Introduction To The Theory Of Quantized Fields [1st edition, second printing 1960]

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INTERSCIENCE MONOGRAPHS IN PHYSICS AND ASTRONOMY Edited by R. E. MARSHAK

Volume

I:

E. R. Cohen, К. M. Crowe, J. W. M. DuMond THE FUNDAMENTAL CONSTANTS OF PHYSICS

Volume

II:

G. J. Dienes and G. H. Vineyard RADIATION EFFECTS IN SOLIDS

Volume III:

N. N. Bogoliubov and D. V. Shirkov INTRODUCTION TO THE THEORY OF QUANTIZED FIELDS

Additional volumes in preparation

INTERSCIENCE MONOGRAPHS IN PHYSICS AND ASTRONOMY

Edited by R. E. MARSHAK University of Rochester, Rochester, New York

VOLUME III

Editorial Advisory Board H. ALFVEN, Royal Institute of Technology, Stockholm, Sweden L. V. BERKNER, Associated Universities, New York, New York H. J. BHABHA, Tata Institute, Bombay, India N. N. BOGOLIUBOV, Academy of Sciences, Moscow, U.S.S.R. S. CHANDRASEKHAR, Yerkes Observatory, Williams Bay, Wisconsin J. W. DuMOND, California Institute of Technology, Pasadena, California L. GOLDBERG, University of Michigan, Ann Arbor, Michigan M. GOLDHABER, Brookhaven National Laboratory, Upton, New York C. HERRING, Bell Telphone Laboratories, Murray Hill, New Jersey J. KAPLAN, University of California, Los Angeles, California C. M0LLER, Institute for Theoretical Physics, Copenhagen, Denmark L. NfiEL, University of Grenoble, Grenoble, France W. К. H. PANOFSKY, Stanford University, Stanford, California R. E. PEIERLS, University of Birmingham, Birmingham, England F. PRESS, California Institute of Technology, Pasadena, California B. ROSSI, Massachusetts Institute of Technology, Cambridge, Massachusetts L. SPITZER, Jr., Princeton Observatory, Princeton, New Jersey

B. SI ROMGREN, Institute for Advanced Study, Princeton, New Jersey G. TORALDO di FRANCIA, University of Florence, Florence, Italy G. E. UHLENBECK, University of Michigan, Ann Arbor, Michigan V. F.WEISSKOPF, Massachusetts Institute of Technology, Cambridge, Massachusetts H. YUKAWA, Kyoto University, Kyoto, Japan

INTRODUCTION TO THE THEORY OF QUANTIZED FIELDS

N. N. BOGOLIUBOV D. V. SHIRKOV Steklov Mathematical Institute, Academy of Sciences, Moscow, U.S.S.R. Joint Institute for Nuclear Research, Dubna, U.S.S.R.

Authorized English Edition Revised and Enlarged by the Authors Translated from the Russian by

G. M. Volkoff Department of Physics, University of British Columbia Vancouver, В. C., Canada

INTERSCIENCE PUBLISHERS, INC., NEWYORK Interscience Publishers Ltd., London 1959

First Published 1959

ALL RIGHTS RESERVED

Library of Congress Catalog Card Number 59-10445 First printing.1959 Second printing (by photo-offset)

. I960

INTERSCIENCE PUBLISHERS, INC., 250 Fifth Avenue, New York 1, N. Y. For Great Britain and Northern Ireland:

INTERSCIENCE PUBLISHERS LTD., 88/90 Chancery Lane, London W. C. 2

Printed in the United States of America

Preface to the First Russian Edition

This monograph is an attempt to give a systematic presen¬ tation of the modern theory of quantized fields from its founda¬ tions right up to its most recent achievements. In writing this book the authors were guided by their intention to present field theory from a unified point of view combining internal logical consistency and closure with completeness of the material covered.

We also aimed, wherever possible, to introduce

the maximum degree of clarity into the basic assumptions of the theory employed at the present stage of its development.

At the

same time, naturally, particular attention was devoted to the mathematical correctness of the arguments, as a result of which the extent of coverage of the applications of the theory to calcula¬ tions of specific physical phenomena can be claimed to be only methodologically complete. The authors also wanted to treat sufficiently fully the most promising approaches developed quite recently.

We hope that

because of this, the book will prove to be useful not only to persons first undertaking the study of quantum field theory, but also to theoreticians working in this domain of physics. The chapter “Dispersion Relations,” which presents the most recent results, was included in the book as a supplement in view of the current interest in this topic. The authors wish to thank the staff and the post-graduate students of the Division of Theoretical Physics of the V. A. Steklov Mathematical Institute of the Academy of Sciences, U.S.S.R., and of the Chair of Statistical Physics and Mechanics of the Physics Faculty of the M. V. Lomonosov Moscow State University, for their remarks and suggestions made during the preparation of the manuscript.

We are particularly grateful to В. V. Medvedev, who V

35830

VI

PREFACE

contributed a number of valuable comments on different parts of the book. February, 1957

N. N.

Bogoliubov

Moscow

D. V.

Shirkov

Translator’s Note

The translator wishes to express his gratitude to the authors for providing him with a pre-publication copy of the manuscript which permitted the translation to be started several months prior to the publication of the Russian edition, and also for supplying some additional new material which does not appear in the Russian edition: a Mathematical Appendix containing the proof of the important theorem quoted without proof in 51.3, a new Section 51.4, two new paragraphs at the end of 49.2, and numerous minor revisions, in particular those on pp. 547, 548, 580, 608, and 609, and corrections of typographical errors. Thanks are also due to Gostekhizdat for airmailing freshly printed copies of the Russian edition for final checking of the translation. This cooperation has made possible the early appearance in English of the present enlarged and corrected edition. July, 1958 Vancouver, Canada

G.

VI 1

Volkoff

Contents

§ 1. Introduction.

1

1.1. A Survey of the Present State of Field Theory . . 1.2. Plan of Presentation. 1.3. Some Notation.

1 7 8

CHAPTER I

Classical Theory of Free Fields

§ 2. Lagrangian Formalism and Field Invariants. 2.1. Fields and Particles. 2.2. Hamiltonian and Lagrangian Formalisms. 2.3. The Lagrangian Function and the Principle of Sta¬ tionary Action.■. 2.4. Transformation Properties of the Field Functions. Tensors and Spinors. 2.5. Noether’s Theorem. 2.6. The Energy-Momentum Vector. 2.7. The Angular Momentum Tensor and the Spin Tensor 2.8. The Charge and the Current Vector.

10 10 11 12 15 19 23 24 26

§ 3. The Scalar Field. 3.1. Lagrangian Formalism for a Real Scalar Field . . 3.2. Momentum Representation and Positive- and Nega¬ tive-Frequency Components. 3.3. The Discrete Momentum Representation. 3.4. The Complex Scalar Field.

28 28

§ 4. The Vector Field. 4.1. Lagrangian, Subsidiary Condition, and Invariants 4.2. Transition to the Momentum Representation ... 4.3. Spin of the Vector Field. 4.4. Klein-Gordon Equations Written in the Form of a System of First-Order Equations.

38 38 43 46

§ 5. The Electromagnetic Field. 5.1. Potential of the Electromagnetic Field. 5.2. Gauge Transformation of the Second Kind and the Lorentz Condition. 5.3. Lagrangian Formalism. 5.4. Transverse, Longitudinal, and Time Components .

49 49 51 53 55

5.5. Spin.

57

§ 6. The Spinor Field. Dirac Matrices and Transformation Laws of Spinor Functions.

58

IX

30 34 35

47

CONTENTS

X

6.1. 6.2. 6.3. 6.4.

Linearization of the Klein-Gordon Operator. ... Dirac Matrices. The Dirac Equation. Transformation Properties of the Spinor Field. . .

§ 7. The Spinor Field. Properties of the Solutions and Dynamic Variables. 7.1. Momentum Representation and Matrix Structure . 7.2. Decomposition into Spin States and Normalization and Orthogonality Relations. 7.3. Lagrangian Formalism and Invariants. § 8. Lagrangian of a System of Fields. 8.1. The Interaction Lagrangian and Invariants of a System of Fields. 8.2. Invariants under Gauge Transformations.

58 60 64 66 72 72 76 79 83 84 85

CHAPTER II Quantum Theory of Free Fields § 9. General Principles of Quantization of Wave Fields .... 9.1. Operator Nature of the Field Functions and the Trans¬ formation Law of the State Amplitude. 9.2. Quantization Postulate for Wave Fields. 9.3. The Physical Meaning of the Positive- and NegativeFrequency Components and of Adjoint Functions . 9.4. The Vacuum State and the Amplitude of the State with a Given Number of Particles.

90

§ 10. Setting Up the Commutation Relations. 10.1. Types of Commutation Relations. 10.2. Normal Product of Operators and the Form of Dynam¬ ic Variables. 10.3. Requirement that Energy Should Be Positive. . . 10.4. Fermi-Dirac and Bose-Einstein Commutation Rela¬ tions . 10.5. Commutation Relations and the Discrete Momentum Representation.

101 101

90 92 96 98

103 105 107 109

§ 11. Scalar and Vector Fields. 11.1. The Real Scalar Field. 11.2. The Complex Scalar Field. 11.3. The Complex Vector Field.. .

112 112 115 116

§ 12. The Spinor Field. 12.1. Fermi-Dirac Quantization and Commutation Func¬ tions . 12.2. Dynamic Variables. 12.3. Charge Conjugation. § 13. The Electromagnetic Field. 13.1. Singularities of the Electromagnetic Field and the Quantization Procedure. 13.2. Indefinite Metric. 13.3. The Form of the Basic Quantities.

120 120 122 124 128 128 130 136

CONTENTS § 14. Green’s Functions. 14.1. Definition. 14.2. "Retarded” and "Advanced” Green’s Functions for a Scalar Field. 14.3. Causal Green’s Function for a Scalar Field .... 14.4. Causal Green’s Functions for Various Fields . . . 14.5. Relation to the Notations of Stueckelberg and of Schwinger.

XI 136 136 138 141 143 146

§ 15. Singularities of the Commutation and Causal Functions . . 15.1. Calculation of the D(+) and T>H Functions .... 15.2. Exphcit Form and Singularities of the Functions D(x) and Dc(x). 15.3. Regularization of Singular Functions by the Method of Pauli-Villars.

147 147

§ 16. Operator Expressions and Singular Functions. 16.1. The Coefficient Functions of Operator Expressions 16.2. Wick’s Theorem for Normal Products. 16.3. Improper Nature of Singular Functions. 16.4. Some Properties of the Pauli-Villars Regularization 16.5. Multiplication of Singular Functions.. . 16.6. Some Properties of Singular Functions. 16.7. Multiplication of Operator Functions. 16.8. Some Definitions.

157 157 159 168 175 177 184 185 190

CHAPTER III

151 153

The Scattering Matrix

§ 17. Fundamental Ideas of the Theory of Interacting Fields . . 192 17.1. Representations of the Schrodinger Equation ... 192 17.2. The Interaction Representation. 194 17.3. The Scattering Matrix. 197 17.4. Relativistic Covariance and Unitary Nature of the S-Matrix. 199 17.5. The Condition of Causality.200 17.6. "Classical Fields” as Arguments of Functionals . . 204 § 18. The Interaction Lagrangian and the S-Matrix.206 18.1. Expansion of the S-Matrix into a Power Series in Powers of the Interaction.206 18.2. Conditions of Covariance, Unitarity, and Causality for Sn.208 18.3. Determination of the Explicit Form of S1(x) and

S2(x,y).211 18.4. Chronological Product of Local Operators.215 18.5. Determination of the Functions S„ for Arbitrary n 217 18.6. Analysis of the Arbitrariness of the Functions Sn and the Most General Form of S(g).220 § 19. Evaluation of Chronological Products.227 19.1. Chronological Pairing.227 19.2. Wick’s Theorem for Chronological Products.... 233

ХІІ

CONTENTS

§ 20. Reduction of the S-Matrix to the Normal Form.235 20.1. Structure of the Coefficients of the Scattering Matrix 235 20.2. Feynman Diagrams and the Rules of Correspondence 239 20.3. Examples.245 20.4. Concluding Remarks.247 § 21. Feynman’s Rules for the Evaluation of Matrix Elements of the Scattering Matrix.'.249 21.1. Transition to the Momentum Representation . . . 249 21.2. Evaluation of the Matrix Elements.252 21.3. Taking Account of Symmetry Properties.257 21.4. Scattering by External Fields.260 21.5. General Structure of the Matrix Elements .... 263 § 22. Probabilities of Scattering Processes and Effective Cross Sec¬ tions .267 22.1. Normalization of the State Amplitude.267 22.2. Calculation of Transition Probabilities.270 22.3. Differential and Total Effective Cross Sections . . 273 § 23. Examples of Calculation of Second-Order Processes .... 275 23.1. Compton Scattering.275 23.2. Annihilation of an Electron-Positron Pair.279 23.3. Bremsstrahlung.282 CHAPTER IV

Removal of Divergences from the S-Matrix

§ 24. On the Divergences of the S-Matrix in Electrodynamics (Second Order).285 24.1. Divergent Diagram with Two External Electron Lines H.286 24.2. Segregation of the Divergent Part of Z.291 24.3. Divergent Diagram with Two External Photon Lines

П.293 24.4. Segregation of the Divergences from П and Gauge Invariance.296 24.5. Construction of an Integrable S2.298 § 25. On the Divergences of the S-Matrix in Electrodynamics (Third Order).301 25.1. Vertex Diagram of the Third Order.302 25.2. Segregation of the Divergence from Г and Gauge In¬ variance .304 25.3. Ward’s Identity.309 25.4. Construction of an Integrable Function S3

...

.

311

§ 26. General Rules for the Removal of Divergences from the SMatrix.

314

26.1. Formulation of the Problem.314 26.2. Graphical Representation of the Subtraction Proce¬ dure and the Operation A(G).316 26.3. Index of the Diagram co(G) and the Degree of Di¬ vergence ..

CONTENTS

26.4. 26.5. 26.6. 26.7.

хііі

Structure of the Exponential Quadratic Form. . . 323 Choice of the Operation A{G).328 Transition to the Limit e->0.331 Generalization of the Prescription for the Construction of A(G).334

§ 27. Analytic Properties of the Coefficient Functions in the Momen¬ tum Representation.336 27.1. Analytic Properties of Sn.336 27.2. Structure of the Functions Hn.337 27.3. Analytic Properties of the Functions Hn.338 § 28. Classification of the Renormalizability of Different Theories 340 28.1. Interactions of the First and Second Kinds. . . . 340 28.2. List of Interactions of the First Kind.344 28.3. Nonlocal Nature of Interactions of the Second Kind 347 28.4. Specification of a Theory of the First Kind by a Finite Number of Constants.349

CHAPTER V

Application of the General Theory of the Removal of Divergences to Special Cases

§ 29. Scalar Field with Nonlinear Interaction.352 29.1. The Hurst-Thirring Field.352 29.2. Finiteness of the Number of Divergent Diagrams . 353 § 30. Spinor Electrodynamics. I. General Form of Counter Terms 355 30.1. Types of Divergent Diagrams and Furry’s Theorem 355 30.2. Gauge Invariance of the Scattering Matrix .... 361 30.3. Counter Terms.376 § 31. Spinor Electrodynamics. II. Mass and Charge Renormalization 378 31.1. Gauge Transformation of the Pairing AA.378 31.2. Nonuniqueness of the Process of Removal of Infinities 380 31.3. The Complete Green’s Functions G, D, and the Vertex Part Г.386 31.4. The Formal Nature of Infinite Renormalizations . 392 31.5. Radiation Corrections to External Lines and the Choice of Finite Constants.395 § 32. Spinor Electrodynamics. III. Radiation Corrections of the Second Order.399 32.1. Correction to the Photon Function.399 32.2. Corrections to the Electron Green’s Function . . . 402 32.3. Corrections to the Vertex Part.405 32.4. Outline of the Calculation of Corrections to the KleinNishina Formula.406 § 33. Pseudoscalar Meson Theory.409 33.1. Isotopic Spin Formalism.409 33.2. Types of Divergent Diagrams and Counter Terms . 411 33.3. Second Charge, Multiplicative Renormalizations, and External Lines.413

CONTENTS

XIV

§ 34. Schwinger's Equations for Green’s Functions.416 34.1. Relation of the Complete Green’s Functions to the Vacuum Expectation Values of Г-Products .... 416 34.2. Generalized Wick’s Theorem.421 34.3. Schwinger’s Equations.424 34.4. Taking Counter Terms into Account.431 N

CHAPTER VI

Schrodinger Equation and Dynamic Variables

§ 35. Schrodinger Equation for the State Amplitude.433 35.1. Equation for — t.

THEORY OF QUANTIZED FIELDS

14

which depends on all four space-time variables.

For brevity we

shall in future refer to u'[x') = Ли(х),

(2-6)

with the matrix of the transformation Л of the functions being completely determined by the matrix of the Lorentz transforma¬ tion L.

We emphasize that the above transformation (2.6) is not

restricted to a replacement of the argument x by x', and describes a transformation from one coordinate system to another, rather than a displacement from one point of space to another. Thus to each Lorentz transformation L there corresponds a linear transformation AL, it being evident that to the identity

THEORY OF QUANTIZED

16

FIELDS

element of the group L there corresponds the identity transforma¬ tion Л — 1, and that to a product of two elements of the Lorentz group there corresponds the product of two transformations ^L\ L2 = Л-ь\Л-ьг-

„

The system of operators Л possessing such properties is denoted in group theory by the term linear representation of the group. The operators A evidently may be represented in the form of matrices whose rank is determined by the number of components of the field function u.

In the case when the number of components

of и is finite, it is said that the group of transformations Л forms a finite-dimensional representation of the Lorentz group, while in the opposite case we obtain an infinite-dimensional represen¬ tation of this group.

In view of the fact that all the principal

physical fields are usually described by functions with a finite number of components, we shall limit ourselves in the following to a discussion of only the finite-dimensional representations of the Lorentz group. Thus we may regard the transformations Л as operators operating in the finite-dimensional space of the components of the field functions and represent them by square matrices of finite rank. Sometimes it happens that the space of the components of the field functions within which the representation Л operates breaks up into subspaces invariant with respect to all the transformations belonging to a given representation (i.e., into subspaces that transform into themselves under the operation Л). representation is said to be reducible. representation is irreducible.

Such a

In the opposite case the

If the process of separating out

invariant subspaces in the space of a reducible representation is carried out to the end, i.e., if the whole space is divided into invariant subspaces which themselves do not contain invariant subspaces, then it is clear that the initial representation will be broken up into irreducible representations operating in the corresponding invariant

subspaces.

Thus

the study of any

reducible representation may be replaced by an equivalent study of the irreducible representations of the given group.

Thus all

2. LAGRANGIAN FORMALISM

17

possible types of wave functions and their laws of transformation (2.6) may be obtained by means of investigating the finite¬ dimensional representations of the Lorentz group. The investigation referred to above forms a special division of the theory of representations of continuous groups and may be briefly summarized as follows.

The finite-dimensional represen¬

tations of the Lorentz group may be single-valued or double¬ valued.

This is related to the fact that the correspondence

L->/.lL need not be necessarily single-valued, since, generally speaking, the field functions are not directly experimentally observable quantities (however, the observable quantities may always be expressed in the form of bilinear combinations of the field'functions).

However, the lack of single-valuedness of the

operator Ль corresponding to the transformation L must always be such that the observable quantities transform in a completely single-valued manner under any arbitrary Lorentz transformation Moreover, it is necessary that the operators Ль should be

L.

continuous functions of the parameters of the transformation L, i.e., that an infinitesimal transformation of the coordinate system should correspond to an infinitesimal transformation of the field functions.

The combined requirements stated above lead to the

representations of the Lorentz group being divided into two categories.

The first category is characterized by the single-

valuedness of the correspondence L -> AL and contains the single¬ valued so-called tensor and pseudotensor 4 representations.

The

field functions which transform in accordance with the tensor representation are called tensors (pseudotensors) and in some cases may be observable themselves (the electromagnetic field). In the second case this correspondence turns out to be double¬ valued: L -> dr ЛьThe transformation law for a (pseudo) tensor of the iVth rank T*i-

.*N

under continuous transformations of coordinates (transformations 4 The distinction between tensors and pseudotensors is related to the transformations of the reflection of space axes and is discussed more fully later.

18

THEORY OF QUANTIZED

FIELDS

involving the reflections of an odd number of axes will be discussed later separately) has the form rh-*'»

■■■>iN(x')

dx'h

dx'iN

dx’i

dx’N'

(Я

Pi.(z)

(2.7)

or using the notation of (2.5) T'*1.N Q4’l _ , _ QiN IN J h.W U)

The double-valued representations are referred to as spinor representations, spinors.

and the corresponding quantities

are called

The transformation law for spinor quantities has a more

complex structure and is given for the simplest spinors in § 6.

We

merely note that the transformation law for tensor quantities which follows from (2.7) u(x)

u'(x') = u(x)

in the case of the transformation of translation x'k = xk -f- ak also holds for spinors. We shall now give the simplest tensor representations and the quantities which correspond to them. which under any continuous accordance with the law

The tensor of zero rank

transformation

transforms

u'(x') = u(x),

in

(2.8)

is an invariant and is called a scalar (pseudoscalar). The tensor of the first rank which under a rotation of coordi¬ nates transforms in accordance with the law u'k(x') = % gmmQkm um(x),

(2.9)

m

is called a contravariant vector (pseudovector). vector associated with it %(ж) = I?im«m(x) m

transforms in accordance with the law

The covariant

2. LAGRANGIAN FORMALISM

19

uk{x') = gkk 2^2kmum(x).

(2.10)

m

Corresponding formulas may be written down without any difficulty for covariant and contravariant tensors of second and higher ranks. As we have pointed out already, relations of the type (2.7)(2.10) establish the transformation laws of tensor quantities only for transformations of a continuous type.

The laws governing

their transformation under a reflection of an odd number of space axes are not determined by these expressions and must be formu¬ lated separately.

Because the consecutive twofold application of

such a reflection must correspond to the identity operator as a result of the single-valuedness of the tensor representation, the transformation laws in question may have only the forms u'[x') = u(x)

(2.11)

u'(x') = — u(x).

(2.12)

or

Those quantities which change sign on reflection, i.e., which transform according to (2.12) in contrast to quantities trans¬ forming in accordance with (2.11), are called pseudoquantities (pseudoscalar, pseudo vector, pseudotensor, etc.). The distinction between the transformation laws (2.11) and (2.12) seems at first glance to have a somewhat formal character. However, as we shall see later (§ 8), the concept of parity defined by these relations plays an essential role in determining the possible forms of interaction between various fields. 2.5.

Noether’s Theorem For the construction of field invariants, we shall use Noether’s

theorem.

The theorem states that to every continuous trans¬

formation of coordinates which makes the variation of the action equal to zero, and for which the transformation law of the field functions is also given, there corresponds a definite invariant, i.e., a combination of the field functions and of their derivatives which remains conserved. To prove Noether’s theorem we shall consider the infinitesimal

THEORY OF QUANTIZED

20

FIELDS

transformation of coordinates xk -> x'k — xk -f- dxk,

bxk =

2 Xk boo1

(2-13)

1 Sj'gs

with s infinitesimal parameters

(г = 1, 2, . .

8

=

-1^ 7 dxk

= 0.

(2.17)

From equations (2.17) one may obtain by means of Gauss

22

THEORY OF QUANTIZED FIELDS

theorem the laws of conservation of the corresponding surface integrals.

Integrating (2.17) over a volume of infinite extent in

the spacelike directions and limited in the timelike directions by spacelike three-dimensional surfaces cq and o2, we shall obtain, after assuming that at the boundaries of the spacelike volume the field is practically equal to zero:

I

f *олѲ*~ 2

к Ul

f

к Jcr*

dokB\= 0.

Here da,c is the projection of an element of the surface a on the three-plane perpendicular to the xk axis.

The equation

which we have obtained shows that the surface integrals

c,

(a) =

2 f dak0\ к

(2.18)

-0-

do not actually depend on the surface a.

In the particular case

when the surfaces a represent the three-planes x° — t = const, the integration in (2.18) takes place over a three-dimensional con¬ figuration space and the integrals C,(*°) - jdxd*

(2 19)

do not depend on the time. It has thus been shown that to each continuous s-parametric transformation of coordinates

(2.13) and of the field function

there corresponds a certain invariant (2.19) C4(f = 1, 2, . . ., s) which is independent of the time. The quantity Ѳk is not unique. pression of the form

To it we may add an ex¬

d

2 dxm f?k subject to the condition that jmk

_ _

jkm

This property is sometimes used for the symmetrization of Ѳ. The degree of arbitrariness indicated above, however, does not affect the value of the integrals (2.18) and (2.19) which are conserved.

2. LAGRANGIAN FORMALISM

23

We now proceed to consider concrete examples of the quantity Ѳ and of the conservation laws (2.19) connected with it. 2.6.

The Energy-Momentum Vector In the case of infinitesimal space-time translations x'k — xk -f- 6xk,

by choosing for the parameters of the transformation dco{ the quantities dxk and by taking into account the law of transformation (2.8) we find

and Ѳ becomes the second rank tensor

In the fully contra variant form this tensor is given by

(2.20) Integrals of the type (2.19) involving Tkl represent a fourvector constant in time P‘ = f T°‘dx.

(2.21)

The zero component of this vector P° in classical mechanics represents the Hamiltonian function, i.e., the energy. It therefore follows from considerations of covariance that the four-vector (2.21) represents energy momentum, and the tensor (2.20) is the energy-momentum tensor. We note here that we shall be interested principally in the integrated dynamic quantities similar to the energy momentum four-vector Pl. The structure of the tensor Tkl, which in our presentation is not even unique, becomes of interest in itself only in a consistent theory which takes into account gravitational effects. However, it is well known that such a unified theory does not exist at present, and therefore we shall not touch upon such questions here.

24

THEORY OF QUANTIZED FIELDS

2.7.

The Angular Momentum Tensor and the Spin Tensor In the case of the infinitesimal four-rotations

in virtue of the antisymmetry of the quantities dconm = — dcomn we may take for the parameters of the transformation the six linearly independent quantities dconm,

n < m.

Here the indices (n, m) denote the plane in which the rotation with the parameter conm takes place. We see that in formula (2.13) the lower index j is replaced by two indices

and we find, taking the antisymmetry of conm into account: дхк = ^XkSco’ = J

2 Х*пт)дсо=]?^х{д(о*1 = ^gllxldcomld*

n>m

= 2ёи^т1дкт + m l

from which it follows that *?«*> = gmmx™dl - gnnxn6km

(n X m).

(2.22)

We represent the total variation of the field function in the form ui(x

)

= ui{x)

+

du0

u, =