e + m C pO 4 .2/ i dp ” I (37.380 ) m2  p 2  i " (8) the motion of a virtual photon from x to y or from y to x with 4-momentum k is associated with factor Z  k k dk 1 g ” C .˛  1/  4 2 2 .2/ i k k C i" k

” μ

νI

(37.39)

k

(9) the vertex of the diagram at which interaction processes (virtual transformation) of particles with 4-momenta p1 , p2 , and k occur is associated with factor p1

i e  .2/4 ı.p2  p1  k/ ”

k

:

(37.40)

p2

Note that the term of a virtual particle means that virtual particles do not satisfy the ordinary relativistic relation (5.4) between the energy p 0 and the momentum p, i.e., p 2 ¤ m2 : In addition to rules (1)–(9), the numerical factor and sign of the analytic expression of contributions of the given process are required for the representation of matrix elements. They can be determined with the use of the Wick theorem. For the detailed analysis (substantiation) and calculation of matrix elements of some actual processes by the Feynman rules (37.33)–(37.40), see [26], Section 20 and 21.

474

Chapter 37

Quantum Electrodynamics (QED)

37.2.2 Coefficient Functions of the S -Matrix in Terms of Creation and Annihilation Operators of Lines of Feynman Diagrams In the case of QED, the decomposition of the S-operator in normal products of free fields has the form Z 1 X X 1 .dx/N .dx 0 /N .dy/M SD .N Š/2 M Š 0 N;M D0

 

.˛/N ;.˛ /N ;./M F2N;M ..˛; x/N I .˛ 0 ; x 0 /N I .; y/M / W N ˛1 .x1 /    0 0 0 .x /A .y1 /    A ˛10 .x1 /    ˛N 1 M .yM / W : N

N ˛N .xN / (37.41)

Note that the vector indices 1 ; : : : ; N in F2N;M should be considered as upper (contravariant). In addition, the equal number of operators N and in decomposition (37.41) is caused by the condition of invariance of the operator S under the operation of charge conjugation (26.58). To rewrite expression (37.26) in terms of the coefficient functions F D ¹F2N;M º1 N;M D0 , as in Section 27.1, we define operations corresponding to the transformation of coefficient functions of operator (37.41) into coefficient functions of the operators W N .x/S W ; W N .x/S W ; W

.x/S W ; W

.x/S W; W A .x/S W ; W A .x/S W :

(37.42)

We propose (Problem 37.4) to repeat calculations performed in Section 27.1 taking into account that the operators and N in the normal product anticommute with operators of Fermi fields. We denote the corresponding operations by N C .x/, N  .x/, C .x/, and a .x/: It is worth noting that the sign “” is separated  C .x/,   .x/, a  from the operation corresponding to the operator W N .x/S W because, in this operation represented with the use of the function S c .x  y/, it is necessary to take into account that N ˛ .x/ ˇ .y/ D  ˇ .y/ N ˛ .x/ D  1 S c .y  x/: i ˇ˛ In calculations, it is necessary to consider the decomposition of WOF of the form (37.41) not taking into account the condition of invariance under the operation of charge conjugation, i.e., the decomposition in the coefficient functions FN;N 0 ;M , where N and N 0 are the numbers of the fields N and in a decomposition of the form (37.41) but N ¤ N 0 , i.e., the sum in (37.41) holds both for N and for N 0 : We obtain the following expressions for WOF (37.42) (see the notation shortening representations): .N ˛C .x/F /N;N 0 ;M ..˛; x/N I .˛ 0 ; x 0 /N 0 I .; y/M / D

N X

.1/j 1 ı˛˛j ı.x  xj /

j D1

 FN 1;N 0 ;M ..˛; x/N n¹˛j ; xj ºI .˛ 0 ; x 0 /N 0 I .; y/M /;

(37.43)

475

Section 37.2 S -Matrix in QED

.N ˛ .x/F /N;N 0 ;M ..˛; x/N I .˛ 0 ; x 0 /N 0 I .; y/M / Z 1 D .1/N dx 0 S˛c0 ˛ .x 0  x/ i  FN;N 0 C1;M ..˛; x/N I .˛ 0 ; x 0 /; .˛ 0 ; x 0 /N 0 I .; y/M /;

(37.44)

.˛C .x/F /N;N 0;M ..˛; x/N I .˛ 0 ; x 0 /N 0 I .; y/M / 0

D

N X

.1/N Cj 1 ı˛˛j0

j D1

 ı.x  xj0 /FN;N 0 1;M ..˛; x/N I .˛ 0 ; x 0 /N 0 n¹˛j0 ; xj0 ºI .; y/M /;

(37.45)

.˛ .x/F /N;N 0 ;M ..˛; x/N I .˛ 0 ; x 0 /N 0 I .; y/M / Z

D

1 c 0 dx 0 S˛˛ 0 .x  x / i  FN C1;N 0 ;M ..˛ 0 ; x 0 /; .˛; x/N I .˛ 0 ; x 0 /N 0 I .; y/M /;

(37.46)

C .a .x/F /N;N 0 ;M ..˛; x/N I .˛ 0 ; x 0 /N 0 I .; y/M /

D

N X

ıj ı.x  yj /

j D1

 FN;N 0 ;M 1 ..˛; x/N I .˛ 0 ; x 0 /N 0 I .; y/N n¹j ; yj º/;

(37.47)

 .x/F /N;N 0 ;M ..˛; x/N I .˛ 0 ; x 0 /N 0 I .; y/M / .a

Z

D

dy 0 iD0 .x  y 0 /

 FN;N 0 ;M C1 ..˛; x/N I .˛ 0 ; x 0 /N 0 I .0 ; y 0 /; .; y/M /:

(37.48)

It is easy to verify that the following commutation relations are true (Problem 37.5): 1 c 0 ŒN ˛C .x/; ˛0 .x 0 /C D ŒN ˛ .x/; ˛C0 .x 0 /C D S˛˛ 0 .x  x /; i C  0 0 .x/; a Œa 0 .x / D iD0 .x  x /:

(37.49) (37.50)

The other commutators or anticommutators are equal to zero. Using these results, we easily establish that the operators T . N .x/S /, T . .x/S /, and T .A .x/S /, where T is the operation of the T -product, are associated with formal operators N .x/ D N C .x/  N  .x/; .x/ D  C .x/ C   .x/; a .x/ D

C a .x/

C

(37.51)

 a .x/;

which, with regard for (37.49) and (37.50), satisfy the relations ŒN ˛ .x/; ˛0 .x 0 /C D 0;

Œa .x/; 0 .x 0 / D 0:

(37.52)

476

Chapter 37

Quantum Electrodynamics (QED)

Also note that, in a Euclidean domain, the formal operators (37.51) and (37.52) are associated with free Euclidean Fermi fields defined by relations (34.21) and (34.22) in Part VI. To rewrite relation (37.26) in terms of coefficient functions, we differentiate (37.26) with respect to e: This yields Z  dS D iT W N .x/  .x/A .x/ W S dx : (37.53) de M4

Using the method described in Section 27.1, we rewrite this equation in terms of the action of operations (37.43)–(37.48) on the sequence of coefficient functions F D ¹F2N;M º1 N;M D0 of the operator S as follows: dF D AF; de Z

where ADi

F jeD0 D 0 D .1; 0; 0; : : :/;  N .x/a .x/ W dx W .x/

(37.54)

(37.55)

and the formal solution of Equation (37.54) F D e eA 0 D

1 X en n A nŠ

(37.56)

nD0

coincides with perturbation series for coefficient functions of the scattering matrix. Analogously, we can write a system of equations of the resolvent type whose iterations reproduce a perturbation series that does not take into account vacuum contributions (see Section 27.2). For details of these equations, see [158].

37.2.3 Furry Theorem In calculation of matrix elements with the use of Feynman diagrams, the Furry theorem is important because it enables one not to take into account diagrams containing fermion closed cycles with odd number of vertices. We formulate this theorem as follows: Theorem 37.2 (Furry theorem [61]). Contributions of Feynman diagrams for the S operator containing closed fermion cycles with odd number of vertices are identically equal to zero.

477

Section 37.2 S -Matrix in QED

Prior to proving this theorem, note that the contribution of this diagram to the decomposition of the S-operator (26.36), (26.37), by perturbation theory, corresponds to the term in Sn .x/n

Z n

D .i e/

T .W N .x1 /b A.x1 / .x1 / W : : : W N .xn /b A.xn / .xn / W/.dx/n ;

in which all Fermi operators (after the application of the Wick theorem) for the T product are paired with each other so that they form a closed cycle, i.e., .x1 / N .x2 /

.x2 / .x3 /   

.xn1 / N .xn /

.xn / N .x1 / :

Taking into account that N .x1 / .xn / D  .xn / N .x1 / D  1 S.xn  x1 /; i

(37.57)

we obtain the following representation for the contribution to the S -operator: cycle

Sn

D .i e/n  Z 1 c 1 c b b b  Tr W A.x1 / S .x1  x2 /A.x2 / : : : A.xn / S .xn  x1 / W .dx/n : i i (37.58)

The trace sign Tr Π   stands for the matrices   in the definition of the operators b A.x/ (see (11.9) or (37.3)) and the matrix function 1i S c .x/ (23.13) (for the definition of p, O see (6.85)). The Feynman diagram for the analytic expression (37.58) has the form x2 x1

:

(37.580 )

xn

Using properties of invariance of the trace under a cyclic permutation of matrices, we get (37.59) Tr A D Tr ŒC 1 AC  for any nondegenerate matrix C .

478

Chapter 37

Quantum Electrodynamics (QED)

Let the matrix C coincide with operator of charge conjugation (6.94), (6.99). Applying (37.59) to expression (37.58) and using the properties A.x/C D b A.x/T ; C 1 b C 1 S c .x/C 1 D S c .x/T ; where T is the sign of transposition of matrices, which are corollaries of property (6.95), representation (23.13), and definition (6.85), we obtain the following relation for (37.58): Z h 1 1 cycle nC1 n Sn D .1/ Tr W b A.x1 / S c .x1  xn /b .i e/ A.xn / S c .xn  xn1 / i i i 1 1 b A.xn1 / : : : b A.x3 / S c .x3  x2 /b A.x2 / S c .x2  x1 / W .dx/n ; (37.60) i i where the properties AT1 AT2    ATN 1 ATN D .AN AN 1    A2 A1 /T and Tr B T D Tr B are used. In (37.60), changing the variables xn ! x2 , xn1 ! x3 ; : : :, x2 ! xn , we obtain the equality cycle cycle Sn D .1/n Sn ; cycle

which yields Sn

 0 for an odd n:



Remark 37.3. It follows from the proved result that the theorem remains true in the case where a certain number of operators of the electromagnetic field A are paired. Remark 37.4. The Furry theorem has another sense in the calculation of matrix elements of the S-matrix for specific processes, i.e., in the case of the use of rules (37.33)–(37.40). In this case, on a diagram of the form (37.580 ), it is necessary to indicate the direction of motion of particles, i.e., momenta, and to take into account the type of particles, i.e., a charge, so that at each vertex, the low of conservation of charge is not violated. Using the Feynman rules (37.33)–(37.40), it is easy to verify that the contribution of a diagram of the form (37.580 ) with odd n does not already become identically equal to zero. Together with this diagram, it is necessary to take into account a diagram any virtual particle of which must be replaced by a virtual antiparticle, which is equivalent to the opposite bypass of the cyclic contour and the corresponding representation of an analytic expression. It turns out that the modulus of this analytic expression coincides with previous one and has the opposite sign and, hence, they compensate one another (Problem 37.6).

479

Section 37.2 S -Matrix in QED

37.2.4 Gauge Invariance for Coefficient Functions of the S -Operator The scattering matrix must be invariant under the gauge transformations ˛ .x/

!

N ˇ .x/ ! A .x/ !

0 ˛ .x/ N 0 .x/ ˇ A0 .x/

D e ie.x/ De

ie.x/

˛ .x/;

(37.61a)

N ˇ .x/;

(37.61b)

D A .x/ C @ .x/

(37.61c)

because the complete Lagrangian (37.2) satisfies these conditions. It is easy to verify invariance of the S-operator under transformations (37.61a)– (37.61c) using its representation (37.26) in the form of a perturbation series. Indeed, invariance under transformations (37.61a) and (37.61b) is obvious and invariance under change (37.61c) is a corollary of continuity equation under the assumption that .x/ decreases at infinity and integration by parts of the expression Z W N .x/  .x/ W @ .x/ (Problem 37.7). To derive conditions for the functions F2N;M , first, it is necessary to rewrite the transformed operator S 0 in terms of the fields N 0 and A0 in the form (37.41). Taking into account that the fields N .x/ and .x/ are taken at the same point x, we get Z 0 W N .x/  .x/A0 .x/ W dx: S D T expŒi e M4

We expand the exponent in a series and use the Wick theorem. After resummation of all contributions of equal normal products of fields, we obtain the relation 0

S D

1 X N;M D0

1 .N Š/2 M Š

X

Z

.dx/N .dx 0 /N .dy/M

.˛/N .˛ 0 /N ./M

M4.2N CM / 0 0 0  F2N;M ..˛; x/N I .˛ ; x /N I .; y/M / 0 0 0 0 .x /A  W N ˛1 .x1 /    N ˛N .xN / ˛10 .x10 /    ˛N 1 .y1 /    AM .yM / N

W; (37.62)

0 differs from F2N;M by the following: the perturbation series of the where F2N;M 0 functions F2N;M is constructed with the use of the functions 0

;c iD .x  y/ D A0 .x/A0 .y/; c .x  y/ by the term @ .x/@ .y/: However, according to which differ from iD   the condition of invariance (37.63) S 0 D S;

480

Chapter 37

at least

Quantum Electrodynamics (QED)

0 D F2N;0 : F2N;0

(37.64)

This follows from equating terms not containing the operators A0 .y/ in decompositions (37.41) and (37.62). Condition (37.64) implies that coefficient functions and matrix elements are independent of the choice of gauge of the photon propagator. To guarantee Equation (37.63), in addition to condition (37.64), which should be extended to all coefficient functions 0 D F2N;M ; F2N;M

the condition @ 

@yj j

F2N;M .: : : I : : : I : : : ; j ; yj ; : : :/ D 0;

j D 1; : : : ; M;

(37.65)

should be imposed. Recall that the summation over the index j is taken from 0 to 3: In the momentum space, condition (37.65) has the form e 2N;M .: : : I : : : I : : : ; j ; yj ; : : :/ D 0 .kj /j F

(37.66)

e 2N;M , M > 2: and, in fact, is the condition of transversality of the functions F

37.3

Equations for Green Functions and Coefficient Functions of the S -Matrix

37.3.1 Schwinger Equation In Section 28.2, we have obtained the equation for the Green functions of scalar fields corresponding to self-interaction (25.3). Analogously, using Equations (37.4), it is easy to derive equations for the Green functions defined as follows: G2N;M ..˛; x/N I .ˇ; x 0 /N I .; y/M / D .ˆ0 ; T . ˛1 .x1 /    ˛N .xN / N ˇ .x10 / 1

0    N ˇN .xN /A1 .y1 /    AM .yM //ˆ0 /:

(37.67)

The first of these equations has the form (Problem 37.8) .i ˛1 ˛ @;x1  m/G2N;M ..˛; x/N j˛1 D˛ I .ˇ; x 0 /N I .; y/M / D e˛1 ˛ G2N;M C1 ..˛; x/N j˛1 D˛ I .ˇ; x 0 /N I .; x1 /; .; y/M / C

N X

.1/N j i ı.x1  xj0 /ı˛1 ˇj

j D1

 G2N 2;M ..˛; x/N n .˛1 ; x1 /I .ˇ; x 0 /N n .ˇj ; xj0 /I .; y/M /;

(37.68)

481

Section 37.3 Equations for Green Functions and Coefficient Functions

where the notation .˛; x/N j˛1 D˛ WD .˛; x1 /; .˛2 ; x2 /; : : : ; .˛N ; xN / is used. In the integral form, Equations (37.68) take the form .N ¤ 0/ G2N;M ..˛; x/N I .ˇ; x 0 /I .; y/M / Z 1  D dx S˛c1 ˛ .x1  x/.i e/˛˛0 i  G2N;M C1 ..˛; x/N j˛1 D˛0 I .ˇ; x 0 /N I .; x/; .; y/M / C

N X

1 .1/N j S˛c1 ˇj .x1  xj0 / i

j D1

 G2N 2;M ..˛; x/N n .˛1 ; x1 /I .ˇ; x 0 /N n .ˇj ; xj0 /I .; y/M /:

(37.69)

Analogously (we recommend the reader to perform it by himself (Problem 37.9)), it is easy to derive a similar relation for the functions G2N;M starting from Equation (37.4). In the Lorentz gauge, it has the form(M > 1) G2N;M ..˛; x/N I .ˇ; x 0 /N I .; y/M / Z  D dy ig 1  D0c .y1  y/.1/N C1 .i e/ˇ ˛  G2N C2;M 1 ..˛; y/; .˛; x/N I .ˇ; y/; .ˇ; x 0 /N I .; y/M n .1 ; y1 // C .1  ı1M /

M X

ig 1 j D0c .y1  yj /

j D2

 G2N;M 2 ..˛; x/N I .ˇ; x 0 /N I .; y/M n ¹.1 ; y1 /; .j ; yj /º/:

(37.70)

Note that the counterterm corresponding to the substitution of the expression ˛ .x/ N ˇ .x/ into the definition of the Green function is equal to zero by virtue of the Furry theorem. The equation corresponding to the second equation of system (37.4) is, in fact, conjugate to (37.68) or (37.69) with regard for the relation  ˇ ˛

G2N;M ..x/N I .x 0 /N I .y/M / D .˝ 0 /N G2N;M ..x 0 /N I .x/N I .y/M / .˝ 0 /N ; where functions (37.67) are rewritten in the matrix form with respect to each of .2N / variables. Equations 37.69 and (37.70) are, in fact, an infinite system of relations between Green functions of an arbitrary number of arguments. For the determination of analytic expressions for Green functions, it is necessary to have closed integral equations

482

Chapter 37

Quantum Electrodynamics (QED)

at least for “main” Green functions. Conditionally, in QED, as “main” functions, the complete Green functions corresponding to the propagation of an electron (positron) G2;0 ..˛; x/, .ˇ; x 0 //, photon G0;2 ..; y1 /, .2 ; y2 // and its radiation (adsorption) G2;1 ..˛; x/, .ˇ; x 0 /, .; y// are taken. Analysis of these equations is considered in what follows.

37.3.2 System of Equations for Self-Energy and Vertex Parts of Green Functions In Section 28.6, we have described one-particle irreducible Green functions, i.e., functions that correspond to the sum of contributions of all strongly connected Feynman diagrams. In the case of QED, there exist two strongly connected 2-point Green functions. The first corresponds to the interaction of electrons or positrons with vacuum and e is called a mass operator. It is denoted by †.p/: By analogy with (28.44)–(28.48), it is connected with complete 2-point electron-positron Green function by the Dyson equation, which, in the momentum representation, has the form e G e 2;0 .p/; e 02I0 .p/ C G e 02I0 .p/ 1 †.p/ e 2I0 .p/ D G G i

(37.71)

where12 1 m C pO e 02I0 .p/ D 1 e G S c .p/ D D ; i i.m2  p 2  i "/ i.m  pO  i "/ and its solution takes the form e 2I0 .p/ D G

1 : e i.m  pO C †.p/  i "/

(37.72)

The second corresponds to the interaction of photons with vacuum and is called a e  .k/: The corresponding Dyson equation polarization operator. It is denoted by … for the photon Green function has the form e 00I2 .k/ C G e 00;2 .k/0 i … e 0  0 .k/G0;2 .k/ 0  ; e 0I2 .k/ D G G

(37.73)

  1 k k g C 2 .˛  1/ i.k 2 C i "/ k

(37.74)

where e 00I2 .k/ D i D e c;el G  .k/ D

is the free 2-point Green function of an electromagnetic field in the ˛-gauge (see (37.25)). 12 For

any nonsingular matrix A, here and below,

1 A

WD A1 :

Section 37.3 Equations for Green Functions and Coefficient Functions

483

Relations (37.71) and (37.73) can be derived based on the same reasonings as in the derivation of relation (28.47) (Problem 37.10). Since the polarization operator e  .k/ coincides with 2-point strongly connected part of the coefficient function …  e F 0;2 , it must satisfy the condition of transversality (37.66). This implies that e  .k/ D .g  k 2  k  k  /.k 2 /: …

(37.75)

e  .k/ in the second order of perturbaIt follows from analysis of the expression for … tion theory (see [26], Section 24) that it satisfies condition (37.73) if the constant independent of M 2 , i.e., the first term, is eliminated. It is easy to verify (Problem 37.11) e 0;2 .k/ preserves the structure that the complete Green function G r e 0;2 .k/ D G e t0;2 e l0;2 .k/ ; G .k/ C G

where

  k k id.k 2 / tr e G 0;2 .k/ D  2 g  2 k C i" k

(37.76)

(37.77)

is the transversal part and e l0;2 .k/ D  G

k2

i ˛ k k C i " k2

(37.78)

e 0;2 : is the longitudinal part of the complete Green function G Then the Dyson equation (37.73) for the functions d.k 2 / takes the form d.k 2 / D 1 C .k 2 /d.k 2 /:

(37.79)

For the Fourier transform of the complete photon Green function, we have the relation   k i k i ˛ k k   e 0;2 .k/ D G ;  2 g  2 2 2 .k Q /  k  i" k k C i " k2 .k Q 2 / D k 2 .k 2 /:

(37.80)

Thus, the electromagnetic interaction affects only the transversal part of the photon Green function, whereas its longitudinal part remains the same as for the free theory. Using the technique in Sections 28.6 and 28.7, we easily obtain from Equations (37.69) and (37.70) an equation for arbitrary connected (truncated) and strongly connected (one-particle irreducible) Green functions. Like Equations (37.69) and (37.70), they establish a connection between different strongly connected Green functions. For the proof of renormalization of QED, an equation that establishes a connection e e  .k/ with strongly connected vertex function of the 2-point functions †.p/ and …  e  2;1 .p; p k/ is important. This connection can also be directly obtained from analysis of Feynman diagrams (see, e.g., [2], Section 20.3). Proposing the reader to perform

484

Chapter 37

Quantum Electrodynamics (QED)

p p p k

p k

k

k

=

+

p−k

p−k

+

+ ...

p−k p−k Figure 37.1. Diagram representation of Equation (37.84).

these simple transformations by himself, we give their final relations Z e2 e 2;0 .p  k/G e e 0;2 .k/ e   .p; p  k/d k; †.p/ D  G .2/4 Z 2 e 2;0 .p/e e  .k/ D e Tr  G e 2;0 .p  k/dp; …   .p; p  k/G .2/4

(37.81) (37.82)

e 0;2 are the total 2-point Green functions and the vertex function is e 2;0 and G where G defined in terms of the strongly connected vertex part by the relation  e   .p1 ; p2 / WD i e.2/4 .  C ƒ .p1 ; p2 //:  2;1 .p1 ; p2 / WD i e.2/4 e

(37.83)

 e 2;0 , Rewriting the equation for e  2;1 , we establish that this equation connects e  , G e and  4;0 .p; k; q/ (see, e.g., [17], Section 144). Thus, the system of equations for e e .k/ , and e †.p/, …   .p; p  k/ is not closed. A closed system can be written staring from graphic reasonings by writing an integral relation for the function e   .p; k/ in the form of an infinite series of skeleton vertex diagrams whose external lines are e 0;2 .k/ and vertices are e 2;0 .p/ or G associated with the complete Green functions G associated with the complete strongly connected vertex functions e   .p; k/: The graphic representation in Fig. 37.1 stands for the nonlinear integral equation with infinite number of terms Z e2   e 2;0 .q/e e 2;0 .q  k/ e dq e   .p; p  q/G   .q; k/G  .p; k/ D  C .2/4 e 0I2 .p  q/ C    : e  .q  k; q  p/G (37.84)

The system of Equations (37.81), (37.82), and (37.84) is closed but Equation (37.84) is not analytically defined because it has infinitely many terms whose non-

Section 37.4 Divergences in QED and Methods for Their Elimination

485

linearity increases to infinity. We use this system only for analysis of divergences of e 0I2 , and e e 2;0 , G  : the functions G In the special case where the arguments of the function e   coincide, the connection e 2;0 is established between the vertex operator e   and the complete Green function G by the Ward identity [210] (see also [181], Chapter 16, Section 5; [2], Section 20.3) i

@ e G 2;0 .p/1 D e   .p; p/: @p 

(37.85)

This relation is easily verified by using the perturbation theory (its exact proof is given in Section 38.3.2). Taking (37.72) and (37.83) into account, the Ward identity is, sometimes, rewritten in the form 

37.4

e @†.p/ D ƒ .p; p/: @p 

(37.86)

Divergences in QED and Methods for Their Elimination

It follows from the analysis of interactions of quantum field theory performed in Section 29.2 that QED belongs to renormalizable theories because, according to relation (29.30), the maximum degree of a vertex of any Feynman diagram in QED is equal to 1 !imax D d  2; 2 max 6 0: Thus, for an actual physical space, the maximum degree i.e., for d 6 4, !i of a vertex !imax D 0: We try to analyze in more detail divergences that appear in perturbation theory and to establish the form of counterterms for their elimination.

37.4.1 Primitively-Divergent Diagrams and Their Regularization For the determination of the number and form of primitively-divergent diagrams, we use relation (29.32). We take into account that, for QED, !imax D 0 and denote the number of external fermion lines by Ne and the number of external photon lines by N : Taking into account that re D 1 and r D 0, we rewrite relation (29.32) in the form .d D 4/ 3 !.G/ D 4  N  Ne : (37.87) 2 It follows from relation (37.87) that the number of primitively-divergent diagrams with !.G/ > 0 is finite. We consider all possible cases.

486

Chapter 37

Quantum Electrodynamics (QED)

1. Ne D N D 0, !.G/ D 4: There is one second-order diagram.

Figure 37.2. Second-order vacuum diagram.

Any higher-order diagram n > 2 with Ne D N D 0 is already not primitively-divergent according to the definition in Section 29.1 because it contains as a subblock the diagram depicted in Fig. 37.2. 2. Ne D 0, N D 1, !.G/ D 3 (Fig. 37.3).

Figure 37.3. Diagram with one external line.

On the one hand, this diagram is divergent but not primitively-divergent because successively breaking one of the internal lines, we again obtain divergent diagrams. However, on the other hand, after the Pauli–Villars regularization (or other regularization), the contribution of this diagram becomes zero by virtue of the Furry theorem (see Section 37.2.3). 3. Ne D 2, N D 0, !.G/ D 1 (Fig. 37.4).

Figure 37.4. Second-order self-energy diagram for electron.

Section 37.4 Divergences in QED and Methods for Their Elimination

487

4. !.G/ D 2 (Fig. 37.5).

Figure 37.5. Second-order self-energy diagram for foton.

5. Ne D 0, N D 3, !.G/ D 1 (Fig. 37.6).

Figure 37.6. Diagram with three external photon lines.

The contributions of these diagrams, just as the diagrams depicted in Fig. 37.3, become equal to zero by virtue of the Furry theorem. 6. Ne D 2, N D 1, !.G/ D 0 (Fig. 37.7).

Figure 37.7. Vertex diagram.

7. Ne D 0, N D 4, !.G/ D 0 (Fig. 37.8).:

Figure 37.8. Scattering of light by light diagram.

488

Chapter 37

Quantum Electrodynamics (QED)

As for the first case (Fig. 37.2), in a similar way as for the scalar self-interaction (25.3), we can select the contributions of all vacuum diagrams in the form of a phase factor in the definition of the S-operator (see relations (26.53) and (26.54) and Remark 27.3) and do not take them into account in the calculation of matrix elements. It remains to analyze divergent diagrams corresponding to the self-energy parts of electrons and photons (the diagrams in Figs. 37.4 and 37.5), the vertex diagram (Fig. 37.7), and the diagram in Fig. 37.8. The detailed analysis of divergences of these diagrams is performed in [26], Section 24. Here, we give only their final form after the selection of the divergent and convergent parts by the method of intermediate Pauli–Villars regularization. The self-energy (Fig. 37.4) electron-positron function has the form .2/ e  2;0 .pI M /

e2 D 8 2

Z1



M 2  p 2 0 d .2m  p / O ln .p/; C †M m2

(37.88)

0 .2/ 0 .p/ is the convergent (as M ! 1) part of the function e  2 .p; M /: Its where †M limit has the form

lim

M !1

0 †M .p/

e2 D † .p/ D 8 2 0

Z1 0



m2  i " d .2m  p / O ln : m2  p 2  i "

Division (37.88) is performed so that †0 .p/ satisfies the conditions †0 .0/ D

@†0 .p/ jpD0 D 0: @p 

This implies that the regularized part of the function e  2;0 .p/ can be determined as a residual term of its expansion in the Maclaurin series up to order n D !.G/ D 1: .2/

R;.2/ e  2;0 .p/

D lim

M !1

.2/ e  2;0 .p; M /

.2/ e  2;0 .0; M /

@e  .2/  jpD0 p  @p 

!

0 D lim †M .p/ M !1

up to the polynomial of the first degree in p: O c1 .pO  m/ C c2 m: Analogously (see [26], Subsections 24.3 and 24.4), we determine the partition for the contribution of the diagram in Fig. 37.4 corresponding to the self-energy photon

Section 37.4 Divergences in QED and Methods for Their Elimination

489

function (!.G/ D 2). We present the following asymptotic value for this contribution for a large value of M : e 2  e2 M2 1   .2/; 2 2 e  0;2 .k; M / D g .M  m /  ln  .k k  g  k 2 / 8 2 4 2 m2 3 e .2/I .k/; C… (37.89) where e .2/I

…

e2   .k/ D .k k  g  k 2 / 2 2



Z1 d .1  / ln 0

m2  .1  /k 2 : m2  i "

As in the case of partition for e  2;0 .p; M /, partition (37.89) is constructed so that .2/

e .2/I .k/ e .2/I .k/ 1 @2 … ˛ e .2/I .0/ D @… … j k D jkD0 k ˛ k ˇ D 0: kD0 @k ˛ 2 @k ˛ @k ˇ This also enables us to determine the regularized part as a residual term of the expan.2/; sion of e  0;2 .k; M / in the Maclaurin series up to order n D !.G/ D 2: R;.2/; e  0;2 .k/ D lim

 .2/; e  0;2 .k; M /

M !1 .2/; e  0;2 .0; M / .2/; @e  0;2 .k; M / jkD0 k ˛  @k ˛

  .2/; .k; M / 1 @2 e ˛ ˇ j k k kD0 2 @k ˛ @k ˇ   D lim …M .k/ D … .k/: 

M !1

Finally, for the vertex function corresponding to the distribution of the diagram in Fig. 37.7 and defined in (37.83), we get   e2  M2 0 .3/; e .p; kI M / D   ln 2  2 C M .p; k/ 2 16 m and, with regard for n D !.G/ D 0, e  R;.3/; .p; k/ D lim Œe  .3/; .p; kI M /  e  .3/; .0; 0I M / M !1

0

D lim M .p; k/: M !1

0

Since the expression for M is rather awkward, we do not present it here.

490

Chapter 37

Quantum Electrodynamics (QED)

It remains to consider the contribution of the diagram depicted in Fig. 37.8. It corresponds to the contribution of the scattering of light by light, i.e., photon-photon interactions. However, this contribution is not convergent, which was proved in the works by Feynman [54, 55], Karplus and Neuman [106], and Ward [209] taking into account the condition of gauge invariance of the theory (see the corresponding reasonings in [181], Chapter 16, and [2], Subsection 23.3). Thus, all divergences of primitively-divergent diagrams can be eliminated by expanding the corresponding contributions in the Maclaurin series up to order n D !.G/ and determining the renormalized amplitude as the corresponding residual term. It is also easy to write the corresponding quasilocal operators in the coordinate space for the construction of counterterms in the expression for the complete Lagrangian. To this end, it is necessary to rewrite relation (37.88) in the form 1e .2/ 2 e1 .M 2 ; m/ C †00 .p/; e O †  2:0 .pI M / D † 0 .M I m/ C .m  p/ M i .2/ which, in fact, is the expansion of e  2;0 .p; M / in the Taylor series at the point pO D m: In addition, the condition of gauge invariance enables us to eliminate the first term in relation (37.89). Higher-order divergencies of perturbation theory are eliminated on the basis of general methods described in Sections 27.3 and 29.4. After the resummation, the counterterm takes the form

L .x/ D .Z  1/ W N .x/.i @O  m/ .x/ W ım W N .x/ .x/ W 1 A.x/ .x/ W :  .ZA  1/ W F .x/F  .x/ W C e.Ze  1/ W N .x/b 4 The constants Z , ZA , ım, and Ze are taken so that m and e are the observable quantities of the mass and charge of electron.

37.4.2 Mass and Charge Renormalization of Electron (Positron) An experimental process of measurement of mass and charge of electrons is necessarily connected with interaction of these particles with electromagnetic field. For this reason, their observable values differ from the mass and charge of free” particles. We denote these values of parameters by m0 and e0 in contrast to their actual values m and e: In the same way as for interaction (25.3) (see Sections 29.3 and 29.4), replacing in the Lagrangian the fields and A and the parameters m and e by their nonrenormalized values 0 , A0 , m0 , and e0 and passing to actual values of field, mass, and charge by the relations 0

1=2

.x/ D Z2

.x/;

1=2

e0 D Z1 Z21 Z3

1=2

A0 .x/ D Z3 A .x/; e;

m0 D m C ım

(37.90) (37.91)

491

Section 37.4 Divergences in QED and Methods for Their Elimination

we obtain the equality L 0 .x;

0

; A0 ; m0 ; e0 / D L 0 .xI ; A; m; e/ C L .x/:

In relations (37.90) and (37.91), we use more conventional notation for constants Z1  Ze ;

Z2  Z ;

Z3  ZA :

In Section 29.4, we have shown that transformations (37.90) and (37.91) are equive 2;0 .pI m0 ; e0 / and alent to renormalizations of the total 2-point Green functions G   e .pIm0 ; e0 / and the strongly connected vertex function e G  .p; p  kIm0 ; e0 / 0I2

e 2;0 .pI m0 ; e0 / D Z2 .e/G e r2;0 .pI m; e/; G e rI .pI m; e/; e  .pI m0 ; e0 / D Z3 .e/G G 0;2

e   .p; p  kI m0 ; e0 / D

0;2 1 e Z1 .e/ r .p; p

(37.92)

 kI m; e/:

Let the renormalized functions also satisfy the Ward identity (37.85). This leads to the equality Z1 D Z2 : e r ; and e er , G  r so that all equations We define the renormalized functions G 2;0 0;2 (37.71) (or (37.72)), (37.73) (or (37.80)), and (37.81)–(37.84) preserve their form. We er .p/, Q r .k 2 /, and ƒr .p; k/ by the relations introduce new regularized functions † er .p/ D †.pI e e e0 .m/; m; e/  †.m/  .pO  m/† Z21 † Z31 Q r .k 2 / D .k Q 2 I m; e/  .0/ Q  k 2 Q 00 .0/; er .p; k/ D ƒ .p; kI m; e/  ƒ0 ; Z11 ƒ

(37.93) (37.94) (37.95)

where, in (37.93) and (37.94), the first two terms of the expansion in the Taylor series at the points pO D m and k D 0, respectively, are subtracted and, in (37.95), the quantity ƒ .p; kI m; e/ at the point p D p0 WD .m; 0; 0; 0/ and k D 0 is subtracted. We choose the constants Z1 , Z2 , and Z3 so that Z21 D 1  †0 .m/; ƒ0 D  ƒ0 .m2 /;

Z31 D 1   00 .0/; e ım D †.m/:

Z11 D 1 C ƒ0 .m2 /;

Under this choice of the constants, using (37.72), (37.80), and (37.83), we get 1 ; er .p/  i "/ i.m  pO C †   k k i k k i r e G 0;2I .kI m; e/ D r g  2 ;  2 Q  k 2  i " k k C i " k2 C i " e  r .p; kI mI e/ D  C ƒr .p; kI mI e/: er2;0 .pI m; e/ D †

er , Q r , and ƒr defined by Equations (37.93)– It remains to show that the quantities † (37.95) do not contain divergences.

492

Chapter 37

Quantum Electrodynamics (QED)

We start from the function ƒr defined by Equations (37.83) and (37.84) with e D e0 and m D m0 : Thus, Z e02 e 2;0 .qI m0 ; e0 / dq e   .p; p  qI m0 ; e0 /G ƒ .p; kI m0 ; e0 / D .2/4 e 2;0 .q  kI m0 ; e0 /e e   .q; kI m0 ; e0 /G  .q  k; q  pI m0 ; e0 / e  G (37.96) 0;2 .p  qI m0 ; e0 / C    : e 0;2 , and e e 2;0 , G   expressed We substitute the strongly connected Green functions G r r r e e e in terms of the regularized functions G 2;0 , G 0;2 , and   by Equations (37.92) into the right-hand side of Equation (37.96). Taking (37.91) into account, we get h e2 Z 1 e r2;0 .qI m; e/ dq e  r .p; p  qI m; e/G ƒ .p; kI m; e/ D Z1 .2/4 e r2;0 .q  kI m; e/e e  r .q; kI m; e/G  r .q  k; q  pI m; e/ i e rI .p  qI m; e/ C    : G (37.97) 0;2 We rewrite (37.97) in the form e r2:0 ; G e r0;2 ; e  r / ƒ .p; kI m; e/ D Z11 ŒI3 .p; kI m; e; G e r2;0 ; G e r0;2 ; e C I5 .p; kI m; e; G  r / C    :

(37.98)

The integrals for I3 , I5 ; : : : contain regularized functions. Thus, divergences can appear only due to integration with respect to dq (each of the integrals for I3 , I5 ; : : : has only one integral with respect to the 4-momentum dq D dq 0 d q). Analysis of perer , er , G turbation theory shows that the asymptotics of the regularized functions G 2;0 0;2 and e  r , for large momenta, is no worse than the asymptotics of the free Green functions. Therefore, the integrals with respect to dq logarithmically diverge. Thus, for their regularization, only one subtraction is sufficient. Therefore, the differences Ii .p; kI : : :/  Ii .p0 ; 0I : : :/;

i D 3; 5; : : : ;

can be rewritten in the form of one convergent integral. Comparing (37.95) with ƒ .p; kI m; e/  ƒ0 , which is determined with the use of (37.98), we can set X ƒr .p; kI e; m/ WD ŒIi .p; kI : : :/  Ii .p0 ; 0I : : :/; (37.99) i

ƒ0

WD Z11

X

Ii .p0 ; 0I : : :/:

(37.100)

i

er .p/ and Analogous arguments are true for the representation of the quantities † r 2 Q .k / by the corresponding differences of integrals (37.81) and (37.82) (Problem 37.12).

Section 37.5 Spectral Representations of 2-Point Green Functions

493

Of course, these arguments cannot be regarded as a rigorous proof of finiteness er , Q r , and ƒr : However, this statement can be verified in each of the quantities † order of perturbation theory. To construct a perturbation theory for arbitrary Green functions, it is necessary to write contributions of the corresponding skeleton diagrams e r , and G e r with vertices and internal associating the regularized functions e  r , G 2;0 0;2 lines, respectively.

37.5

Spectral Representations of 2-Point Green Functions

In Section 28.7, we have described the general scheme of the constriction of the spectral Källén–Lehmann representation for the 2-point Green function of a real scalar field based on general foundations of quantization of fields. This scheme remains true for fields coupled by the electromagnetic interaction. For this reason, we dwell only on several nuances typical of QED. We start from the 2-point photon Green function. In Section 37.3.2, we have established that the structure of this function is defined by division (37.76)–(37.78) or (37.80), i.e., due to the interaction of photons with electromagnetic field, only the transversal part of this function changes. Then it is necessary to write the spectral tr : It is obvious that representation only for the transversal part G0;2 d.k 2 / D k2 C i "

Z1 d2 0

.2 / : k 2  2 C i "

(37.101)

We again assume that the spectrum has a gap connected with 1-photon stable state. Thus, (37.102) .2 / D Z3 ı.2 / C .2 /: Using the same reasonings as for the function of a scalar field, we easily establish that the constant Z3 satisfies conditions (28.89) and (28.90). Using relation (37.89), we can determine the form of the spectral function ./ and the constant Z3 up to the second order of perturbation theory (for details, see [26], Subsection 32.1). Comparing (37.89) with (37.75), for .k 2 /, we get e2 M2 e2 ln C I.k 2 /; .k 2 / D  12 2 m2 4 2  2 Z 1 m  .1  /k 2 2 I.k / D 2 d .1  / ln : m2 0

(37.103) (37.104)

In (37.104), changing the variable D 12 .1  x/ and integrating by parts, we obtain 1 I.k / D  k 2 3

Z1

2

0

x 2 .3  x 2 / dx: 4m2  k 2 .1  x 2 /

494

Chapter 37

After the change 1  x 2 D 1 I.k / D  k 2 3

4m2 , 2

Z1

2

4m2

Quantum Electrodynamics (QED)

this relation takes the form

p .1 C 2m2 =2 / 1  4m2 =2 2 d : 2 .2  k 2  i "/

(37.105)

Taking (37.79) and (37.103) into account, we establish that, in the second order of perturbation theory (see (37.101)), e2 M2 e2 ln C I.k 2 /: (37.106) 12 2 m2 4 2 Comparing (37.105), (37.106) with (37.101), (37.102), we obtain the following explicit expressions for the spectral function ./ in the second order of perturbation theory:  s e2 4m2 2m2 .2/ 2 1  2 .2  4m2 / 1C 2  . / D 2 2 14    d .2/ .k 2 / D 1 

and the constant Z3

e2 M2 ln ; 12 2 m2 which is divergent in the second order of perturbation theory as M ! 1: For completeness, we write the remainder term for the spectral representation for the transversal part of the 2-point photon Green function .2/

Z3 D 1 

tr .; xI ; y/ G0;2

D

c;tr Z3 D  .x

Z1  yI 0/ C

c;tr d2 .2 /D  .x  yI /;

0

where c;tr .x D 

1  yI / D .2/4

Z

  e ik.xy/ k k g   2 dk 2 :   k2  i " k

In conclusion, we obtain the spectral representation for the electron-positron Green function (in notation of (37.67)): Z 1 e 2;0 .p/˛ˇ : G2;0 .˛; xI ˇ; y/ D dp e ip.xy/ G .2/4 We start from the 2-point Wightman function rewritten in the matrix form W2;0 .x; y/ D .ˆ0 ; .x/ N .y/ˆ0 /: Repeating the arguments presented in Section 28.7, by analogy with (28.82) and (28.85), we get Z 1 dp .p 2 /.p 0 / .p/e ip.xy/ ; (37.107) W2;0 .x; y/ D w.x  y/ D .2/3

495

Section 37.5 Spectral Representations of 2-Point Green Functions

where the matrix .p/ is defined by the decomposition XX .p 2 /.p 0 / ˛ˇ .p/ D .2/3 .ˆ0 ; ˛ .0/ˆp;˛.n/ /.ˆp;˛.n/ ; N ˇ .0/ˆ0 / n ˛.n/

and ˆ.p.n/ ;˛.n/ / , ˆ.0;0/ D ˆ0 is the complete collection of states that satisfy (28.78) with P  corresponding to QED. The matrix structure of .p/ can be determined with the use of the decomposition of the matrix .p/ in the complete system of linearly independent matrices i defined in relation (6.9), i.e., .p/ D

16 X

i i .p/:

iD1

We rewrite this decomposition for the matrix function w.x/ taking into account structure (6.9) of the matrices i as follows: T .x/C 5 W P .x/CD  WA .x/; (37.108) w.x/ D 1W S .x/Ci   WV .x/C  W

where letters S, V , T , P , and A denote scalar, vector, tensor, pseudoscalar, and axialvector parts of the matrix function W2;0 : To verify this, it is necessary to use the laws of transformation of the fields and N with respect to the homogeneous Lorentz transformations (see (17.13)) with a D 0 and property (6.22) and to establish the corresponding properties under the Lorentz transformations for the expansion coefficients (37.108). It is easy (see Exercise 37.13) to show that W T D W P D W A  0: We finally rewrite (37.108) in the form W2;0 .x/ D 1  W S .x/ C i   WV .x/: Under the Lorentz transformations, the quantities W S and W V have the properties W S .x/ D W S .ƒx/; WV .x/

D

(37.109)

.ƒ1 / W .ƒx/:

(37.110)

Equality (37.109) means that W S .x/ D w S .x 2 /;

2

x 2 D x 0  x2 :

Equality (37.110) means that WV has the form of the 4-gradient of a Lorentz-invariant function [see (5.70 )], i.e., WV .x/ D @ w V .x 2 /:

496

Chapter 37

Quantum Electrodynamics (QED)

Then we can rewrite representation (37.107) in the form Z 1 dp .p 2 /.p 0 /Œ1  wQ S .p 2 / C pO wQ V .p 2 /e ip.xy/ : W2;0 .x; y/ D .2/3 Using again representation (28.83) and definitions (18.7) and (19.6)–(19.9), we get Z1 W2;0 .x; y/ D

 1 1 d2 1 .2 / S  .x  yI / C 2 .2 /1  D  .x  yI 2 / ; i i

0

(37.111) where 1 .2 / D wQ V .2 /; 2 .2 / D wQ S .2 /  wQ V .2 /:

(37.112)

Now we proceed to the representation for the 2-point electron-positron Green function G2;0 .x; y/ D .ˆ0 ; T . .x/ N .y//ˆ0 / D .x 0  y 0 /.ˆ0 ; .x/ N .y/ˆ0 /  .y 0  x 0 /.ˆ0 ; N T .y/ T .x/ˆ0 /: (37.113) Here, we use the transposition sign T for the spinors and N to preserve the matrix structure of both terms. Recall that the product of spinors N is a matrix and N is a scalar because is a column and N is a raw. If average (37.113) is rewritten in terms of the components of the fields and N , then the sign T can be omitted because T T D N N ˛ and ˇ D ˇ : ˛ The vacuum average of the first term is the right-hand side of relation (37.111). To write an analogous representation for the vacuum average of the second term, we use the rules of transformation of the operators and N with respect to the operation of charge conjugation (21.5) and (21.6) and invariance of vacuum under these transformations (21.9). Taking (6.95) into account, we finally obtain the representation (Problem 37.14) Z1

 1 c 2 1 c 2 2 d 1 . / S .x  yI / C 2 . /1  D .x  yI  / : i i 2

G2;0 .x; y/ D 0

(37.114) By analogy with (28.87), we select the point of the spectrum corresponding to a stable 1-particle (electron) state. To this end, in representation (37.114), we perform the changes 1 .2 / D Z2 ı.2  m2 / C C .2 / C  .2 /; 2 .2 / D 2 .2 /:

Section 37.5 Spectral Representations of 2-Point Green Functions

497

We write the following final relation for the Fourier transformation of the electron-positron 2-point Green function: e 2;0 .p/ D Z2 .pO C m/ C G i.m2  p 2  i "/ Z1 d2

C m

Z1 d2 m

i.2

pO C  C .2 /  p 2  i "/

pO    .2 /: i.2  p 2  i "/

To obtain restrictions on Z2 and the spectral functions ˙ .2 /, we write the spectral representation for the vacuum average of the anticommutator of the fields and N : In fact, this representation can be derived from relations (37.113) and (37.114) by replacing on the right-hand side of (37.113) and in the definitions of the functions S c and D c the function .x 0  y 0 / by 1 and the function .y 0  x 0 / by 1: The corresponding representation takes the form .ˆ0 ; Œ .x/; N .y/C ˆ0 / D

Z1

h 1 d2 1 .2 / S.x  yI /C i

0

i 1 C 2 .2 /1  D.x  yI 2 / ; i

(37.115)

where S.x  yI / and D.x  yI 2 / are the commutator functions defined by relations (18.8) and (19.8), (19.9). Setting x 0 D y 0 in (37.115) and taking into account (17.20) for a scalar field and (17.44) for spinor fields, we obtain the equality Z1

Z1 2

2

d2 ŒC .2 / C  .2 /:

d 1 . / D Z2 C

1D 0

(37.116)

m

Equation 37.116 restricts values of the normalization constant and guarantees renormalization of the spectral functions ˙ :

Chapter 38

Quantization of Gauge Fields

Using, as an example, quantum electrodynamics, which is invariant under the simplest gauge group G D U.1/, in Section 37.1, we have shown that quantization of an electromagnetic field encounters considerable difficulties because, first of all, the standard canonical method of quantization of fields cannot simultaneously guarantee the conditions of covariance (relativistic invariance) and gauge invariance of the theory. For example, for the construction of a photon propagator (see Section 37.1.3), it is necessary to fix a gauge counterterm (37.22), i.e., to violate the gauge invariance of the Lagrangian. In the case of QED, it turns out that this procedure is sufficient to correctly define the photon propagator for an arbitrary gauge (see (37.25)) and, in the ordinary way, to construct a covariant perturbation theory for the scattering matrix (or for Green functions) on the basis of the interaction Lagrangian (see Section 37.2.1). The main problem of the application of this scheme to the construction of the propagator for the gauge field Wa is caused by the fact that even the Lagrangian of the free Yang–Mills field 1 a  1  a 0 a .@ W /.@ W0 /; Fa  (38.1) LYM .x/ D  F 4 2˛ a which is modified with the use of a nontrivial term of the form (37.22), with F defined by relations (12.11) does not lead to a simple propagator of the form (37.25). The reason is that Lagrangian (38.1), in addition to ordinary quadratic in Wa , terms, contains the terms of the form g.@ Wa /W˛b Wˇc or g 2 .Wa /2 .Wb /2 , a ¤ b ¤ c: This implies that the fields Wa necessarily interact with each other with interaction constant g (a parameter of the corresponding group of gauge transformations). For g D 0, Lagrangian (38.1) describes N 2 1 (if the gauge invariance is described by the S U.N / group) independent identical massless fields, which, in fact, are identical to an electromagnetic field. In this case, the propagator coincides with relation (37.25) for each of these fields. So, for the construction of the complete propagator for the fields Wa , it seems quite natural to use a perturbation theory with respect to the parameter g: However, as was noted by Feynman in [57], this approach leads to the breakdown of unitarity of the S -operator. This breakdown is formed by the diagrams depicted in Fig. 38.1. Feynman also noted that this problem can be eliminated by subtracting contributions of topologically similar diagrams that describe the interaction of the field Wa with a certain fictitious field from the contributions of these diagrams. However, it is not clear how to realize this interaction so that all requirements of the theory are preserved and to take into account similar subtractions in higher orders of perturbation

Section 38.1 Path Integral for Green Functions in QED (Coulomb Gauge)

499

+

Figure 38.1. Diagrams that lead to the breakdown of unitarity of the S -operator.

theory. This problem was independently solved in 1967 by de Witt in [34, 35, 36] and Popov and Faddeev in [47]. It turns out that the method of path integration is the most convenient method for quantization of non-Abelian gauge fields because it preserves covariance of a theory and gives correct results. In Section 30, we have described in detail this method and illustrated its action in quantum nonrelativistic mechanics, the theory of self-interacting scalar field, and the theory with Yukawa interaction (13.35). Prior to proceeding to quantization of the Yang–Mills field, we briefly dwell on main the Faddeev–Popov ideas and illustrates its action in the simplest case of QED. We start from the form of a path integral in QED in the Coulomb gauge, which has been described in detail in Sections 37.1.2 and 37.1.3, lest one think that these new methods are completely disconnected with previous scheme of quantization.

38.1

Path Integral for Green Functions in QED (Coulomb Gauge)

In Section 30.3, we have briefly described the method of canonical quantization of a mechanical system with the use of a path integral for a system with additional constraints. In the case of QED, conditions imposed on the 4-vector of the potential A .x/ are analogs of these constraints (for details, see Sections 20.2.3 and 37.1.2). With the use of the Coulomb gauge, we can most clearly formulate the process of canonical quantization that preserves invariance under the group of space rotations and (which is shown in what follows) establish a beautiful connection with other gauges. Recall that (38.2) '1 .A/  r  A.x/ D 0 is the primary constraint and ? '2 .A /  1 .A /  r  A .x/ D 0;

where

? .x/ WD A .x/ C r A0 D @0 A; A

is the secondary constraint (see (37.9)).

(38.3)

500

Chapter 38

Quantization of Gauge Fields

Then, according to (30.67) and (30.68), with regard for Remark 30.4, the expression for the generating functional of the Green functions of the fields N , , and A, takes the form Z N GŒ; N ; j  D GŒ0; 0; 01 DDAD N D e iA. ; ;AIj / R Y N .x/C N .x/.x/  e i dxŒ.x/ ı.r  A.x//ı.r  .x//; (38.4) x2M

where A. N ; ; AI j / D

Z M

 1 dx W .x/@0 A.x/  .x/2 2

i 1  .r  A.x//2  .J.x/ C j.x//  A.x/ C L .x/ W Z 2  dx 0 HCoul .x 0 I j /; Z Z 1 W .J 0 .x/ C j 0 .x//.J 0 .y/ C j 0 .y// W HCoul .x 0 I j / D ; dx dy 2 4jx  yj R3

(38.5) (38.6)

R3

and L .x/ is the Lagrangian of a free spinor field. In addition, the expression for the integration measure in (38.4) is understood as a product of all components of the vectors  and A and the spinors N and : In integral (38.4), as a canonical momentum, ? .x/ D A .x/ C r A0 .x/ we take not the operator A (see (37.8)) but the operator A and denote the corresponding integration variable in (38.4) by .x/: For this reason, constraint (37.9) in integral (38.4) has a simpler form. We denote the integration variable A.x/ by the same letter as the operator of the vector potential A.x/: It is also necessary to explain why the classical source j  .x/ N ; j , is con(i.e., j 0 .x/), which is the argument of the generating functional GŒ; tained in the functional in the bilinear form (see (38.6)). The point is that classical external sources must be introduced in the Lagrangian as follows: L .x/ ! L .xI ; N ; j / D L .x/ C .x/ N .x/ C N .x/.x/ C A .x/j  .x/: Then, according to the form of L .x/ (see (37.2) and (37.3)), the function j  .x/ is always an additive summand to the current J  .x/ and, hence, in the quantization in the Coulomb gauge (see Section 37.1.2, relations (37.10)–(37.18)), j 0 .x/ is bilinear in HCoul : Finally, det k¹'1 ; 1 ºk in (38.4) disappeared (see (30.68)) because the Poisson bracket Z Z ı'1 .x/ ı1 .x/ D dy.@k ı.x  y//2 ¹'1 ; 1 º.x/ D dy ıAk .y/ ı.A? /k .y/

501

Section 38.1 Path Integral for Green Functions in QED (Coulomb Gauge)

is independent of the integration variables and the infinite constant det k¹'1 ; 1 ºk cancels with the same constant in the normalization factor GŒ0;0;0: We make several transformations in (38.4). First of all, we select theQsquare in the first two terms in (38.5). Taking into account the first ı-function in x ı.r  A.x//, we rewrite the integral with respect to the variable  in the form Z R Y i 2 ıŒr ..x/  @0 A.x//: (38.7) De  2 dx..x/@0 A.x// x

Taking into account that the dependence on  is concentrated only in (38.7) and it, in fact, is independent of A.x/, we cancel it with analogous integral in the factor GŒ0; 0; 01 : Now using relation (30.7) with C.x; y/ D i ı.x 0  y 0 /.4/1 jx  yj1 and C 1 .x; y/ D i x ı.x  y/ and z.x/ D i.J 0 .x/ C j 0 .x//, we represent the exponent of the Coulomb part of the Hamiltonian in the form Z R R 1 0 2 0 0 0  2i dx 0 HCoul .x 0 /  12 Tr ln C DA0 e i dxWŒ 2 .r A .x// CA .x/.J .x/Cj .x// W : De e (38.8) (It is supposed that integration is taken for functions that vanish at infinity.) Substituting (38.8) into integral (38.4) and taking into account that  Z 1 1 1 .@0 A.x//2 C .r A0 .x//2  .r  A.x//2 W dx W 2 2 2  Z 1 D dx  W F .x/F  .x/ W ; 4 we obtain the final expression Z N 1 GŒ; N ; j  D GŒ0; 0; 0 DAD N D e iŒA. ; where

;A/C N CN CAj

Q  A/; (38.9) ı.r

Z

h 1 dx W  F .x/F  .x/ C L .x/ 4 M4 i C A .x/J  .x/ C N .x/.x/ C .x/ N .x/ C A .x/j  .x/ W (38.10)

A. N ; ; A/ C N  C N

and

 Aj D

Q  A/ D ı.r

Y

ı.r  A.x//:

(38.11)

x

Integral (38.9) contains only the noncovariant ı-function that fixes the Coulomb gauge.

502

Chapter 38

Quantization of Gauge Fields

Using relation (38.10), it is easy to obtain propagators (2-point free Green functions) that must be associated with internal lines of Feynman diagrams in perturbation theory in constant e: For example, setting e D 0 and j  0, we easily take the Gauss integral with respect to the Grassmann variables N and (see (30.87)). Then the corresponding Green function ı 2 GŒ; N ; 0 1 c S .x  y/ D .i /2 j0 N i ı.y/ı .x/ N Z pO C m 1 dp e ip.xy/ 2 : D 4 .2/ i m  p2  i " Analogously, but somewhat more complicated (Problem 38.1), the propagator of a photon line is determined. To this end, it is necessary to use representation (30.19) for the ı-function and relations of the form (30.7). We get ı 2 GŒ0; 0; j  1 c;el, Coul .x  y/ D .i /2  D jj 0 i ıj .x/ıj  .y/ Z 1 e  .k/e ik.xy/ ; D d kD .2/4 i e 0i .k/ D D e i0 .k/ D 0; e 00 .k/ D 1 ; D D k2   1 ki kj e D ij .k/ D 2 ıij  2 : k C i" k

(38.12)

(38.13)

Remark 38.1. The Coulomb gauge is not convenient because constraints (38.2) and (38.3) lead to the non-Lorentz-covariant ı-function (38.11) in integral (38.9), which yields the noncovariant 2-point Green function (38.12), (38.13). The method described in what follows eliminates this disadvantage.

38.2

Covariant Gauges: Popov–Faddeev–de Witt Method

In the previous chapter, we have shown that, with the use of the technique of path integration, Green functions can be represented in the form of an integral of expŒi A. N ; ; A/, where A. N ; ; A/ is a completely covariant gauge-invariant expression of action that is independent of the choice of gauge. The specific type of gauge is fixed only by the corresponding ı-function in the integral. This fact was established for the Coulomb gauge. In 1967, independently Faddeev and Popov in [47] and de Witt in [34, 35, 36] proposed a scheme that simply realizes the passage from one gauge to other. We briefly comment the main idea of the method in [47] (see also [164]). Let S! .x/ be an element of the gauge group Gx (see Section 2.4) that is considered as a function on M: For the Yang–Mills theory, we denote the system of fields , A ,

Section 38.2 Covariant Gauges: Popov–Faddeev–de Witt Method

503

or Wa on which a representation of the group Gx is realized by letter B D B.x/: Let B ! be the result of the action of the element S! .x/ of the group on the field B (see (2.29) and (2.30)). For every x 2 M, the quantity !.x/ is a parameter that completely defines the element S! : A collection of the fields B ! , where B is fixed and S! runs through the gauge group G, is called an orbit of a gauge group. The expression for the generating functional of Green functions obtained in Section 38.1 is expressed by the path integral (38.9) of gauge-noninvariant factors containing the external sources , N , and j as functional arguments of the functional GŒ; N ; j  and ı-functions that fix a gauge. This is quite natural because Green functions themselves must change their form depending on the type of gauge but the measure R Y dB.x/ (38.14) dŒB WD e iA.B/ DB  e i L .x/dx x

is invariant under transformations of the gauge group G dŒB !  D dŒB:

(38.15)

In the case where the complete Lagrangian depends on all fields ˛i , A , and Wa , where ˛ D 1; 4 are spinor indices ,  D 0; 3 are vector indices, and .i; a/ are indices connected with fundamental representation of the group S U.N / (i D 1; N , a D 1; N 2  1), the measure DB has the form Y DB D D N ˛ D ˛ DA DWa ; (38.16) ˛;;i;a

where each differential in the product has form (30.75). Invariance of the measure (38.14) is a corollary of gauge invariance of the complete Lagrangian L and measure (38.16). The last statement is not trivial. Indeed, invariance of the measure D N D under transformations (2.30) [or (2.18) and (2.19)] is a corollary of det S D 1 for the Jacobian of transform (2.30). Invariance of the measure DA under the gradient transformations (2.29) is also obvious because, for every x 2 M, transformation (2.29) is a translation by a certain fixed constant !.x/ D g.x/: To prove invariance of the measure DW , it is necessary to use the law of transformation of gauge fields (2.32). For every x and , the Jacobian of this transformation J! D det k

@W!Ia k D det .x/ D 1 @Wb

(38.17)

by virtue of (2.36)–(2.39). Usually, this measure cannot be finite because its invariance under the transformations B ! B ! implies that the integral of an arbitrary gauge-invariant functional F .B/ .F .B ! / D F .B// with respect to the measure dŒB is proportional to the “volume of orbit,” i.e., Z Z F .B/dŒB

D!;

(38.18)

504

Chapter 38

Quantization of Gauge Fields

where D! is defined by the same relation of the form (30.8) and is an invariant measure on the group G: It is clear that the last integral is divergent. This can be explained using, as an example, an ordinary Lebesgue measure on the plane XOY perturbed by a translation-invariant function d .x; y/ D .x  y/dx dy;

2 C 1 .R1 /:

Then, for any function F 2 where is the weight, Z Z Z F .x  y/d .x; y/ D F .z/ .z/dz  dy:

(38.19)

L1 .R1 /,

R2

R1

(38.20)

R1

Here, the gauge group is a translation group. According to the main Faddeev–Popov idea, the path integral of the functional F .B/ with respect to the variables B is considered as an integral with respect to classes of fields formed with the use of an additional gauge transformation. For example, in QED, all fields A C @ !.x/ are united in one class with field A .x/: In fact, due to the gauge invariance, the action A.B/ and measure (38.14) are concentrated on classes of fields. For this reason, it is necessary to replace the integral over all possible fields by an integral over a certain surface (in the space of all fields B) that once intersects orbits of a gauge group and the integral along the orbit. In fact, this implies that it is necessary to determine the process of selection of factor (38.18), i.e., the volume of the orbit. In example (38.19), (38.20), it is necessary to integrate over the “surface” (line) x  y D z and to select integration along the orbit that is the straight line R1 : The integration measure over the surface already depends on the form of the surface but all physical results must not depend on the choice of surface. This idea is realized as follows: Let '.B/ D 0 with a certain surface such that the equation '.B ! / D 0 has a unique solution with respect to ! (or S! ) for any B: We define the functional ' ŒB by the condition Z ! Q ' ŒB ı.'.B //D! D 1:

(38.21)

It is easy to verify that the functional ' ŒB is gauge-invariant. Indeed, we rewrite (38.21) with a certain B instead of B (respectively, S 2 G). Then we have Z Z !C !C Q Q //D! D ' ŒB  ı.'.B //D.! C / 1 D ' ŒB  ı.'.B Z ' ŒB  ! Q D ' ŒB  ı.'.B : //D! D ' ŒB

Section 38.2 Covariant Gauges: Popov–Faddeev–de Witt Method

505

Thus, ' ŒB  D ' ŒB:

(38.22)

As a rule, we are interested in path integrals corresponding to average values of certain gauge-invariant functionals F ŒB with respect to the measure dŒB, i.e., R F ŒBdŒB : (38.23) hF i D R dŒB To select a factor of the form (38.18) in the numerator and the denominator of expression (38.23), we insert the identity (the left-hand side of (38.21)) into these integrals. We interchange the integrals with respect to D! and dŒB and perform the change B ! B ! in the integral with respect to dŒB, which yields ! Q Q ı.'.B // ! ı.'.B//:

By virtue R of (38.15), (38.22) and invariance of the functional F ŒB, we select the integral D! in the numerator and the denominator and represent average (38.23) in the form R Q F ŒB' ŒBı.'.B//dŒB : (38.24) hF i D R Q ' ŒBı.'.B//dŒB The main point of the proposed structure is independence of the quantity hF i of the surface '.B/ D 0: To illustrate this, under the integral sign in the expressions in relation (38.24), we insert other “identity” Z ! Q //D!; 1 D  ŒB ı..B where .B/ D 0 is the equation of other surface that once intersects orbits of the group G, i.e., the equation .B ! / D 0 has a unique solution with respect to ! for any B: We interchange the order of integration with respect to dŒB and D!, perform the change dŒB in the integral with respect to B ! B ! , use invariance of the functionals F ŒB,  ŒB, and  ŒB and the measure D!, and again interchange the order of integration. This yields the relation R Q F ŒB ŒBı..B//dŒB : (38.25) hF i D R Q  ŒBı..B//dŒB Remark 38.2. The proposed method enables one to pass to an integral of a more gen! // is Q eral form than (38.24). For example, if, in relation (38.21), the functional ı.'.B replaced by other gauge-noninvariant functional ˆŒB such that integral (38.21) exists and the corresponding gauge-invariant functional is defined by the relation Z (38.26) ˆ ŒB ˆŒB ! D! D 1;

506

Chapter 38

Quantization of Gauge Fields

then the same process leads to an integral of the form R F ŒBˆ ŒBˆŒBdŒB : hF i D R ˆ ŒBˆŒBdŒB

(38.27)

Thus, this method enables one to pass in a path integral from one surface to other, i.e., from one gauge to other. This implies that, for all gauge-invariant functionals F ŒB corresponding to observable physical quantities, integral (38.24) is independent of the choice of the type of gauge and, for gauge-noninvariant quantities, relation (38.24) enables one to obtain the corresponding relation in one or other gauge changing the form of the surface '.B/ D 0:

38.3

Covariant Quantization of Electromagnetic Interaction

In Section 38.1, relation (38.4) for the generating functional of Green functions in the Hamilton form and the fixed Coulomb gauge is written based on the formalism described in Sections 30.2 and 30.3. Representation (38.4) in the form of a path integral enables one to successively reduce it to the almost covariant form (38.9). According to Faddeev-Popov method, only the surface of integration corresponding to the Coulomb gauge '1 .A/ D r  A.x/ D div A.x/ D 0 is noncovariant. This relation enables one to construct a perturbation theory for Green functions. To this end, it is necessary to expand the exponent Z expŒi e dx W N .x/b A.x/ .x/ W in expression (38.9) in a series in the interaction constant e and, for each integral with respect to the measures DA and D N D , to use relations of the form (30.14) c;el,Coul .x  y/ with respect to the variable A and for momenta with C.x; y/ D 1i D a similar relation with respect to the Grassmann variables N and , which can be obtained from representation (30.88). However, this process is rather awkward and gives a noncovariant expression. To obtain the covariant form of perturbation theory, it is necessary to use the equation of surface invariant under the Lorentz transformations. This surface is the Lorentz condition (8.8) @ A .x/ D 0: Similar calculations lead to the function (Problem 38.2)   Z e ik.xy/ i k k 1 tr D .x  y/ D ı  2 dk 2 : i  .2/4 k C i" k

(38.28)

Section 38.3 Covariant Quantization of Electromagnetic Interaction

507

To obtain function (37.25), it is necessary, in relation (38.27), to take ˆŒB .B D A / in the form R  i 2 (38.29) ˆŒA D e  2˛ .@ A .x// dx : It is easy to verify that the functional ˆ ŒA is independent of A (Problem 38.3) and, hence, can be taken outside the integral sign and canceled with the same functional in the denominator of (38.27). This process must also be performed in the expression Q  A/ (for e D 0 and for the generating functional (38.9) with ˆŒA instead of ı.r N D   0). After integration, we get  Z 1 el j  .x/D .x  y/j  .y/dx dy ; exp 2 el .x  y/ coincides with expression (37.25) (Problem 38.4). where D We obtain three different schemes for the construction of a perturbation theory, furthermore, they lead to the same physical corollaries.

38.3.1 Connection between Different Gauges Representation (38.9)–(38.11) enables one to connect expressions for Green functions in different gauges. We rewrite expression (38.9) for the Coulomb gauge in the notation of the previous chapter R Q  A/ dŒBe B ı.r ; (38.30) G Coul Œ D R Q  A/ dŒBı.r Z

where B  D

dxŒ.x/ N .x/ C N .x/.x/ C j  .x/A .x/:

Following the Faddeev–Popov method, for every value of the field A .x/, we define the functional ŒA by the equality Z Y ŒA ı.@ A .x/ C .x//d .x/ D 1: (38.31) x

Using relations (30.22) and (30.23), we easily verify that the integral on the left-hand side of (38.31) does not depend on A .x/ along with the functional ŒA: We substitute identity (38.31) under the integral sign of the numerator and the denominator in (38.30), interchange the order of integration, and perform the following changes in these integrals: .x/ ! e ie .x/ .x/; N .x/ ! e ie .x/ N .x/; A .x/ ! A .x/ C @ .x/:

(38.32)

508

Chapter 38

Quantization of Gauge Fields

After these changes, the ı-function ı.@ A .x/ C .x// is transformed into ı.@ A .x// and the ı-function ı.r  A.x// is transformed into ı.r  A.x/ C .x//: Then we again interchange the order of integration and, using relation (30.23), take the integral over D: According to (30.23), instead of .x/, it is necessary to substitute the function  Z  Z 1 k Q.xI A/ D ./1 .x; y/r  A.y/dy D 1 @x Ak .x 0 ; y/d y; 4 jx  yj which is a solution of the equation (see (30.22)) r  A.x/ C .x/ D 0: Canceling by ŒA, which, in fact, is independent of A, and taking into account invariance of the measure dŒB under transformations (38.32), we obtain the following relation: R Q  B  ı.@ A .x// dŒBee x Coul R Q ; (38.33) Œ D G dŒB ı.@ A .x// x

where e D B

Z

Q Q dxŒ.x/ N .x/e ie .xIA/ C N .x/.x/e ie .xIA/

Q A//: C j  .x/.A .x/ C @ .xI

(38.34)

Using relations (38.33) and (38.34), one can easily obtain the connection between Green functions in the Coulomb gauge and Green functions in the Lorentz gauge.

38.3.2 Ward Identity Using the representation of Green functions in the form of a path integral, one can easily derive the Ward identity (37.85). We write the relation for the 2-point electron Green function in an arbitrary gauge '.A/ D 0;

R

G2I0 .x  y/˛ˇ D i

Q dŒB ˛ .x/ N ˇ .y/ı.'.A// : R Q dŒBı.'.A//

(38.35)

Recall that the notation B and dŒB is identical to (38.14). In the integral of the numerator in (38.35), we change the variables .x/ !

.x/e ie .x/ ;

N .x/ ! N .x/e ie .x/ :

(38.36)

Section 38.3 Covariant Quantization of Electromagnetic Interaction

509

By virtue of this change, L .x/ ! L .x/  e W N .x/  .x/ W @ .x/ and, together with exponents of fields (38.36), the measure dŒB in the numerator is multiplied by the exponent Z expŒi e.x/  i e.y/  i e W N .x/  .x/ W @ .x/dx: After this, we use the operator of variational differentiation ı=ı.z/ for both sides of (38.35) and set   0: Then, according to definitions (30.91)–(30.93), we obtain the equation ı.x  z/G2;0 ..˛; x/; .ˇ; y//  ı.y  z/G2;0 ..˛; x/; .ˇ; y// 

D ı 0 ı @zI G4;0 ..ı; z/; .˛; x/; .ı; z/; .ˇ; y//:

(38.37)

The right-hand side can be expressed in terms of the Green function G2;1 : To this end, we use the Schwinger equation 37.70 rewritten in the differential form z G2:1 ..˛; x/I .ˇ; y/I .; z// 0

0

D eg  ı 0 ı G4;0 ..ı; z/; .˛; x/I .ı 0 ; z/; .ˇ; y//:

(38.38)



We act by the operator @z on (38.38), sum over , and substitute (38.37) into the left-hand side of (38.38). In the obtained equality, we pass to the strongly connected parts according to Section 28.6. We obtain the following relation (in the matrix form): 1 1 G2;0 .x  y/ı.z  y/  ı.x  z/G2;0 .z  y/ Z 1  dz 0 @ z G0;2 ..; z/; . ; z 0 //2;1 .x; y; z 0 /: D e

Taking into account that, by virtue of the gradient invariance, k … .k/ D 0 (see (37.75)), we obtain the following relation from Equations (37.73) and (37.74): 0 c 0 @ z G0;2 ..; z/; . ; z // D @z; D0 .z  z /:

This relation yields the Ward identity 1 1 1  .x  y/ı.z  y/  ı.y  z/G2;0 .z  y/ D @;z 2;1 .x; y; z/ G2;0 e

(38.39)

or, in the momentum space, 1 1 G2;0 .p/  G2;0 .q/ D .p  q/   .p; q/:

(38.40)

510

Chapter 38

38.4

Quantization of Gauge Fields

Quantization of Yang–Mills Fields Interacting with Matter Fields

We consider the non-Abelian gauge group of transformations S U.N / (see Section 12.3). The corresponding field W .x/ is a function on M with values in the Lie algebra of the group SU.N /: It can be represented in terms of the fields Wa corresponding to exchange bosons and generators of the group S U.N / by the relation W D Wa T a : According to the gauge principle (see Section 11.3), we consider the complete interaction Lagrangian in the form L .x/ D L .x/ C LYM .x/;

(38.41)

L .x/ D L0 . ; D /

(38.42)

where is the Lagrangian of the free Dirac field in which the ordinary derivatives @ are replaced by the covariance derivatives D D @  igWa T a

(38.43)

and 1 a .x/Fa .x/; LYM .x/ D  F 4 a F .x/ D @ Wa  @ Wa C gf abc Wb Wc :

(38.44) (38.45)

Lagrangian (38.41) is invariant under the gauge transformations (see Chapter 12) !

a

.x/ D S! .x/ .x/ D e ig!a .x/T .x/; N .x/ ! N ! .x/ D N .x/S!1 .x/ D N .x/e ig!a .x/T a ; i W .x/ ! W! .x/ D S! .x/W .x/S!1 .x/ C S! .x/@ S!1 .x/: g .x/ !

(38.46) (38.47) (38.48)

We have shown in Chapter 2 (see Section 2.4) that the last transformation (38.48) can be directly written for the gauge fields Wa [see (2.32)–(2.40)] Wa .x/ ! .W ! /a .x/ D ab .x/Wb .x/ C @ ! a .x/;

(38.49)

where the matrix .x/ D expŒig !.x/ Q and !Q is defined by relation (2.37). In what follows, we use this transformation in the infinitesimal form, i.e., for the infinitely small functions-parameters ! a .x/ Wa .x/ ! .W ! /a .x/ D Wa .x/ C @ ! a .x/ C gf abc Wb .x/! c .x/:

(38.50)

Section 38.4 Quantization of Yang–Mills Fields Interacting with Matter Fields

511

For the construction of a perturbation theory for Green functions or scattering matrix, we use the methods presented in Section 38.2. For the functional F ŒB  F Œ N ; ; W  invariant under the gauge transformations (38.46)–(38.49) and averages (38.23) or (38.27), the gauge-invariant measure dŒB has the form R YYY d Wa .x/; dŒ N ; ; W  D e i L .x/dx D N D DW; DW D a



x

which is a corollary of (38.17). Analogs of constraints (or, in terminology of Section 38.2, surfaces of integration in Q the phase space of fields) are the functionals ˆŒB in (38.29) (or ı.'.B// in (38.24)) that fix a gauge. By analogy with electromagnetic interaction, the following functionals are considered: (1) the Coulomb gauge Q k Wk /  ˆ1 ŒB  ˆ1 ŒW  D ı.@

YY x

ı.@k Wka .x//I

(38.51)

ı.@ Wa .x//I

(38.52)

a

(2) the covariant Lorentz gauge Q  W /  ˆ2 ŒB  ˆ2 ŒW  D ı.@

YY x

a

(3) the temporal gauge Q 0/  ˆ3 ŒB  ˆ3 ŒW  D ı.W

YY x

ı.W0a .x//I

(38.53)

ı.W3a .x//I

(38.54)

a

(4) the axial gauge Q 3/  ˆ4 ŒB  ˆ4 ŒW  D ı.W

YY x

a

(5) the ˛-gauge of the general form i

ˆ5 ŒB  ˆ5 ŒW  D e  2˛

R

@ Wa .x/@ Wa .x/dx

;

(38.55)

which, for ˛ D 1, defines the diagonal Feynman gauge (see (37.25)). Unlike the electromagnetic field, the nontrivial functional ˆ ŒB  ˆ ŒW  defined by equality (38.26) is important. In the case of an Abelian group of transformations, ˆ ŒB is independent of the field B and cancels with the same functional in the denominator. For the calculation of the functionals ˆ ŒW , it is necessary to make

512

Chapter 38

Quantization of Gauge Fields

a comment. In examples (38.51)–(38.54), the integrals in (38.27) are taken over the surfaces 'i .W a .x// D 0;

i D 1; 2; 3; 4;

a D 1; N 2  1;

x 2 M;

(38.56)

that correspond to the arguments of the ı-functions in (38.51)–(38.54). This implies that only the variables W a lying on this surface make a contribution to the corresponding path integral. Then, in the definition of the functional 'i ŒB by relation (38.21), the integration is performed only with respect to infinitesimal transformations rather than with respect to all elements B ! of the class B: This enables one to use relation (38.50) for the corresponding .W ! /a .x/ and conditions (38.56) for the calculation of a functional. For 'i ŒW , we get Z Q i .W ! //D!; 'i ŒW 1 D ı.' (38.57) inf

where inf is a set of infinitesimal gauge transformations. Passing in integral (38.57) to the new variables !Q ia .x/ D 'i ..W ! /a .x//; we obtain 'i ŒW  D det M'i ; where .M'i .x; y//ab D

ı'i ..W ! /a .x// : ı! b .y/

(38.58)

Since the integration is performed over the surface inf , matrix (38.58) is independent of !a .x/ because W ! .x/ contains only terms linear in !a .x/: Matrix elements of the matrix-operator M'i [recall that M'i is a matrix of only the discrete variables a and b and an integral operator with kernel (38.58)] can easily be determined (Problem 38.5). We have (1) for the Coulomb gauge, .M'1 .x; y//ab D .ı ab @k @k  gf abc Wkc .x/@k /ı.x  y/I

(38.59)

(2) for the covariant Lorentz gauge, .M'2 .x; y//ab D .ı ab   gf abc Wc .x/@ /ı.x  y/I

(38.60)

(3) for the temporal gauge, .M'3 .x; y//ab D ı ab @0 ı.x  y/I

(38.61)

Section 38.4 Quantization of Yang–Mills Fields Interacting with Matter Fields

513

(4) for the axial gauge, .M'4 .x; y//ab D ı ab @3 ı.x  y/:

(38.62)

It is easy to see that, in cases (3) and (4), the matrices M'i , i D 3; 4, are independent of the fields. Therefore, the corresponding 'i ŒW  are constants (although infinite) and, in the same way as in the case of the electromagnetic interaction, they cancel with corresponding constants in the denominator of (38.27). Thus, quantization of the fields Wa does not have difficulties but the corresponding 2-point Green functions (for g D 0) are not Lorentz-covariant. Now we consider ˛-gauge (5) (see (38.55)). We use the same method as in the passage from one gauge to other considered in Section 38.3.1. We begin with relation (38.27) in the generalized Lorentz gauge @ Wa .x/ D ' a .x/; where ' a .x/ 2 inf : Then the corresponding ' ŒW  defined by the relation Z YY ' ŒW 1 D ı.@ .W ! /a .x/  ' a .x//d! a inf

a

x

is calculated in the same way as '2 ŒW  and is identical to it ' ŒW  D '2 ŒW  D det M'2 : In this case, we rewrite relation (38.27) in the form Z Q  W  '/dŒB hF i  ŒW ı.@ Z Q a W  '/dŒB: D F ŒB ŒW ı.@

(38.63)

Since hF i is independent of ', we multiply both parts of (38.63) by the functional " # Z 1 X a 2 exp i ' .x/ dx 2˛ a and integrate with respect to the measure D': We obtain the following relation for hF i : 1 hF i D N

Z F ŒB det M'2 e

1 i 2˛

PR a

.@ Wa .x//2 dx

dŒB;

where the normalization factor N is equal to the integral of F ŒB  1:

(38.64)

514

Chapter 38

Quantization of Gauge Fields

In the same way as for the electromagnetic interaction (see Section 38.3), substituting, instead of F ŒB, the not gauge-invariant functional Z F ŒB D expŒi Wa .x/ja .x/dx and setting g D 0, we obtain the following relation for the 2-point Green function of W -bosons (Problem 38.6): Z 1 ab i 1 c;YM ab ab D d k e ik.xy/ D .x  y/  D .x  y/ D .k/; i  i .2/4 (38.65)  k k ı ab ab g  2 .1  ˛/ : D .k/ D 2 k C i" k C i" The next task is the construction of a covariant perturbation theory for Green functions or scattering matrix.

38.5

Faddeev–Popov Ghosts

For g D 0, for each type of gauge, it is easy to obtain 2-point Green functions, which completely coincide with corresponding functions in the model of QED. However, any attempts to use the ordinary perturbation theory in g by expanding the exponent in series lead to difficulties caused by the factor det M'2 in (38.64). In [47], it is proposed to use the continual analog of relation (30.86) by introducing some new fields a .x/  and  a .x/  N a .x/ corresponding to fictitious (ghost) “scalar fermions.” Assuming that a .x/ and N a .x/ are continual Grassmann variables, we represent det M'i in the form Z R i N a .x/M'ab .xy/b .y/ N j D D: det M'j D e We consider the Lorentz-covariant case j D 2: M'2  M: In terms of Lagrangian, this implies that it is necessary to add the term LFP .x/ D @ N a .x/@ a .x/ C g@ N a .x/f abc b .x/Wc .x/

(38.66)

to the total Lagrangian L .x/: This Lagrangian corresponds to propagation of “spinless fermions” and their interaction with gauge fields Wa : Prior to writing Feynman rules for the construction of a perturbation series for Green functions or matrix elements of the S-matrix, we write an expression for the complete action corresponding to a system of the interacting matter fields .k/ with the use of the gauge group S U.N /, i.e., k D 1;2; : : : ;N ; a D 1; 2; : : : ; N 2  1: Thus, the complete action in the path integral of the generating functional for the Green function Z  N C N Cja Wa / iAŒ N ; ;W; e D N D DW D (38.67) GŒ; N ; j  D GŒ01 e i.

515

Section 38.5 Faddeev–Popov Ghosts

with fixed ˛-gauge is constructed with the use of Lagrangian (38.41)–(38.45), the term fixing the gauge LW D 

1 W @ Wa .x/@ Wa .x/ W ; 2˛

(38.68)

and the ghost Lagrangian (38.66). We write the complete action with regard for boundary conditions (decrease at infinity) for the variables , W , and  in the form (Problem 38.7) i A Π; W;  D i A0 Π; W;  C i AI Π; W; ;

(38.69)

i A0 Œ ; W;  D i A0 Œ  C i A0 ŒW  C i A0 Œ;

(38.70)

i AI Œ ; W;  D i AI Œ I W  C i AI ŒW I W  C i AI ŒW I ; Z  .k/ i A0 Œ  D i dx N ˛.k/ .x/.i ˛ˇ @  mı˛ˇ / ˇ .x/;   Z i ˛1   dx Wa .x/ g   C @ @ Wa .x/; i A0 ŒW  D  2 ˛ Z i A0 Œ D i dx N a .x/a .x/; Z  .k/ i AI Œ I W  D ig dx N ˛.j / .x/˛ˇ Tjak ˇ .x/Wa .x/; Z h i AI ŒW I W  D i dx gf abc g  W .@ W a /.x/Wb .x/Wc .x/ W

(38.71) (38.72) (38.73) (38.74) (38.75)

i 1  g 2 f abc f ade g  g  W W b .x/Wc .x/Wd .x/W e .x/ W ; 4 (38.76) Z i AI ŒW I  D igf abc dx.@ N a /.x/b .x/Wc .x/: (38.77) Using these relations, we simply establish the correspondence rules (Feynman rules) between analytic expressions and graphic representations. Internal lines are associated with Feynman propagators, which are Green functions of differential operator-matrices between the corresponding variables , W , and  in relations (38.72)– (38.74). Vertices of diagrams are associated with matrices or structure constants ahead of the corresponding fields [see (38.75)–(38.77)] and external lines are associated with corresponding fields or their derivatives (see (38.75)–(38.77)). We represent these rules in the form Table 38.1 writing them in the momentum space and performing antisymmetrization of the coefficient of the product of vector fields in relation (38.76) (the first term) and symmetrization of the coefficient in relation (38.76) (the second term) for the corresponding antisymmetry and symmetry of these coefficients to be explicit. These rules are supplemented with rule according to which each closed loop of dashed lines, i.e., lines corresponding to ghosts, is associated with factor .1/: We also note that expansions do not have diagrams with

516

Chapter 38

Quantization of Gauge Fields

external dashed lines, i.e., they are contained in diagrams only in the form of closed loops. In Table 38.1, the following notation is used: e .3/ .k1 ; k2 ; k3 / D .2/4 ı.k1 C k2 C k3 / V  1 Œ.k1  k2 / g  C .k2  k3 / g 3Š C .k3  k1 / g  ; 

e .4/Iabcd .k1 ; : : : ; k4 / V 

(38.78)

4

D .2/ ı.k1 C    C k4 / 1 eab ecd Œf f .g  g  g g / 4Š C f ecb f ead .g g  g g /



C f eac f ebd .g  g  g g :

38.6

(38.79)

BRST-Invariance

QED illustrates that the gauge invariance is, on the one hand, a method for the solution of many problems in the quantization of gauge fields and, on the other, restricts the choice of counterterms in the elimination of divergences. Fixing a gauge, we, hence, “hide” the initial gauge invariance of the theory and more restrict the form of counterterms. The Faddeev–Popov method briefly presented above preserves the Lorentz-covariance of the theory but fixes a gauge and, in fact, fixes the form of counterterms in the Lagrangian that guarantee renormalization of the theory. In what follows, we show that, for the ˛-gauge, this method does not violate renormalization of the theory but, in a more general case, the problem remains open. It turns out (!) that there exists a symmetry (even for the fixed ˛-gauge) for which path integrals, in terms of which Green functions are expressed, remain invariant. This symmetry was independently established by Becchi, Rouet, and Stora in [9, 10] and Tyutin in [202]. It is constructed with the use of a certain parameter  (the same for all x 2 M), which is a Grassmann variable i.e.,  2 D 0;

(38.80)

and anticommutes with the fields of ghosts  a and N a and the matter fields and N : To preserve the gauge invariance of Lagrangian (38.41)–(38.45) in the construction of a new symmetry, in relations (38.46)–(38.49), we choose ! a .x/ D   a .x/:

(38.81)

517

Section 38.6 BRST-Invariance Table 38.1. Feynman rules for gauge fields.

No.

1.

2.

Name

Graphic

Analytic

of contribution

representation

expression

Propagator of Fermi particles

Propagator of vector bosons

j, α

k, β p

k

a, μ

b, ν

k

3.

Propagator of “ghosts”

a

b

ıj k .2/4 i

Z

.m C p/ O ˛ˇ dp 2 2 m  p  i"

Z dk ıab 4 2 .2/ i k C i" h k k i  g  .1  ˛/ 2 k C i" ıab  .2/4 i

Z

dp C i"

p2

a, μ

4.

k

Quark-gluon vertex (38.75)



p

ig˛ˇ Tjak .2/4 ı.q C k  p/

q

α, j

β, k b, μ

5.

k2

3-gluon vertex (38.76) — the 1st term

.3/

a, λ

c, ν k2 k3

b, μ

6.

4-gluon vertex (38.76) — the 2nd term

e gf abc V .k ; k ; k /  1 2 3

k3

k1

c, ν

e ig 2 V 

.4/Iabcd

a, λ

k1

.k1 ; k2 ; k3 k4 /

k4 d, ρ

c, μ

7.

k

Gluon-ghost vertex

q a

g.2/4 ı.p C k  q/q f abc

p b

518

Chapter 38

Quantization of Gauge Fields

This choice of local parameters guarantees convergence of infinitesimal and complete gauge transformations (38.46)–(38.49) by virtue of relations (38.80). However, in addition to Lagrangian (38.41)–(38.45), the Faddeev–Popov theory contains two terms, namely, Lagrangian (38.66) and term (38.68) that fixes a gauge, i.e., a certain additional effective Lagrangian Lef .x/ D LFP .x/ C LW .x/:

(38.82)

Thus, the complete Lagrangian of the theory of interacting matter fields and the Yang– Mills fields has the form L

IW .x/

D L .x/ C LYM .x/ C Lef .x/;

(38.83)

and the corresponding action has the form A

IW

D A C AYM C Aef :

(38.84)

Using condition (38.80), we write arbitrary BRST-transformations in the infinitesimal form B ! B  D B1 C ı B; ı B D ıB; B 2 ¹

˛;

N ˇ ; Wa ;  b ; N c º;

(38.85)

where D ig a .x/T a ˛ .x/; ı N ˇ .x/ D ig N ˇ .x/ a .x/T a ; ı

˛ .x/

ıWa .x/

a

abc b

(38.86) (38.87) .x/Wc .x/;

D @  .x/ C gf  1 ı a .x/ D  gf abc  b .x/ c .x/; 2 1 a ı N .x/ D  @ Wa .x/: ˛

(38.88) (38.89) (38.90)

In relation (38.85), 1 is the identity operator in the representation where the Grassmann variable  is realized. The first two terms of Lagrangian (38.83) are invariant under transformations (38.85) with ıB defined in (38.86)–(38.88)  .x/: L .x/ C LYM .x/ D L  .x/ C LYM

This is consequence of the general gauge invariance of the theory with local parameters of the group in the form (38.81). The effective Lagrangian Lef .x/, under transformations (38.85) with ıB defined in (38.88)–(38.90), is transformed as follows: Lef .x/ ! Lef .x/ D Lef .x/ 

1   @ J .; W /; ˛

(38.91)

519

Section 38.6 BRST-Invariance

where J .; W / D .@  a .x/ C gf abc  b .x/Wc .x//@ Wa .x/:

(38.92)

We recommend the reader to perform himself rather awkward transformations and to verify relation (38.91) (Problem 38.8). It follows from relation (38.91) that the Lagrangian Lef .x/ is not invariant under the BRST-transformations (38.85)–(38.90). However, the form of term (38.92) that appears after this transformation enables us to state that the corresponding action Aef ! Aef D Aef is invariant provided that all fields vanish at the boundary of integration. Thus, the theory is invariant under the BRST-symmetry. Transformation (38.86)–(38.89) of the fields ˛ ; N ˇ ; Wa , and  b is nilpotent. This implies that, for the operation Q acting on the field B 2 ¹ ˛ ; N ˇ ; Wa ;  b º (as a functional variable in the integral for action) by the relation QB WDıB; 2

Q B D0:

(38.93) (38.94)

This property is easily proved by direct calculations taking into account the anticommutative properties of the fields of matter and ghosts and the Jacobi identity for structure constants (see also [213], Section 15.7). For property (38.94) to be true for all fields including the field N a , it is necessary to change the form of the effective Lagrangian. To this end, it is necessary to rewrite the exponent of the ˛-gauge term (38.68) in the form of the Fourier transform Z R  a R R 1 i˛  a  a e i 2˛ W@ W .x/@ W .x/Wdx D C e 2 ha .x/ha .x/dx e i ha .x/@ W .x/ Dh; (38.95) where C is the renormalization constant. Then, in the expression for the effective Lagrangian, it is necessary to replace term (38.68) by the sum of terms in exponents on the right-hand side of Equation (38.95) without imaginary unit i: Now the BRST-symmetry of the new action holds for transformations (38.86)–(38.89), the new transformation of the field N a ı N a D ha ;

(38.96)

and the additional transformation of the field ha ıha D 0:

(38.97)

Transformations (38.86)–(38.89), (38.96), and (38.97) belong to supersymmetric transformations. For example, even for g D 0, transformation (38.88) transforms

520

Chapter 38

Quantization of Gauge Fields

a gauge boson into ghosts, transformation (38.89) annihilates a 1-particle state of ghosts, and transformation (38.96) transforms a 1-particle state of antighosts into the b In the BRST-quantization, field quantum ha : The operator for a supercharge is Q: b the commutator or anticommutator of the supercharge Q with corresponding fields is defined by the action of the transformation Q (38.93) on these fields. The nilpotent b plays an important role in the determination of the physically admissible operator Q states ‰ in a Hilbert space of states. These states are defined so that all physically b i.e., admissible states belong to the kernel of the operator Q, b D 0: Q‰

(38.98)

This condition guarantees the absence of ghost states in a set of physically admissible b sets up a correspondence between nonphysical states. In addition, the operator Q polarization states of gauge bosons and ghosts. On the one hand, the condition of BRST-invariance of the theory is more profound because gives one considerably wider possibilities for the choice of counterterms for the elimination of divergences and, on the other, transformation (38.85)–(38.89) is nonlinear, which results in a very nontrivial problem of determination of the corresponding transformations for Green functions. We recommend the reader the work by Weinberg ( [213], Chapters 15 and 17) in which the foundations of the BRSTsymmetry and its generalizations are presented in detail and the proof of renormalization of the most general non-Abelian gauge theory is given (see also [155]).

Chapter 39

Standard Models of Interactions

The most developed (electromagnetic) theory has been described in Chapter 37. In the present chapter, we briefly dwell on models constructed on the basis of the theory of gauge fields and their interaction with matter fields, i.e., fermion fields, corresponding to mass particles with spin s D 1=2: First of all, it refers to QCD that studies the interaction of hadrons within the framework of the theory of strong interactions and the model of electroweak interaction. A quantum theory of these models can be constructed on the basis of the formalism described in Chapter 38. The problem of elimination of ultraviolet divergences is worthy of special attention. Difficulties that arise in the application of the methods described above depend on the choice of a specific model, i.e., Lagrangian. The main Lagrangian used for the construction of models is the gauge-invariant Lagrangian (38.41)–(38.45), which describes the interaction of gauge massless fields Wa with matter fields : For specific models, there is a series of additional problems caused by a nonzero mass of some type of exchange bosons. The introduction of a mass term violates the gauge invariance and leads to additional difficulties in the process of elimination of divergences. In what follows, we consider problems of description of mass bosons, which, on the classical level, briefly have been discussed in Chapter 13.

39.1

Renormalization of Gauge Theories

In Section 38.4, we have considered the problem of modification of Lagrangian (38.41)–(38.45) for the construction of a covariant perturbation theory for the S matrix and Green functions. This modified Lagrangian is contained in the expression for action (38.69)–(38.77): Z e .x/ W AŒ ; W;  D dx W L eI; e DL e0 C L L e 0 D N .i   @  m/ L

(39.1)   1 ˛1   @ @ Wa C N a a ;  Wa g    2 ˛ (39.2)

e I D g N   T a W a C gf abc g  @ W a W b W c L    1 2 abc ade   b c d e  g f f g g W W W W C gf abc @ N a b Wc : 4

(39.3)

522

Chapter 39

Standard Models of Interactions

Contributions of Feynman diagrams are constructed with the use of three types of e between propagators that are Green functions of the operators in the expression for L a a N N the fields and , W and W , a and a and vertices of four types corresponding to each term of the interaction Lagrangian (39.3) (see Table 38.1 and relations (38.78) and (38.79)). According to relations (29.30) and (29.31), the maximum degrees of each vertex (for d D 4) !1max D !2max D !3max D !4max D 0:

(39.4)

Thus, the theory is renormalizable with the use of a finite number of counterterms defined by primitively-divergent diagrams and some additional relations. To determine the number and form of primitively-divergent diagrams, we rewrite relation (29.32) in terms of the number of fermion external lines L , the gluon external lines LW , and the external lines corresponding to ghost particles L : 3 !.G/ D 4  L  LW  L : 2

(39.5)

Relation (39.5) is written with regard for (39.4) for d D 4: Thus, divergent (primitively-divergent) diagrams are associated only with the following triples: .L ; LW ; L / D (0,0,0), (2,0,0), (0,2,0), (0,0,2), (2,1,0), (0,1,2), (0,3,0), (0,4,0), (0,0,4), (0,2,2). However, the number of these diagrams is greater because they can be constructed with the use of vertices of different types. Here, we present their form but do not perform rather complicated and awkward calculations of the corresponding integrals and selection of divergent terms. For the calculation of contributions of several specific diagrams, see [48, 49, 6, 208]. For the calculation of the renormalized contribution to the self-energy function of gauge bosons .0; 2; 0/, it is necessary to use the methods described in Chapter 29 and Section 37.4 for contributions of the diagrams depicted in Fig. 39.1 .!.G/ D 2/: Divergences of these diagrams can be compensated by counterterms of the form 1 1 .2/  ZW W W .g    @ @ /Wa W  Z˛.2/ W Wa @ @ Wa W; 2 2˛ .2/

.2/

.2/

(39.6)

.2/

where ZW D ZW;a C ZW;ı C ZW;b , and only the diagram in Fig. 39.1 (a) .2/

makes the contribution to Z : For the renormalization of Fermi fields and the corresponding masses of particles of matter, we consider contributions of the diagrams in Fig. 39.2 .!.G/ D 1/, .2; 0; 0/: The corresponding contribution to the counterterm of the Lagrangian has the form Z

.2/

N .i   @  m/

 ım Q .2/ N :

(39.7)

The fields of ghosts a .x/ must also be renormalized. The corresponding counterterm associated with contribution of the diagram in Fig. 39.3 .!.G/ D 2/, .0; 0; 2/ has the form .2/ (39.8) Z a a :

523

Section 39.1 Renormalization of Gauge Theories

a, μ

a, μ

b, ν

b, ν

(a)

(b) a, μ

b, ν

(c) Figure 39.1. Self-energy diagrams of gauge bosons. q

p

p j, α

p−q

k, β

Figure 39.2. Self-energy diagram of matter field.

Vertex diagrams are also divergent. We associate vertex diagrams with diagrams with three or four external lines. Divergent diagrams with three external lines are divided into three groups. The first group includes diagrams corresponding to the gluon vertex .0; 3; 0/, !.G/ D 1, which are depicted in Fig. 39.4. The contribution to the corresponding counterterm has the form .3/

Z1 gf abc g  W @ W a Wb Wc W; .3/

.3/

.3/

.3/

.3/

Z1 D Z1;a C Z1;b C Z1;c C Z1;d :

(39.9)

The second group includes diagrams corresponding to the gluon-ghost vertex (Fig. 39.5), .0; 1; 2/; .!.G/ D 1/: The corresponding counterterm has the form Z2 gf abc @ N a b Wc ; .3/

.3/

Z2 D Z2;a C Z2;b :

(39.10)

The third group includes diagrams corresponding to the quark-gluon vertex .!.G/ D 0/, (Fig. 39.6), (2,1,0). The counterterm has the form .3/ Z3 g N   T a Wa ;

.3/

Z3 D Z3;a C Z3;b :

(39.11)

524

Chapter 39

Standard Models of Interactions

q

p

p p−q

a

b

Figure 39.3. Self-energy diagram of ghost field. b, ν

a, μ

b, ν

c, ρ

a, μ

(a)

c, ρ

(b) b, ν

b, ν

a, μ

c, ρ

(c)

a, μ

c, ρ

(d)

Figure 39.4. Gluon vertex diagrams.

Finally, diagrams with four external lines with !.G/ D 0 are 4-order diagrams with respect to g: Diagrams belonging to a 4-gluon vertex are depicted in Fig. 39.7 (0,4,0). The corresponding counterterm has the form 1 .4/  Z4 g 2 f abc f ade g  g  W b Wc Wd W e : 4

(39.12)

Two diagrams that belong to a ghost-ghost vertex and a gluon-ghost 4-vertex (Fig. 39.8) (0,0,4) and (0,2,2) are also divergent .!.G/ D 0/: However, since ghost particles are not observed in actual interactions, there diagrams occur only in higher orders of perturbation theory and contribute to the counterterms ZW and Z . In the same way as in the case of QED, the vacuum diagrams (0,0,0) are not considered

525

Section 39.1 Renormalization of Gauge Theories c, μ

c, μ

a

b

a

b

(a)

(b)

Figure 39.5. Gluon-ghost vertex diagrams. c, μ

c, μ

α, j

β, k

α, j

(a)

β, k

(b)

Figure 39.6. Quark-gluon vertex diagrams.

because they can be eliminated by formal resummation and renormalization of a vacuum state. After resummation in all orders of perturbation theory, we obtain the final expression for counterterms in the form of (39.6)–(39.12) with constants .ZW  1/, Q .Z 1/, and .Zk 1/, k D 1; 4, and the complete Lagrangian .Z˛ 1/, .Z 1/, ı m, has the form 1  ZW Wa .g    @ @ /Wa 2

e C L e D Z N .i   @  m/ L

1 Z˛ Wa @ @ Wa C Z1 gf abc g  @ W a Wb Wc C Z N a a  2˛ C Z2 gf abc @ N a b Wc C Z3 g N   T a Wa 1  Z4 g 2 f abc f ade g  g  W b Wc Wd W e : (39.13) 4 To obtain Lagrangian (39.13) from the nonrenormalized Lagrangian e. e0 D L L

0

; .W 0 /a ;  0 I m0 ; 0 ; g0 /;

(39.14)

526

Chapter 39 b, μ

c, ν

a, λ

d, ρ

b, μ

c, ν

d, ρ

a, λ

(a)

Standard Models of Interactions

(b)

b, μ

c, ν

a, λ

d, ρ

(c)

b, μ

c, ν

b, μ

c, ν

a, λ

d, ρ

a, λ

d, ρ

(d)

(e)

Figure 39.7. 4-gluon vertex diagrams.

b

c

b, μ

c, ν

a

d

a

d

(a)

(b)

Figure 39.8. 4-ghost-ghost vertex diagram (a); 4-gluon-ghost vertex diagram (b).

it is necessary to set 0

DZ

1=2 1=2

;

 .W 0 / a D ZW Wa ;

(39.15) (39.16)

527

Section 39.1 Renormalization of Gauge Theories 1=2 N N 0 D Z ;

(39.17)

1=2

 0 D Z ;

(39.18)

1 1 1 0 D .1 C Z /ZW  ;

m0 D m C ım; 3=2

g0 D Z1 ZW g0 D g0 D g02 D

ım D Z

(39.19) 1

ı m; Q

g;

(39.20) (39.21)

1=2 Z2 Z1 ZW g; 1=2 Z3 Z 1 ZW g; 2 2 Z4 ZW g :

(39.22) (39.23) (39.24)

Equalities (39.21)–(39.24) must guarantee renormalization of the interaction constant g0 : Thus, to guarantee the gauge-invariant renormalization of the Lagrangian e 0 , it is necessary that the equalities L 3=2

Z1 ZW

1=2

D Z2 Z1 ZW D Z3 Z 1 ZW 1=2

1 D Z4 ZW 1=2

(39.25)

be true. These equalities decrease the number of independent constants because they yield ZW D Z12 Z41 ;

Z D Z1 Z2 Z41 ;

Z D Z1 Z3 Z41 :

(39.26)

However, equalities (39.25) are not obvious and, in fact, mean renormalizability of e : Sometimes, it is said that the gauge-invariant theory described by the Lagrangian L a system of counterterms has the gauge-invariant structure Z2 Z3 Z4 Z1 D D D : ZW Z Z Z1

(39.27)

Relations (39.27) are also called the Slavnov–Taylor identities [186, 196]. They guarantee universality of the renormalized interaction constant g: They can be verified by the direct calculation of contributions of the Feynman diagrams depicted in Figs. 39.1–39.7, i.e., in lower orders of perturbation theory (in a so-called onecomponent approximation). To this end, it is necessary to use the gauge-invariant intermediate regularization (e.g., with the use of the introduction of an arbitrary dimension (see Section 29.7) or the Pauli–Villars regularization (Section 29.1)) and to select divergent terms (after elimination of the intermediate regularization) in the constants Z: In the general case, i.e., beyond the framework of perturbation theory, analytic expressions for the constants Z are not obtained. However, it is possible to obtain a relation between these constants, i.e., in fact, to prove (39.27) using so-called generalized Ward–Takahashi identities.

528

Chapter 39

Standard Models of Interactions

In Section 38.3.2, we have introduced this relation in the case of an Abelian theory of QED, which is the Ward identity (38.39), (38.40). In the case of a non-Abelian gauge-invariant theory, the corresponding analogs are relations between Green functions, which, for the first time, were introduced in [34, 35, 34, 34] and later were independently obtained by Taylor in [196] and Slavnov in [186], which are called the Slavnov–Taylor identities or the generalized Ward–Takahashi relations. Here, we do not present these relations and recommend the reader, e.g., [49] , Chap, IV, Section 6. From the point of view of the practical use of the renormalization theory of gaugeinvariant non-Abelian theories, the scheme, which has been considered in detail in Sections 29.1–29.6 for the ' 4 interaction and in Sections 37.4.1 and 37.4.2 for QED, remains true. In conclusion, we present this scheme for interaction (39.1)–(39.3). The following three actions are necessary for the realization of this scheme: 1. Using the R-operation, we select divergences (first, introducing an intermediate regularization) in the following strongly connected parts of the corresponding Green functions: e ab (a) …  .k/ is the gluon self-energy operator (Fig. 39.1); e ab .k/ is the self-energy operator of ghosts (Fig. 39.3); (b) … e (c) †.p/ is the self-energy operator of quarks (Fig. 39.2); eabc (d) ƒ  .p; k; q/ is the 3-gluon vertex operator (Fig. 39.4); eabc (e) ƒ  .p; k; q/ is the gluon-ghost vertex operator (Fig. 39.5); k eaIj .p; k; q/ is the quark-gluon vertex operator (Fig. 39.6); (f) ƒ 

eabcd .p; k; q; l/ is the 4-gluon vertex operator (Fig. 39.7) (g) ƒ  e eabc (†.p/ and ƒ  .p; k; q/ are 4  4 matrices and the others are ordinary functions). By the choice of the point of subtraction, we define the renormalized operators er abcd e r /ab .…  .k/; : : :, .ƒ / .p; k; q/ by analogy with corresponding relations for QED (see relations (37.92)–(37.95)). Remark 39.1. In the same way as in the case of QED, the self-energy operator of gluons must satisfy the condition of transversality (37.66), i.e., have the form 2 2 e ab …  .k/ D ıab .g k  k k /.k /:

(39.28)

Equation 39.28 is a corollary of one of the Ward–Takahashi identities. An analoe r /ab gous relation must be true for the renormalized operator .…  .k/:

529

Section 39.1 Renormalization of Gauge Theories

Recall that the construction of the operators e r /ab er abcd .…  .k/; : : : ; .ƒ / .p; k; q/ means, in lower orders of perturbation theory, i.e., in the one-loop approximation, the simple subtraction of terms of the expansion of the corresponding contributions of Feynman diagrams depicted in Figs. 39.1–39.7 in the Taylor series and, in higher orders, the application of the R-operation. 2. Based on the renormalized operators er abcd e r /ab .…  .k/; : : : ; .ƒ / .p; k; q/; we construct the complete renormalized Green functions   k k ab k k i i ı ab r ab e i.D / .k/ D r 2 g  2  2 ı ; 2 Q .k /  k  i " k C i" k C i " k2 C i " Q r .k 2 / D k 2  r .k 2 /;

(39.29)

which corresponds to the gluon propagator in Feynman diagrams; 1 1 er S q .p/ D

; Q r .p/  i " i i m  pO C †

(39.30)

which correspond to the complete renormalized propagator of a quark or an antiquark of mass m; 1 er ıab S  .q/ab D ; r 2 i i ŒQ  .q /  q 2  i "

e r .q/ D ıab Q r .q 2 /; … ab

(39.31)

which corresponds to the complete renormalized propagator of a ghost particle; abc .3/ er /abc V .p; k; q/ C .ƒ .e  r /abc  .p; k; q/ D f  .p; k; q/;

(39.32)

.3/

where V is defined in (38.78) and relation (39.32) corresponds to the complete renormalized operator of the 3-gluon vertex; e .4/Iabcd .p; k; q; l/ C .ƒ er /abcd .p; k; q; l/; .e  r /abcd  .p; k; q; l/ D V  

(39.33)

e .4/ is defined in (38.79) and this relation corresponds to the complete renorwhere V malized operator of the 4-gluon vertex; 4 abc er /abc C .ƒ .e  r /abc  .p; k; q/ D .2/ ı.p C k  q/q f  .p; k; q/;

(39.34)

which corresponds to the complete gluon-ghost vertex, and, finally, er /a .p; k; q/; .e  r /a .p; k; q/ D .2/4 ı.p C k  q/  T a C .ƒ which corresponds to the complete renormalized quark-gluon vertex.

(39.35)

530

Chapter 39

Standard Models of Interactions

Like relation (37.92) in QED, the renormalized propagators and vertex operators are connected with nonrenormalized quantities by the multiplicative relations with the corresponding constants Z that satisfy Equations (39.27). 3. To write contributions to amplitudes of actual physical processes, it is necessary to take into account skeleton diagrams in which each internal line and each vertex are associated with complete renormalized propagators and vertices (39.29)–(39.35), and the order of perturbation theory is defined by the renormalized interaction constant 3=2

g D Z12 Z4

39.2

g0 :

(39.36)

On the Masses of Gluons and Spontaneous Symmetry Breakdown

39.2.1 Connection of the Radius of Interaction and the Mass of Exchange Bosons The constructed theory of gauge fields has rather considerable difficulties in its application to specific models of interactions of elementary particles. The electromagnetic interaction can serve as a standard of this application. However, the weak interaction, which is close to the electromagnetic interaction, already cannot be described within the framework of the theory of non-Abelian gauge fields based on the formalism described above. The reason for these difficulties is a very small radius of weak interactions as compared with radius of the electromagnetic interaction. Within the framework of the nonrelativistic approach, the interaction between two particles can be described with the use of the Schrödinger equation  „2 @ .1 C 2 / C V .jx1  x2 j/ ‰.t I x1 ; x2 /; (39.37) i „ ‰.t I x1 ; x2 / D  @t 2m where V .jx1  x2 j/ is the interaction potential of two particles. The short range of interaction means a fast decrease in the potential as jx1  x2 j ! 1: In turns out that the behavior of the potential can be established in lower orders of perturbation theory. If the interaction between particles of mass m occurs with the use of a certain exchange particles of mass , then V .jx1  x2 j/

e jx1 x2 j : jx1  x2 j

(39.38)

In quantum electrodynamics, an exchange particle is a photon with  D 0: Therefore, the interaction is long-range (Coulomb) and decreases according to the Coulomb rule.

Section 39.2 On the Masses of Gluons and Spontaneous Symmetry Breakdown p1

q

531

p1

p2

p2

Figure 39.9. Scattering diagram of second order.

To verify (39.38), we consider, e.g., the interaction of particles within the framework of the local Yukawa model (13.35). Thus, the main assumption is the condition according to which the scattering amplitude determined with the use of the Schrödinger equation (39.37) coincides with scattering amplitude of the S-matrix of the Yukawa model in the nonrelativistic range of energies. Let particles with momenta p1 and p01 before scattering have the momenta p2 and p02 after scattering. Then, e.g., in the center-of-mass system .p1 D p01 /, scattering from the state ‰k1 with relative momentum k1 D 12 .p1  p01 / into the state ‰k2 with relative momentum k2 D 12 .p2  p02 / is defined by the matrix element of the S -matrix. It is convenient to express this element in terms of the T -operator (see, e.g., [197], Chapter 8, Section 4) .

k0 ; S ‰ k1 /

D ı.k1  k2 /  2 i ı.E2  E1 /.‰k2 ; T ‰k1 /; 2

p0

p2

(39.39)

k2

where Ei .k/ D 12 mi C 12 mi D mi and the matrix element of the T -operator satisfies the well-known Lippmann–Schwinger equation Z .‰k ; T ‰k1 / ; (39.40) .‰k2 ; T ‰k1 / D .‰k2 ; V ‰k1 /  d k.‰k2 ; V ‰k / E.k/  E1 .k1 / where V is the interaction operator that stands for the potential V .jx1  x2 j/ in the Schrödinger equation and the interaction Lagrangian (13.35) in the theory of S-matrix. We consider Equation (39.40) in the first order of perturbation theory in .‰k2 V ‰k /, i.e., the first term of the Born series (the Born approximation). Then the integral term can be eliminated. We have .‰k2 ; V ‰k1 /  .‰k2 ; T ‰k1 /:

(39.41)

For the determination of the matrix element of the S -matrix (39.39), we restrict ourselves to the second order in the interaction constant g: The contribution to the righthand side of (39.41) is given by the diagram in Fig. 39.9. Using rules (37.33)–(37.35) for external fermion lines and taking into account that a wavy line is associated with propagator of an exchange boson, which, in the case of interaction (13.35), is a scalar particle, we obtain the analytic expression Z .2/4 ı.p1 C q1  p2 /.2/4 ı.p10  p20  q/ g 2 vN ;C .p2 /v ; .p1 / dq .2/4 i.q 2  2  i "/ 0

0

 vN  ;C .p02 /v  ; .p01 /:

(39.42)

532

Chapter 39

Standard Models of Interactions

In the nonrelativistic approximation .jpi j m/, ! q p2i 0 2 2 D m.1 C Ei .pi //: pi D pi C m  m 1 C 2m

(39.43)

Taking into account the explicit form of spinors (see Section 6.3.2), we easily reduce relation (39.42) in the center-of-mass system and in the approximation pi0  m to the form i.2/4 g 2 1 ı.k2  k1 /ı.E2  E1 / : (39.44) m .k2  k1 /2 C 2 Comparing (39.39), (39.41) with (39.44), we obtain .‰k2 ; V ‰k1 /  .2/3

g 2 ı.k2  k1 / : m .k2  k1 /2 C 2

(39.45)

We obtain the following relation for the effective interaction potential between particles in the coordinate space: V .jx1  x2 j/ 

g 2 e jx1 x2 j ; m jx1  x2 j

(39.46)

which agrees with notions of the character of interaction.

39.2.2 Are Theories with Nonzero Mass of Exchange Bosons Renormalizable? Thus, for the construction of models described by gauge fields with short-range interaction, it is necessary to assume that exchange bosons (gluons) have masses. This assumption is confirmed by experimental data for processes with weak interactions. This requires the introduction of a mass term of the form (13.1) into the complete interaction Lagrangian, which violates the gauge invariance. However, the violation of gauge invariance is not the worst feature of a model whose Lagrangian contains the mass term (13.1). A more serious problem is nonrenormalizability of the theory. This nonrenormalizability is caused by the replacement of the propagator  1 ıab k k 1 e cIab D  .kI 0/ D g  .1  ˛/ (39.47)  i .2/4 i .k 2 C i "/ k2 in perturbation theory by the propagator for vector mass bosons (see (23.14)) 1 e cIab ıab g  k k =m2W : D  .kI mW / D i .2/4 i m2W  k 2  i "

(39.48)

Thus, the degree of the polynomial of the numerator of the propagator rl D 2: Then the maximum index of vertex (see (29.30) and (29.31)) !imax > 0; which leads to nonrenormalizability of the theory (see Section 29.2).

(39.49)

Section 39.2 On the Masses of Gluons and Spontaneous Symmetry Breakdown

533

Of course, the term proportional to k k =m2W is contained in the longitudinal part of the propagator and possibly (!?) does not lead to serious difficulties (as in QED). However, this statement is not obvious and, perhaps, not true.

39.2.3 Spontaneous Breakdown of the U.1/-Symmetry On the classical level, the mechanism of spontaneous breakdown of a gauge symmetry has been briefly described in Sections 13.1–13.3. We recommend the reader to reread these chapters. Here, we do not repeat main ideas of this mechanism and only make some corrections concerning quantum theory according to which the fields , H , G, and W are operator-valued functionals. According to the main idea, the universe is filled with a certain scalar field .x/ with vacuum average .ˆ0 ; .x/ˆ0 / D u0 ¤ 0:

(39.50)

By virtue of the condition of translation invariance of vacuum, u0 D const: Using relation (28.7), we easily rewrite the quantity u0 in the form u0 D

1 . 0 ; T .0 .x/S / 0 /; S0

(39.51)

where 0 is the field .x/ in the interaction representation. Substituting the S-operator of theory (13.2) in the form of a perturbation series into (39.51), we establish that the corresponding series for u0 u0 D

1 X

.n/

n u0

(39.52)

nD0 .n/

is identically equal to zero because u0  0 for all n: Thus, the effect of spontaneous symmetry breakdown is a nonperturbative effect. It leads to the appearance of the mass terms (13.18) and (13.33) in Lagrangians (13.17) and (13.31). Lagrangians (13.17) and (13.31) lost the explicit gauge invariance in terms of the new fields H.x/ and W .x/: However, since the initial Lagrangian (13.12) has this invariance, the theory must remain gauge-invariant, which became a reason for the term spontaneous breakdown of invariance. Perhaps, it is more correct to say about the lost of the explicit gauge invariance and that the gauge invariance of Lagrangians (13.17) and (13.31) is latent. Let us return to the problem of renormalizability of the theory. First, we consider the case of U.1/ symmetry (see Section 13.2). The theory corresponding to Lagrangian (13.12) is renormalizable because this is, in fact, scalar electrodynamics. The addition of the Lagrangian of matter with replacement of ordinary derivatives by covariant ones, i.e., the Lagrangian LM .x/ D N .x/.iD    m/ .x//;

(39.53)

534

Chapter 39

Standard Models of Interactions

to Lagrangian (13.12), does not break renormalizability of the theory. However, after transformations (13.13)–(13.16), renormalizability of the theory with Lagrangian (13.17) is not obvious by virtue of the arguments presented above. Recall that, for the construction of the photon propagator for theory (13.12) [or (13.12), (39.53)], we fix a gauge by adding term (37.22) to the complete Lagrangian. To obtain the corresponding propagator in the case with broken symmetry, we use the term that fixes a gauge proposed by ’t Hooft in [198, 199, 201] L .x/ D 

1 .@ B  .x/  ˛gu0 G.x//2 ; 2˛

(39.54)

where u0 D 2 = and G is the field of a Goldstone boson (see (13.7)). Then Lagrangian (13.12), together with term that fixes gauge (39.54) and is expressed in terms of the fields B contained in the definitions of the covariant derivatives (13.10) and the field '.x/ D H.x/ and G.x/ contained in definition (13.7) has the final form 1 L .x/ C L .x/ D ŒW @ H.x/@ H.x/ W 2u20 W H.x/2 W 2 1  2 G.x/2 W C ŒW @ G.x/@ G.x/ W  W ˛mB 2  1 ˛1   2   2  W .@ B .x//.@ B .x//  mB B .x/B .x/  .@ B .x// W 2 ˛

$

C g W B  .x/.H.x/ @  G.x// W C g 2 u0 W B .x/B  .x/H.x/ W 1 C g 2 W B .x/B  .x/.H.x/2 C G.x/2 / W  u0 W H.x/.H.x/2 C G.x/2 / W 2 1 4 C gu0 @ .B  .x/G.x//:   W .H.x/2 C G.x/2 /2 W C (39.55) 4 4 The first three terms define the character of Feynman propagators. The first term corresponds to the Higgs field H.x/: The self-action of this field generates a particle of mass mH D .2u20 /1=2 : The second term corresponds to a Goldstone boson of mass 2 1=2 .˛mB / appeared due to the gauge term L .x/: Since this boson corresponds to none actual physical particle, it can be eliminated from the theory after calculations by setting ˛ D C1: Finally, the third term corresponds to the gauge boson B with 2 D g 2 .u20 C v02 /: In a similar way as in Section 37.1.3, we easily determine the mB propagator for the field B .x/: In the momentum space, this propagator has the form (Problem 39.1) ! p p i   e  .p/ D g C .˛  1/ 2 : (39.56) G 2 2 mB  p2  i " p  ˛mB

Section 39.2 On the Masses of Gluons and Spontaneous Symmetry Breakdown

535

e  .p/ for Thus, according to (39.56), for ˛ < 1, the behavior of the propagator G 2 large p guarantees renormalizability of the theory. Note the following: considering the field B .x/ as an electromagnetic field, it is necessary to consider mB as a mass of photon generated by interaction of this field with Higgs field.

39.2.4 Spontaneous Breakdown of the Local SU.N /-Symmetry In the previous section, for scalar electrodynamics, we have shown that the spontaneous breakdown of a vacuum symmetry leads to a mass of a gauge boson absorbing a Goldstone boson. Furthermore, this procedure preserves renormalizability of the theory. We briefly give arguments for the preservation of this mechanism in a more general case for a theory with non-Abelian gauge invariance. On the classical level, this has been performed in Section 13.3 for a local symmetry. In the general case, we assume that the Higgs field .x/ consists of N -complex scalar fields 1 0 1 .x/ C B (39.57) .x/ D @ ::: A : N .x/ The model Hamiltonian of the self-interacting Higgs field has form (13.20). The requirement of local SU.N /-invariance leads to Lagrangian (13.23) with b  D 1  @ C ig T a Wa ; D

a D 1; N 2  1

(39.58)

and a D @ Wa  @ Wa  gf abc Wb Wc ; F

(39.59)

where T a are generators of the corresponding Lie algebra and f abc are its structure constants (see (2.20)). By virtue of the breakdown of the vacuum symmetry ˆ0 , the average (39.60) .ˆ0 ; .x/ˆ0 / D 0 ¤ 0: Then it is necessary to consider the perturbation of this field '.x/ D .x/  0 :

(39.61)

The value 0 corresponds to a minimum of the potential u./, which, in the multicomponent case, is degenerate and forms a hypersurface of the form (13.21). As in the case of S U.2/ (see relations (13.25)–(13.29)), it is convenient to take 0 in the form 0 1 0 :: C 1 B 2 : 2 B > 0; (39.62) D  u 0 D p @ C 0  2 0A u0

536

Chapter 39

Standard Models of Interactions

and to rewrite the field .x/ in the form of the perturbation 1 0 0 : a B :: C 1 i G a .x/T u0 C: B .x/ D p e A @ 0 2 u0 C H.x/

(39.63)

The choice of the vacuum state in the form (39.62) enables us to draw an important conclusion. The subgroup SU.N 1/ of the group S U.N / has .N 1/2 1 generators that act on the first .N  1/ components of vector (39.57). Thus, the action of these generators on the vacuum vector 0 gives zero. Hence, in the construction of the perturbed field (39.63), in the exponent, we can take into account only the action of n D .N 2 1/Œ.N 1/2 1 D 2N 1 operators that do not transform the vector 0 into zero. The substitution of vector (39.63) into the Lagrangian of Higgs field (13.2) leads to the appearance of the mass mH D 2u20 only for the fields H.x/, whereas the other 2N  1 fields G a .x/ remain massless. Note that these not completely rigorous arguments are a corollary of the well-known Goldstone theorem [76] (see also [208], Part 13). In what follows, as in the previous cases, we use the gauge invariance of Laa in the form (39.58) and (39.59) b  and F grangian (13.23) with corresponding D and choose a fixed gauge in (38.46)–(38.48) with !a .x/ D 

1 Ga .x/; u0 g

a D 1; 2N  1:

(39.64)

Then fields obtained with the use of the gauge transformations (38.46)–(38.48) with !a .x/ (39.64) have the form 1 0 0 :: C 1 B : C ! .x/ D p B (39.65) A @ 0 2 u0 C H.x/ and

i S! .x/@ S!1 : (39.66) g To determine the spectrum of masses formed due to the symmetry breakdown, it is necessary to rewrite Lagrangian (13.23) in terms of the fields-perturbations H.x/ and Ga : This yields only 2N  1 Goldstone bosons. Therefore, the number of gauge fields Wa that absorb these Goldstone bosons and get the corresponding masses is equal to 2N  1: The other N.N  2/ gauge bosons are massless. A specific expression for the masses mW a D ma can be obtained after the representation of the Lagrangian L .! ; W ! / in terms of the fields H.x/, Ga .x/, and  Wa : The coefficients of H.x/2 , Ga .x/2 , and Wa Wa (summation over a is absent) give the corresponding masses (see the arguments in [208], Part 13). W! D .W ! /a T a D S! .x/Wa T a S!1 C

Section 39.3 Models of Interactions of Elementary Particles

537

Gauge (39.64) is very convenient for the description of the mechanism of absorption of Goldstone bosons by gauge fields that get the corresponding masses. In the same way as in the case of U.1/-symmetry, it leads to the propagator corresponding to the vector mass field (39.48), which results in “nonrenormalizability of theory.” Of course, this “nonrenormalizability” is not actual and, finally, all divergences must cancel. However, it is impossible to follow this process. As in the previous section, we can select the counterterm Lgf .x/ that fixes a gauge. In this case, an analog of (38.68) is the Lagrangian Lgf .x/ D 

1  a .@ W  i ˛g'C T a a /.@ Wa  i ˛g'C T a a /: 2˛

(39.67)

This counterterm enables one to define the Feynman propagator by relation (39.56) for gauge bosons that get a mass. However, in this case, in the quantization of a gauge field, it is necessary to change the corresponding Faddeev–Popov ghost Lagrangian. This Lagrangian has the form (see [6], Section 13.6) LFP D .@ N a /.@ a / C gf abc .@ N a /b Wc  ˛ N a .M 2 /ab b  ˛g 2 N a b 'T b T a 0 ; W

(39.68)

2 /ab D g 2 C T a T b  is the mass matrix. where .MW 0 0

39.3

Models of Interactions of Elementary Particles

The theory described in the previous chapters of this part is a mathematical foundation for the construction of models aimed at the calculation of characteristics of specific physical processes and their comparison with experimental data. The construction of models is a very complicated and long-term process that requires a lot of experimental data, analysis of theoretical and phenomenological approaches and their generalization in one theoretical model. Each model is based on the Lagrangian that defines the dynamics of all possible processes within the framework of the chosen type of interaction. In Chapter 14, we have briefly described the form of Lagrangians for strong, weak, and electroweak interactions and a possible version of their Grand unification. In this chapter, we briefly dwell on main problems caused by quantum theory, namely, the specific feature for each model (the general problem has been considered above). We also propose reviews of these problems.

39.3.1 Strong Interaction. Model of QCD It is conventional to connect the advances in QCD (a mysteriously exact agreement of the results of theoretical calculations with experimental data) with a rather small

538

Chapter 39

Standard Models of Interactions

value of the fine structure constant E2 1 e2  ; (39.69) D E1 4„c 137 where E2 is the energy of two electrons for a unit distance between them and E1 is the rest energy of an electron. This constant is dimensionless and independent of the choice of the system of units. In the range of low energies, ˛ is a real constant that can be measured in experiments (see, e.g., [87], Chapter 1, Section 5). The quantity ˛ is a parameter of expansion of scattering amplitudes in a perturbation series. Since ˛ 1, we can assume that, after elimination of divergences and a certain magic cancellation of contributions of diagrams (since their number increases in QED with order of perturbation theory as .nŠ/3=2 ), the series is convergent or, at least, asymptotic. Then a small value of ˛ guarantees this excellent agreement with experiment. A similar experiment of measurement of an analogous constant (or quantity) in the interaction of protons and -mesons shows that ˛H  1: This result immediately destroys every hope of the successful use of the method of perturbation theory for the calculation of processes within the framework of strong interactions. In this connection, an idea of a complicated structure of a proton and -meson (and, hence, the other particles) was suggested. Possibly, this idea is based on an analogy with chemical bonds in molecules. Indeed, atoms of certain matters being electrically neutral rather weakly interact with each other until their electron orbits overlap. In this case, atoms unite in one molecule and the bond becomes very strong. Therefore, the character of interaction between quarks by exchange of gluons must be defined by a certain quantity ˛s , which can be defined by the analogous relation g2 ; (39.70) ˛s D 4„c where g is the “color” charge of quark. The notion of the “color” interaction appeared after the experimental detection of the  C p-resonance CC with electric charge QCC D 2e: This means that it must consist of three u-quarks (uuu). However, u-quarks are Fermi particles, which contradicts the Pauli principle. The way out was found by assuming that each quark has an additional quantum number called a “color” charge or simply a “color.” Here, we do not discuss in detail the “color” structure of hadrons and recommend the first chapter of [87] where a clear substantiation of a small value of the quantity ˛s for short distances between quarks and its increase with increase in this distance are given. This phenomenon is called an asymptotic freedom, which means a decrease in the interaction with an asymptotic approach of quarks. The asymptotic freedom was discovered by Gross and Wilczeck in [81] and Politzer in [162] in 1973. Their calculations explain a rather amazing experiment in 1968 (see, e.g., [19]) on deeply inelastic electron-nucleon scattering that reveals the effect of a decrease in strong interactions for high energies. The asymptotic freedom can be explained by a so-called phenomenon of “confinement” (impossibility of escape for a “color” particle). For the present stage of power ˛D

Section 39.3 Models of Interactions of Elementary Particles

539

possibilities of accelerators, free quarks and gluons are not observed. However, several indirect experiments unambiguously indicate their existence. It is also of interest that gluons are “color” and, hence, unlike neutral photons in QED, interact with each other, which, in fact, agrees with free equations of Yang–Mills fields (12.18) containing a term responsible for the interaction of gluons of different types. Thus, for the calculation of physical processes within the framework of strong coupling, a base Lagrangian is the gauge-invariant Lagrangian (14.3). In addition, in quantum theory, it a with the use of counterterm (39.67) is necessary to fix a gauge of the field Wa  G and, to guarantee the unitary property of the theory, to add the Faddeev–Popov Lagrangian of “ghosts” (39.68). Finally, to eliminate ultraviolet divergences in the calculation in perturbation theory, it is necessary to introduce the counterterm Lren , i.e., the same Lagrangian with corresponding renormalization constants. The complete Lagrangian of QCD has the final form e QCD D LQCD C Lgf C LFP C Lren : (39.71) L For some applications of QCD to the description of specific physical processes, see [208], Part III, [87], Chapters 10–11. Remark 39.2. Lagrangian (39.71) does not include LHigg because the interaction with scalar field introduces a positive constant in the calculation of an effective constant of strong interaction and can violate the asymptotic freedom. Thus, gauge bosons corresponding to processes of strong interaction are taken massless. This agrees with theory of spontaneous breakdown of the S U.3/-symmetry (see the remark at the end of Section 13.3) for which three gauge bosons are massless and five gauge bosons that absorbed Goldstone bosons are mass.

39.3.2 Weak and Electroweak Interactions Experiments of ˇ-decay of nuclei give the phenomenological coupling constant (see (14.9)) 105 (39.72) G 2 ; mN where, for convenience, it is normalized to the square of the nucleon mass because the dimensions of ŒG D .mass/2 : Thus, a small value of G gives a hope for a well convergence of perturbation series. By virtue of the arguments presented below (see also [208], Chapter 10, Section 10.3), the phenomenological constant G is connected with constant in the S U.2/-invariant Lagrangian (14.16) by relation (14.12) and is small due to a large mass of the gauge boson W responsible for the weak interaction. It is easy to see that the phenomenological Hamiltonian (14.9) corresponds to the 4-fermion interaction. This means that, in quantum theory, the structure of the corresponding Feynman diagrams has vertices with four incoming fermion lines with rc D 1 (see Section 29.2). Thus, !imax D 2 and

540

Chapter 39

Standard Models of Interactions

the theory constructed on the basis of Lagrangian (14.9) is nonrenormalizable. At the end of Section 14.2, we have considered the Lagrangian of weak interactions (14.16). This model is already a renormalizable theory and ultraviolet divergences are eliminated with the use of methods presented in previous chapters (see Section 39.1). In conclusion of consideration of weak interactions, note that the phenomenological Glashow–Iliopoulos–Maiani Lagrangian (14.9) appears in a more consequent scheme of gauge fields as a certain approximation for small values of the boson momentum. Indeed, we rewrite the complete Lagrangian in the form 1 2 a  W Wa C LW Iint : Lw D L0 .l; qj¹@ º/ C LYM C MW 2

(39.73)

Then, in terms of generatrices of functionals for Green functions rewritten in the form of the path integral (38.67), we obtain the relation Z  a N GŒ; ; N j  D GŒ01 D N D DW e i.N C Cja W /  ei

R

L0 .l;qj¹@ º/dx 2i

R

2 dxWa Œg  .MW /C@ @ Wa C ig 2

R



Ja .x/Wa .x/dx

; (39.74)

where D N D is understood as integration with respect to all fields of quarks and leptons. Of course, it is worth noting that the self-interaction of the fields Wa is neglected and the mass term is introduced as a result of the interaction with Higgs field. It is easy to verify that the integral in (39.74) with respect to the variables Wa is a Gauss integral and can be integrated with respect to DW: We get Z R N GŒ; ; N j  D GŒ01 D N D e i L0 .l;qj¹@ º/Ci.N C /CiAphen ŒJ Cj ; (39.75) where Aphen ŒJ C j  D

 Z g2 2 1 c dx dy Ja .x/ C ja D .x  yI MW / 8 g i  2  Ja .y/ C ja .y/ ; (39.76) g

and c D .x

1  yI MW / D .2/4

Z dk

2 / .g  k k =MW 2 MW  k2

e ik.xy/ :

2 Assuming that the integration is performed only for k 2 MW , we get c .x  yI MW / D

1 ı.x  y/: 2 MW

(39.77)

Section 39.3 Models of Interactions of Elementary Particles

541

Under condition (14.12), we obtain Z Aphen ŒJ  D i .phen/

.phen/

dxLW;int .x/;

(39.78)

where LW;int .x/ is identical to (14.9). For the construction of a unified electroweak theory, it is necessary to take into account a series of important aspects, which have been briefly described in Section 14.3. Thus, the classical Lagrangian of electroweak interaction takes the form (14.18). It corresponds to a renormalizable theory and can be analyzed within the framework of the approach given in Section 39.1. For this analysis, see [213], Section 21.3, [6], Section 14.1.

Chapter 40

Problems to Part VII

Problem 37.1.

Write the complete Hamiltonian of QED in terms of canonical variables.

Problem 37.2.

Prove that the fields .x/, N .x/, and A .x/ that are defined by relations (37.19), (37.11), and (37.10) and satisfy the equal-time commutation relations (19.1) and (20.48) are solutions of Equations (37.4).

Problem 37.3.

Calculate the determinant of the matrix e 0 D k 2 ı  k g k : …

Problem 37.4.

Calculate the action of the operations N ˙ .x/ and  ˙ .x/ that transform the coefficient functions of FOW of the operator S into the coefficient functions of FOW W N .x/S W, W N .x/S W, W .x/S W, W

.x/S W :

Problem 37.5.

Deduce the commutation relations (37.49) and (37.50).

Problem 37.6.

Prove the Furry theorem for contributions written in the momentum space.

Problem 37.7.

Prove invariance of the S-matrix in the form (37.26) under the gauge transformations (37.61).

Problem 37.8.

Deduce Equation (37.68) using the field equations 37.4.

Problem 37.9.

Deduce the Schwinger equations corresponding to the second and third equations of system (37.4).

Problem 37.10.

Deduce the Dyson equation for the 2-point electron-positron Green functions.

Problem 37.11.

Prove that the 2-point photon Green function satisfies representation (37.76)–(31.78).

Problem 37.12.

er and Q r defined by relations (37.93) Prove that the functions † and (37.94) do not contain divergences.

543

Chapter 40 Problems to Part VII

Problem 37.13.

Prove that the tensor, pseudovector, and axial-vector parts of the function w.x/ (37.108) turn identically into zero.

Problem 37.14.

Write a representation for the 2-point Green function G2;0 using representation (37.111), (37.112) for the Wightman function W2;0 :

Problem 38.1.

Prove the relation R R Z i dxWŒ 14 F .x/F  .x/W C dxA .x/j  .x/ 1 Q  A/ e M M DAı.r K R

D e

M

dx

R

M

c;el,Coul

dyj  .x/ 1i D

.xy/j  .y/

;

where K is chosen from the condition that, for j D 0, the left-hand side of the equality becomes equal to one. Problem 38.2.

Determine the propagator of photon line (38.28) in the Coulomb gauge by the method of path integration.

Problem 38.3.

Using (38.26), show that functional (38.29) is independent of A:

Problem 38.4.

Determine the propagator of photon line (37.25) in the ˛-gauge by the method of path integration.

Problem 38.5.

Determine elements of the matrix operator M'i , i D 1; 2; 3; 4, for the Coulomb, Lorentz, temporal, and axial gauges, i.e., prove relations (38.59)–(32.62).

Problem 38.6.

Find an expression for the 2-point Green function of W -bosons in the ˛-gauge.

Problem 38.7.

Write an expression for the complete action corresponding to the Yang–Mills fields and matter fields with regard for the Faddeev– Popov field of “ghosts.”

Problem 38.8.

Establish the form of the Lagrangian after the BRST-transformation of fields (38.85)–(38.90).

Problem 39.1.

Find an expression for the 2-point Green function of the field B on the basis of Lagrangian (39.55).

Appendix

Hints for the Solution of Problems

Part I Problem 1.1.

Apply the Schwarz inequality to the expression x  y and use condition 2.

Problem 1.2.

Set x 0 1 D 0 and

Problem 1.3.

For the determination of Mk0 , see Problem 1.2 (see also [181], Part 2).

Problem 1.4.

Use the law of transformation of spinors of rank 2 (1.38), decomposition (1.39), and relation (1.29).

Problem 2.1.

Use the Jacobi identity (2.24) and relation (2.20).

Problem 2.2.

The reality of the matrices S! .x/ follows from the expansion of the exponential in (2.19) in a series and relation (2.27). The orthogonality follows from the Hermitian property of T a :

Problem 2.3.

Verify the group property of transformations (2.28).

Problem 2.4.

Verify the group property of transformations (2.30) and (2.31).

x1 x0

D

v c

D ˇ in (1.10) and use Equations (1.12).

Part II Problem 4.1.

The first statement follows from (4.7), (4.6), and (4.8), and the second follows from (4.8) (see also [4], Chapter 15).

Problem 5.1.

Use the relation ı.f .x// D

X k

1 jf

0 .x

k /j

ı.x  xk /

(see [206], Section 1.9). Problem 6.3.

See the determination of P 0 for a scalar field (relation (5.59)).

Problem 6.4.

See the determination of P 0 for a scalar field (relation (5.59)).

Appendix Hints for the Solution of Problems

Problem 6.5.

See the proof of relation (5.42).

Problem 7.1.

See the determination of P 0 for a scalar field (relation (5.59)).

Problem 8.1.

Use definition (8.4) and relation (8.15).

545

Part III Problem 11.1.

First, choose 1 .x/ so that, for the new field B0 .x/, the condition 00 B00 .x/  0 is satisfied and then 2 .x/ so that B0 .x/  0 and 00 @k Bk .x/  0:

Problem 12.1.

Use the explicit form of the Pauli matrices and the definition of scalar and vector products in R3 :

Problem 12.3.

Use definition (12.1) and prove the Jacobi identity for the Pauli matrices.

Part IV Problem 20.3.

Use reasoning analogous to that applied in the case of a scalar field (Section 17.4.1).

Problem 20.4.

See Section 4.1.4.

Problem 21.1.

Use relations (21.5) and (21.13) and the property C  0 D  0 C (see (6.99)).

Problem 21.3.

Transform condition (6.148) and take into account the properties of the matrices   and C (see (6.99)).

Problem 21.4.

Use relations (21.5), (21.6) and (21.12), (21.13).

Problem 23.1.

See the determination of D ret .x/ in [26], Section 14.1.

Problem 23.2.

See the hint for Problem 20.1.

Problem 23.4.

Use Theorem 20.2 and the properties of vacuum averages of the normal product of operators of free fields.

Part V Problem 25.1.

See relations (25.12)–(25.15).

546

Appendix Hints for the Solution of Problems

Problem 25.2.

Use the representation of a field in the form of the Fourier transform (18.2) and relation (5.58).

Problem 26.2.

(a) Use condition (26.56), representation (26.43), and condition (17.13) for a scalar field; (b) write a representation of the form (26.43) and perform the same operations as for interaction (a).

Problem 26.3.

See the proof of Proposition 26.4.

Problem 27.1.

See the derivation of equations of the resolvent type (Section 27.2).

Problem 27.2.

Proceed by analogy with a scalar field (see Section 27.1).

Problem 27.3.

Follow Sections 27.2 and 27.3, taking into account the statistic and Problem 3.

Problem 28.1.

Use the representation of Green functions, relation (28.8) with Lagrangian (25.3), and Theorem 23.8, expanding the exponential in a formal series.

Problem 29.7.

Use relation (29.29), taking into account that m D m1 C    C mk and n D n1 C    C nk , where mi and ni are the corresponding values of the block Gi :

Problem 29.8.

Prove Lemma 29.9 (Section 29.4). See [26], Section 18.6.

Problem 30.5.

Use relation (30.31) with f D

m X

˛i fi ; g D

iD1

n X

ˇj gj ;

j D1

differentiate with respect to the corresponding ˛i and ˇj , and equate them to zero. Problem 30.6.

Rewrite the left-hand side of (30.76) with the use of (30.14) and the right-hand side using the generalized Wick theorem.

Problem 30.7.

Expand the exponential in a series and take into account properties (30.81), (30.83), and (30.84) and the skew symmetry of the matrix B:

Part VI Problem 32.1.

Use relations (32.2) and (32.1).

Appendix Hints for the Solution of Problems

547

Problem 32.2.

Use the expressions for Fourier transforms of generalized functions (see [206], Chapter II).

Problem 32.3.

See the proof in [23], Chapter 3, Section 2.

Problem 32.4.

See the proof in [23], Chapter 3, Section 2.

Problem 32.5.

See an analogous proof for the Fock space (Chapter 22).

Problem 33.2.

See the proof in [23], Chapter 4, Section 1.

Problem 33.3.

See the proof in [23], Chapter 4, Section 1.

Problem 33.4.

Use the expression for the S-operator in the form of T -exponential and determine the expression for the current in lower orders of perturbation theory (see also [23], Chapter 4, Subsection 3.3).

Problem 33.5.

Equate the formal expressions of perturbation series on the righthand side and the left-hand side of relation (33.23).

Problem 34.1.

Proceed in the same way as for the Fock space (Chapter 22).

Problem 34.2.

See the proof of relation (22.23).

Problem 34.3.

Use the definition of matrices (34.17) and properties (34.16).

Part VII Problem 37.1.

See [212], Section 8.3.

Problem 37.2.

Proceed by analogy with interaction (25.1) (see Section 25.1).

Problem 37.6.

See [2], Section 17.4.

Problem 37.11.

See [2], Section 21.2.

Problem 38.1.

First, prove the relation Z   1 dx W F .x/F  .x/ WD  4 M4 Z 1 D dx W ŒA.x/A.x/ C A0 .x/.x /A0 .x/ W : 2 M4

Then use the representation of ı-function (30.19) and relations of the form (30.7).

548 Problem 38.8.

Appendix Hints for the Solution of Problems

Use property (38.80) for the Grassmann variables  and  and properties of the structure constants f abc :

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Index

Abelian group 23 Action in classical mechanics 40 in field theory 52 Adiabatic limit 314 switching of interaction 296, 314 Alpha gauge 238, 471, 482, 483, 511, 515 matrices 85, 444 Ampére law 135 Analytic continuation of Feynman amplitudes 440 of Schwinger equation 480, 481 of Wightman functions 453–455 Angular momentum orbital 58 spin 58 Anomalous magnetic moment 120 Anti-time-ordered product 446, 449 Anticommutation relations of field operators 223, 262, 264, 265, 267 Anticommutator viii Antiparticle 20, 24 Asymptotic completeness 300, 425, 431, 434, 436 conditions 299 fields 431, 434 freedom 538 state 299, 300, 302, 431 Axioms Bogoliubov–Medvedev–Polivanov (BMP) 435 LSZ 433 of quantization 201, 204, 205, 207, 212 Osterwalder–Schrader (OS) 455 Wightman 423 Bargman–Hall–Wightman theorem 454

Baryons 26, 28 Basis in the Fock space 271–273 Beta-decay 123, 185 Bianchi identities 169 Big Bang 182 Bogoliubov–Parasiuk operation (Roperation) 368 Bohr magneton 120 Born approximation 531 Bose–Einstein statistics 208 Bose-field 208 Bosons 208 BRST-transformation 516–519 Canonical commutation relations 208, 218, 220, 222, 229, 238 transformations 41 variables 41, 60 Causality axiom 207 principle 207, 312 Charge and vector of current 59 for the spinor field 110 of complex scalar field 80, 221 of spinor field 224 of the field 114 of vector field 129 Charge conjugation of Dirac matrices 106 of electromagnetic field 146 of scalar field 72 of spinor field 106 of the vector field 134 transformation 24, 241, 242 of electromagnetic field 242 of scalar field 241 of spinor field 242 of vector field 241 Chebyshev–Hermite functions 271

563

Index Chirality 22 Coefficient functions of scattering matrix 308 Commutation relations canonical 208 equal-time 218, 222, 229, 237 Commutator viii Commutator function 219, 222, 223, 226 Cone future 10 lateral 10 light 9 past 10 Conjugation rules for components of the field 65, 78, 101 for spinors 86, 102 Constraints 43, 511 primary 43 secondary 44, 468 Counterterm of Lagrangian 374–377 of gauge theory 522–525 of quantum electrodynamics 490 scalar interactions 383 Coupling constant 291 renormalized 383, 490, 527 Covariance of Gaussian measure 392 Covariant derivatives 162, 163, 165, 169 d’Alembert operator 62 Degree of divergence of diagram 366, 370 of vertex 367 Dionic solutions 194 Dirac brackets of classical mechanics 46 equation 82 conjugate 86 for charge-conjugate spinor 106 solutions 95, 97 gamma-matrices 83–85 properties 83, 86, 106 picture 204 Dirac gamma-matrices properties 90 Domain of definition of field operators 257 tube 453, 454

Dynamic characteristics of the first kind 45 of the second kind 45 Dynamic invariants 54, 56, 58, 59, 80, 111, 113, 114, 129 Dyson equation 347, 349, 482, 483 Effective scattering cross-section 322, 323 Electromagnetic field canonical variables 138 potential 136 quantization 230, 231, 235 strength 135 Energy-momentum tensor 57 vector 125 of electromagnetic field 140, 231 of scalar field 80 of spinor field 111, 224 of vector field 129 Equations for coefficient functions of S matrix 327 evolution type 334 of QED 476 resolvent type 331–333 for Green functions 338–340, 480, 481 in terms of functional derivatives 342, 343 one-particle irreducible 347, 349, 353 truncated 344, 346, 347 Everywhere dense set 253 Evolution operator 302 Exchange (gauge) bosons 181 Faddeev–Popov “ghosts” 514, 515 Fermi-field 208 Fermions 208 left 122 right 122 Feynman diagram 317 disconnected 347 for coefficient functions 317, 319 for gauge fields 522–525

564 for Green functions 340 for matrix elements 319, 321, 322 for quantum electrodynamics 471– 473 of quantum electrodynamics 486, 488 of scattering operator 317 self-energy 347 strongly connected 347 vertex 317, 359 weakly connected 347 rules for coefficient functions 318, 319 for gauge theory 517 for quantum electrodynamics 471– 473 for self-interaction ' 4 317 Field electromagnetic 135, 136, 141 pseudoscalar 19 Rarita–Schwinger 150 scalar 19 vector 25 with spin 2 151 with spin 3/2 149 Flavors of quarks 181 Fock space 250, 252, 258, 262, 266, 271 Euclidean 443, 445 Foldi–Wouthuysen transformation 445 Functional integral 391, 392 Furry theorem 476–478 Gauge axial 148 Coulomb 147 Lorentz–Landau 147 temporal 148 Gauge (exchange) bosons 176, 179, 181 Gauge invariance for coefficient functions 479, 480 principle 161, 162 transformations 161 Gauge transformations 29, 30, 32, 161 local 29 of the first kind 24, 29 of the second kind 30 infinitesimal 165 Gauss law 135

Index Gell-Mann matrices 27 Generalized coordinates 39, 52 momenta 41 velocities 39, 41, 52 Generators of a group 14, 15, 20, 21, 28, 168, 169 Gluons 182 Goldstone bosons 174, 176 Goldstone theorem 536 Grassmann variables 412, 502, 516 Great Unification 183, 189 Green function 274 advanced 275 causal 275 Euclidean 442, 444, 451, 456 free electromagnetic fields 277 free N -point 285 free scalar fields 274, 276 free spinor fields 277 free vector fields 277 of interacting fields 337, 338 photon 277, 470, 471 retarded 275 spectral representation 356, 493–497 Group Abelian unitary U.1/ 23 gauge 30 Lorentz 8, 11 of complex transformations 15 of proper orthochronous transformations (restricted) 13 orthochronous 12 Lorentz(general) 11, 13 of complex unimodular matrices 17 of internal internal symmetries 23 Poincaré 14 special unitary S U.2/ 25 S U.3/ 27 S U.5/ 190 S U.n/ 24 Gupta–Bleuler formalism 232, 236, 268 for interacting electromagnetic field 466 Haag theorem 295 Hamiltonian

565

Index interaction of real scalar fields 292 of classical mechanics 41 with constraints 43 of free field complex scalar 80, 81, 221 electromagnetic 140, 231 real scalar 292 spinor 111, 224 vector 128, 129, 225 of particle in the external electromagnetic field 71 Hamiltonian equations 41 Hamiltonian formalism 39 of classical mechanics 41 of field theory 60, 80, 115, 129 Heaviside function 66 Heisenberg equation 203 picture 202, 203 for field operators 212 Higgs field 172 Higgs mechanism 172, 176, 180 Hypercharge 26, 28 In-field operators 431, 434 In-states 299, 302, 431 Indefinite metric 9, 233 Infinitesimal transformations 14, 20, 165 Instantons 194 Integral Feynman 392, 400, 401, 405 of motion 54 Wiener 391 Interaction between scalar and electromagnetic fields 71 electromagnetic 161, 162, 188 electroweak 189 strong 183, 190 united 190 weak 184, 188 Interaction picture 204 Interaction potential 530–532 Interaction representation 202 Isotopic spin 25, 26

Jacobi identity 31, 42 Jost points 454 Jost theorem 454 Källén–Lehmann representation of 2-point Green function 353, 356, 493 Klein–Fock–Gordon equation 61, 63 solutions 64, 66 Lüders–Pauli theorem 73, 246 Lagrange equation 41 Lagrange formalism 53, 60 Lagrange–Euler equations 53 Lagrangian 52 Glashow–Illiopoulos–Majani 185 of classical field theory 52 of classical mechanics 40 of Faddeev–Popov “ghosts” 515 of free field complex scalar 78 electromagnetic 139 real scalar 291 spinor 125 vector 128 of Great unification 189 of Higgs field 172, 174, 177 of the Yang–Mills fields 167 phenomenological 184, 187 total of complex scalar selfinteraction 172 of electromagnetic and spinor fields 161 of electroweak interaction 189 of QCD 183 of real scalar field 291 of real scalar self-interaction 291 of weak interaction 187 Yukawa 180, 187 Lagrangian formalism 39, 40, 52, 53 Leptons 185, 186 Locality 52 Longitudinal component of electromagnetic field 141 of vector field 131 Lorentz transformation 12

566 complex 15 improper 13 nonorthochronous 13 of scalar field 62, 63 of spinor field 86, 87, 91, 92 of the vector field 130 orthochronous 13 proper 13 LSZ axiomatics 432 Magnetic moment of electron 120 of neutron 120 Majorana spinors 126 Mass nonrenormalized (“bare”) 383, 490, 491 of Great Unification 190 renormalization 383, 491 counterterm 383, 491 renormalized (physical) 491 surface 65, 341, 434, 437 Maxwell equations 135 Mesons pseudoscalar 19 scalar 19 Minkowski space 8, 9, 17 Moment of Gaussian measure 394 Momentum canonical 41 spin 58 Noether’s theorem 54 Nonrelativistic approximation 69, 71 Norm of a vector 251 Normal form of scattering operator 305, 308 Normal product operators 279 with pairing 279, 280 Operator annihilation of external line of Feynman diagram 327, 328, 330, 474, 475 of particle (antiparticle) 217 antilinear 244 antiunitary 244

Index creation of external line of Feynman diagram 327, 328, 330, 474, 475 of particle (antiparticle) 217 Euclidean field 443–445 evolution 302 of symmetrization 271 projective 121 Orbit of gauge group 503 Orthogonality of spinors 103 Osterwalder–Schrader theorem 458, 459 Out-field operators 431, 434 Out-states 299, 302, 431 Parameter of a group 164, 190 Path integral 391, 392, 394 Pauli matrices 17 principle 69, 117, 210 Pauli–Jordan function 219, 220 Pauli–Villars regularization method 358, 359, 364 Phase transformations of fields 23 Photon propagator in arbitrary gauge 469, 470 Photons longitudinal 142 time 142 transverse 141 Plane waves 64 Poisson brackets 42 Polarization operator 482 Positron 107 Primitively-divergent Feynman diagrams 358 Pseudo-Euclidean metric 9 Pseudo-Maxwell equation 466 Pseudoscalar 63, 93, 94 Pseudovector 94 Quantization of free electromagnetic field 229, 236 of gauge field (Yang–Mills field) 238, 498, 499, 503, 504, 506, 508, 509, 511, 513, 514, 516, 519, 520 of interacting electromagnetic field 466–470

567

Index of vector field 225 primary 43, 201 second 202 Quarks 181 Regularization analytic (Speer) 387 by projection-iterative method 387, 389 dimensional 387 Pauli–Villars 358, 359, 364 with respect to lines (Slavnov) 387, 389 Relativistic invariance of equations Dirac 86 Klein–Fock–Gordon 62 Weyl 124 Renormalizable and nonrenormalizable theories 368 Renormalization analytic (Speer) 387 dimensional 387 Renormalization of mass and charge 381, 383–385 Representation adjoint 31 fundamental 32 of anticommutation relations 262, 265 of commutation relations 250, 252, 253, 259 of electromagnetic field 268 of scalar field 252 of the Lorentz group 17 finite-dimensional 16 reducible(irreducible) 91 Rules of bypass of poles 275 Scalar 19, 53, 63, 93 Scattering amplitude 313, 315–317, 324–326 matrix 305, 309 causality conditions 312 law of conservation of energy 312 perturbation theory 302, 304, 305 matrix(operator) 298, 301, 302, 306, 307, 310, 312 theory 298, 299, 301 Scattering cross-section 322

differential 323, 326 effective 323 total 323 Scattering matrix coefficient functions of 306, 307 unitarity 310 Scattering operator normal form of 306, 307 Schrödinger equation 202 picture 202 Schwinger equation 338, 339 Soliton 193 Space of states 66 n-particle 76 one-particle 68 Space reflection 13 of scalar field 72 of spinor field 108 of the electromagnetic field 146 of vector field 134 Spacelike surface 75 Spectral property 426 Spectral representation of 2-point Green function 353, 356 in EQD 493–497 Spectrum of the operator of energymomentum 207 Spin 16 Spinor 16, 18 Spontaneous breaking of symmetry 172, 173 of the vacuum 173 Stability of vacuum 437 of single-particle state 437 Structural constants of the group 26, 27 Tensor of angular momentum 58 of electromagnetic field 136 of energy-momentum 58 of spin 57 Theory of Great Unification 183, 189 Time component of electromagnetic field 141 Time reversal 13 of scalar field 73

568 of spinor field 108 of the electromagnetic field 146 of the vector field 134 Time-ordered pairing 282 Time-ordered product 281 Total 323 Transversality condition 138 Transverse component of the electromagnetic field 141 Unitary representation 20 Vacuum 207 average 232, 278, 284, 286 for free fields (“bare”) 211, 257, 336, 337 for interacting fields (“physical”) 213, 336, 337 Variational derivative 312, 342 Variational principle 39, 40 Vector axial 495 contravariant 8 covariant 8 cyclic 258 exponential 258 isotropic 9 of current for the spinor field 110 of scalar field 66 of the vector field 129 of energy-momentum 56

Index of Fock space 251 of spin 58 of state 202, 250 spacelike 9 timelike 9 unit 132 Vertex diagram 359 function 359, 362 of gauge theory 523–525 Virtual particle 473 Virtual state 320 Ward identity 485, 508, 509 Wave function of electron(positron) 104, 107 of photon 145 of scalar boson 68 Wave operator 62, 299, 300 Wave packet 64 Wave-particle duality 1 Wick product 212 theorem 279, 281 for a time-ordered product 281 generalized 284 Wightman functions 425, 426 Wigner transformation 73 Yang–Mills equations 168 fields 162, 165, 167