An Introduction To Relativistic Quantum Field Theory [1st, edition, second corrected printing]

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AN INTRODUCTION TO RELATIVISTIC QUANTUM FIELD THEORY

AN INTRODUCTION TO RELATIVISTIC QUANTUM FIELD THEORY

Silvan S. Schweber

Brandeis University

Foreword by Hans A. Bethe Cornell University

HARPER & ROW, PUBLISHERS New York, N.Y.

Second Printing, 1962 Copyright © 1961 Harper &

Row,

Publishers,

Incorporated

All rights reserved for all countries, including the right of translation 6675

MANUFACTURED IN THE UNITED STATES OF AMERICA

OltUtR

To Myrna

34744

Table of Contents Foreword.xi Preface.xiii

Part One: The One-Particle Equations 1. Quantum Mechanics and Symmetry Principles ... 3 a. Quantum Mechanical Formalism. 3 b. Schrodinger and Heisenberg Pictures. 8 c. Nonrelativistic Free-Particle Equation. 9 d. Symmetry and Quantum Mechanics.13 e. Rotations and Intrinsic Degrees of Freedom.18 f. The Four-Dimensional Rotation Group.33 2. The Lorentz Group.36 a. Relativistic Notation.36 b. The Homogeneous Lorentz Group.38 c. The Inhomogeneous Lorentz Group.44 3. The Klein-Gordon Equation.54 a. b. c. d.

Historical Rackground.54 Properties of Solutions of K-G Equation.55 The Position Operator.60 Charged Particles.63

4. The Dirac Equation. a. b. c. d. e. f. g. h.

65

Historical Rackground.65 Properties of the Dirac Matrices.70 Relativistic Invariance.74 Solutions of the Dirac Equation.82 Normalization and Orthogonality Relations: Traces ... 85 Foldy-Wouthuysen Representation.91 Negative Energy States.95 Dirac Equation in External Field—Charge Conjugation . . 99

viii

TABLE OF CONTENTS

5. The Zero Mass Equations.108 a. The Two-Component Theory of the Neutrino.108 b. The Polarization States of Mass Zero Particles . .113 c. The Photon Equation.116

Part Two: Second Quantization 6. Second Quantization: Nonrelativistic Theory a. b. c. d. e. f. g. h. i. j. k.

121

Permutations and Transpositions.121 Symmetric and Antisymmetric Wave Functions .... 123 Occupation Number Space. 126 The Symmetric Case. 128 Creation and Annihilation Operators. 130 Fock Space. .134 The Antisymmetric Case.137 Representation of Operators. . 140 Heisenberg Picture. . . . 146 Noninteracting Multiparticle Systems ... 148 Hartree-Fock Method. 150

7. Relativistic Foek Space Methods a. b. c. d. e. f.

...

....

156

The Neutral Spin 0 Boson Case.156 Lorentz Invariance.164 Configuration Space.167 Connection with Field Theory.183 The Field Aspect.193 The Charged Scalar Field.195

g. Conservation Laws and Lagrangian Formalism .... 207 h. The Pion System.211 8. Quantization of the Dirac Field.218 a. The Commutation Rules.218 b. Configuration Space.224 e. Transformation Properties.231 d. The Field Theoretic Description of Nucleons.236 9. Quantization of the Electromagnetic Field

....

.

240

a. Classical Lagrangian.240 b. Quantization: The Gupta-Bleuler Formalism.242 c. Transformation Properties.252

TABLE OF CONTENTS

IX

Part Three: The Theory of Interacting Fields 10. Interaction Between Fields.257 a. b. c. d. e. f. g.

Symmetries and Interactions.257 Restrictions Due to Space-Time Symmetries.264 Electromagnetic Interactions.272 The Meson-Nucleon Interaction.280 The Strong Interactions.285 The Weak Interactions.294 The Equivalence Theorem.301

11. The Formal Theory of Scattering.308 a. b. c. d. e. f.

Potential Scattering.309 The Lippmann-Schwinger Equations.315 The Dirac Picture.316 Unitarity of S Matrix.325 The Reactance Matrix.328 The U Matrix.330

12. Simple Field Theoretic Models.339 a. b. c. d.

The Scalar Field.339 The Lee Model.352 Other Simple Models.370 The Chew-Low Theory.372

13. Reduction of S Matrix

415

a. Formal Introductions.415 b. The Scattering of a Neutral Meson by a Nucleon. . 426 c. Wick’s Theorem.435 d. The Representation of the Invariant Functions .... 442 14. Feynman Diagrams

447

a. Interaction with External Electromagnetic Field .... 447 b. Feynman Diagrams for Interacting Fields.466 c. Momentum Space Considerations.4(2 d. Cross Sections.484 e. Examples: 1. Compton Scattering.487 2. Pion Photoproduction 493 3. Pion Decay’.495 4. /3 Decay of the Neutron.498 f. Symmetry Principles and S Matrix.501

X

TABLE OF CONTENTS

15. Quantum Electrodynamics.507 a. b. c. d. e. f. g. h.

The Self-Energy of a Fermion.508 Mass Renormalization and the Nonrelativistic Lamb Shift . 524 Radiative Corrections to Scattering.531 The Anomalous Magnetic Moment and the Lamb Shift . 543 Vacuum Polarization.550 Applications.561 The Furry Picture.566 Renormalization in Meson Theory.575

16. Quantitative Renormalization Theory.584 a. b. c. d. e. f. g. h.

Primitively Divergent Diagrams.584 The Renormalizability of Quantum Electrodynamics . 607 The Separation of Divergences from Irreducible Graphs . 615 The Separation of Divergences from Reducible Graphs . 619 The Ward Identity.625 Proof of Renormalizability.629 The Meaning of Charge Renormalization.638 General Remarks.640

Part Four: Formal Developments 17. The Heisenberg Picture.649 a. b. c. d. e. f.

Vacuum Expectation Values of Heisenberg Operators 650 The Lehmann Spectral Representation.659 The Magnitude of the Renormalization Constants 677 The Matrix in the Heisenberg Picture.683 Low Energy Theorems.696 The Round State Problem.705

18. The Axiomatic Formulation .... a. b. c. d. e.

721

Wightman Formulation.723 The LSZ Formulation of Field Theory.742 Integral Representations of a Causal Commutator . 764 Dispersion Relations.776 Outlook.826

Problems and Suggested Further Reading.828 References.

853

Index.

$91

Compendium of Useful Formulae.907

Foreword It is always astonishing to see one’s children grow up, and to find that they can do things which their parents can no longer fully understand. This book is a good example. It was first conceived by Dr. Frederic de Hoffmann and myself as merely a short introduction to the rather simple-minded calculations on % mesons in Volume II of the old book Mesons and Fields, published in 1955. In Dr. Schweber’s hands Volume I, even then, had developed into a thorough textbook on renormalization in field theory. It has now become a comprehensive treatise on field theory in general. In the six years since the publication of the two-volume Mesons and Fields field theory has made spectacular progress. Some of this progress was stimulated by experiment, e.g., by the discovery that parity is not conserved in weak interactions. Much of it, however, consisted in a deeper search into the foundations of field theory, trying to answer the central question of relativistic quantum theory which Schweber poses himself in Chapter 18 of this book: Do solutions of the renormalized equations of quantum electrodynamics or any meson theories exist? This search has led tb the axiomatic approach to quantum field theory which is probably the most promising and solid approach now known, and which is described in Chapter 18. About half of the present book is devoted to the interaction between fields. This new book contains a thorough discussion of renormalization theory, starting from the general principles and leading to quantitative results in the case of electrodynamics. I do not know of any other treat¬ ment of this subject which is equally complete and rigorous. The physicist who is interested in applications of field theory will be happy about the good discussion of the theory of Chew and Low of 7r-meson scattering, which theory has been so successful in explaining the 7r-meson phenomena at low energy and which has superseded the methods presented in Vol¬ ume II Mesons of the older book.

Xll

FOREWORD

The book emphasizes general principles, such as symmetry, invariance, isotopic spin, etc., and develops the theory from these principles. It is never satisfied with superficial explanations. The student who really wants to know and understand field theory, and is willing to work for it, will find great satisfaction in this book. H. A. Bethe Ithaca, N. Y. March 1961

Preface The present book is an outgrowth of an attempted revision of Volume I of Mesons and Fields which Professors Bethe, de Hoffmann and the author had written in 1955. The intent at the outset was to revise some of the contents of that book and to incorporate into the new edition some of the changes which have occurred in the field since 1955. Unfortunately, due to the pressure of other duties, Drs. Bethe and de Hoffmann could not assist in the revision. By the time the present author completed his revi¬ sion, what emerged was essentially a new text. With the gracious consent of Drs. Bethe and de Hoffmann, it is being published under a single authorship. The motivation of the present book, however, is still the same as for the volume Fields on which it is based, in part: to present in a simple and self-contained fashion the modern developments of the quantum theory of fields. It is intended primarily as a textbook for a graduate course. Its aim is to bring the student to the point where he can go to the literature to study the most recent advances and start doing research in quantum field theory. Needless to say, it is also hoped that it will be of interest to other physicists, particularly solid state and nuclear physicists wishing to learn field theoretic techniques. The desire to make the book reasonably self-contained has resulted in a lengthier manuscript than was originally anticipated. Because it was my intention to present most of the concepts underlying modern field theory, it was, nonetheless, decided to include most of the material in book form. In order to keep the book to manageable length, I have not included the Schwinger formulation of field theory based on the action principle. Similarly, only certain aspects of the rapidly growing field of the theory of dispersion relations are covered. It is with a mention of the Mandelstam representation for the two-particle scattering amplitude that the book concludes. However, some of the topics not covered in the chapters proper are alluded to in the problem section.

Notation For the reader already accustomed to a variety of different notations, an indication of our own notation might be helpful. We have denoted by an overscore the operation of complex conjugation so that a denotes

XIV

PREFACE

the complex conjugate of a. Hermitian conjugation is denoted by an asterisk: (a*) = ajl. Our space-time metric is such that goo = — gn = — g22 = — g33 = 1, and we have differentiated between covariant and contravariant tensors. Our Dirac matrices satisfy the commutation rules Yyv -J- yvy>1 = 2g^. The adjoint of a Dirac spinor u is denoted by u, with u = u*7°.

Acknowledgments It is my pleasant duty to here record my gratitude to Drs. George Sudarshan, Oscar W. Greenberg and A. Grossman who read some of the early chapters and gave me the benefit of their criticism, and to Professor S. Golden and my other academic colleagues for their encourage¬ ment. I am particularly grateful to Professor Kenneth Ford, who read most of the manuscript and made many valuable suggestions for improv¬ ing it. I am indebted to Drs. Bethe and de Hoffmann for their consent to use some of the material of Volume I of Mesons and Fields, to the Office of Naval Research for allowing me to undertake this project in the midst of prior commitments and for providing the encouragement and partial support without which this book could not have been written. I am also grateful to Mrs. Barbara MacDonald for her excellent typing of the manuscript; to Mr. Paul Hazelrigg for his artful execution of the engravings; and to The Colonial Press Inc. for the masterly setting and printing of a difficult manuscript. I would like to thank particularly the editorial staff of the publisher for efficient and accurate editorial help and for cheerful assistance which made the task of seeing the manuscript through the press a more pleasant one. Above all, I am deeply grateful to my wife, who offered constant warm encouragement, unbounded patience, kind consideration and understand¬ ing during the trying years while this book was being written. For the second printing of this first edition, I have had an invaluable list of corrections from Professor Eugene P. Wigner, of Princeton Univer¬ sity, and from others, for which I am sincerely thankful. Most of these have been incorporated in this edition. Silvan S. Schweber

Lincoln, Mass. August, 1962

Part One

THE ONE-PARTICLE EQUATIONS

1 Quantum Mechanics and Symmetry Principles la. Quantum Mechanical Formalism Quantum Mechanics, as usually formulated, is based on the postulate that all the physically relevant information about a physical system at a given instant of time is derivable from the knowledge of the state function of the system. This state function is represented by a ray in a complex Hilbert space, a ray being a direction in Hilbert space: If |\k) is a vector which corresponds to a physically realizable state, then |\k) and a constant multiple of |'k) both represent this state. It is therefore customary to choose an arbitrary representative vector of the. ray which is normalized to one to describe the state. If I'k) is this representative, the normalization condition is expressed as ('h | 'k) = 1, where (x | T) = ('k | x) denotes the scalar product of the vectors |x) and |'k).1 If the states are normalized, only a constant factor of modulus one is left undetermined and two vectors which differ by such a phase factor represent the same state. The system of states is assumed to form a linear manifold and this linear character of the state vectors is called the superposition principle. This is perhaps the fundamental principle of quantum mechanics. A second postulate of quantum mechanics is that to every measurable (i.e., observable) property, a, of a system corresponds a self-adjoint oper¬ ator a = a* with a complete set of orthonormal eigenfunctions |a') and real eigenvalues o', i.e., a | a') — a' | a') (a' | a") = 8a'a"

(1)

(2) (3)

The symbol ba’a" is to be understood as the Kronecker symbol if a' and a" lie in the discrete spectrum and as the Dirac b function, 2), etc. are normalized representatives of these rays, by Schwartz’s inequality |(^i, \k2) |2 < 1, he., the transition probability from |Ti) to |T2) is less than one, which cannot be if they represent the same state. Therefore |^i), |T2), etc. must be constant multiples of each other. It may, however, be the case that there exist rays in Hilbert space which do not correspond to any physically realizable state. This situa¬ tion occurs in relativistic field theories or in the second quantized formula¬ tion of quantum mechanics. In each of these cases the Hilbert space of rays can be decomposed into orthogonal subspaces 3CA, • • • such that the relative phase of the component of a vector in each of the sub¬ spaces is arbitrary and not measurable. In other words, if we denote by | A, l) the basis vectors which span the Hilbert space 3Ca, and by |B, j) the basis vectors which span 3CS, etc., then no physical measurement can differentiate between the vector

bj | B,j) ® ■■■

at | A, l) @ i

j

and the vector ^ i

die™ | A, l) © 2 J

I

j) © • • ‘

6

QUANTUM

MECHANICS AND SYMMETRY PRINCIPLES

[la

where a, /3, • • • are arbitrary phase factors. The phenomenon responsible for the breakup of the Hilbert space into several incoherent orthogonal subspaces is called a superselection rule [Wick (1952), Wigner (1952a), Bargmann (1953)]. A superselection rule corresponds to the existence of an operator which is not a multiple of the identity and which commutes with all observables. If the Hilbert space of states, 3C, decomposes for example into two orthogonal subspaces, 3Ca and 5Cb, such that the relative phases of the components of the state vector in the two subspaces is com¬ pletely arbitrary, then the expectation value of a Hermitian operator that has matrix elements between these two subspaces is likewise arbitrary when taken for a state with nonvanishing components in 3Ca and 3CbNow for a quantity to be measurable it must surely have a well-defined expectation value in any state. Therefore, a Hermitian operator which connects two such orthogonal subspaces cannot be measurable. An ex¬ ample of this phenomenon is the Hilbert space which consists of the states of 1, 2, 3, • • • , n, ■ ■ ■ particles each carrying electric charge e. The orthogonal subsets then consist of the subspaces with definite total charge and a Hermitian operator connecting subspaces with different total charge cannot be observable. The superselection rule operating in this case is the charge conservation law, or its equivalent statement: gauge invariance of the first kind (Sec. 7g). An equivalent formulation of the above consists in the statement that all rays within a single subspace are realizable but a ray which has com¬ ponents in two or more subspaces is not. If not all rays are realizable, then clearly no measurement can give rise to these nonrealizable states. They cannot therefore be eigenfunctions of any Hermitian operator which corresponds to an observable property of the system. To be observable a Hermitian operator must therefore satisfy certain conditions (super¬ selection rules). Ordinary elementary quantum mechanics operates in a single coherent subspace, so that it is possible to distinguish between any two rays and all self-adjoint operators are then observable. Quantum mechanics next postulates that the position and momentum operators of a particle obey the following commutation rules:

[qh p,]

=

ihdij

(l,j = 1,2,3)

(4)

For a particle with no internal degrees of freedom, it is a mathematical theorem [Von Neumann (1931)] that these operators are irreducible, meaning that there exists no subspace of the entire Hilbert space which is left invariant under these operators. This property is equivalent to the statements that any operator which commutes with both p and q is a multiple of the identity and that every operator is a function of p and q. The description of the system in terms of the observables p and q is complete.

la]

7

QUANTUM MECHANICAL FORMALISM

finally, quantum mechanics postulates that the dynamical behavior of the system is described by the Schrodinger equation (5)

ihdt \]t) = H\]t)

where dt = d/dt and H, the Hamiltonian operator of the system, corre¬ sponds to the translation operator for infinitesimal time translations. By this is meant the following: Assume that the time evolution of the state vector can be obtained by the action of an operator U(t, to) on the initial state | ; to) such that | t) = U (t, t0) | t0) (6a) U(t0, to) = 1

(6b)

Conservation of probability requires that the norm of the vector | t) be constant in time: (t\t) = (to \ to)

= (to | U*(t, to) U(t, to) | to)

(7)

and therefore that U*(t, to) U(t, to) = 1

This does not yet guarantee that U is unitary. the following equation must also hold:

(8a) For this to be the case,

U(t, to) U*(t, to) = 1

(8b)

This condition will hold if U satisfies the group property: U(t, k) U(k, to) = U(t, to)

(9)

If, in Eq. (9), we set t = t0, and assume its validity for t0 < k, we then obtain U(U, k) U(k, k) = 1 (10a) whence U(to, k) = U-Kk, to) (10b) and multiplying (10a) on the left by U*(to, k) using (8) we obtain U(k, to) = U*(to, k) - U-\k, to)

(10c)

so that U is unitary. If we let t be infinitesimally close to to, with t — t0 = dt then to first order in dt we may write U(to + St, to) = 1 - | HSt

(11)

In order that U be unitary, H must be Hermitian. The dimension of H is that of an energy. Equation (6a) for the infinitesimal case thus reads |f0 + St) — \to) = — Ji Hdt |[f0)

(12a)

8

QUANTUM MECHANICS AND SYMMETRY PRINCIPLES

which in the limit as

[la

8t —» 0 becomes Eq. (5) since, by definition, lim (5f)-1(|f +

dt) — 1f)) = dt | f)

(12b)

8t —>0

lb. Schrodinger and Heisenberg Pictures In the previous remarks about quantum mechanics, we have defined the state of the system at a given time f by the results of all possible experi¬ ments on the system at that time. This information is contained in the state vector | t)s — |\Srs(Q). The evolution of the system in time is then described by the time dependence of the state vector which is governed by the Schrodinger equation

Hs | ¥s(f)> = ihdt | ¥*(

(13)

The operators corresponding to physical observables, Fs, are time-inde¬ pendent; they are the same for all time with dtFs = 0. This defines the Schrodinger picture and the subscript S identifies the picture [Dirac (1958)]. Although the operators are time-independent, their expectation value in any given state will in general be time-dependent. Call

(Fs) = ('ks(f) | Fs | >7s(£))

(14)

then

ih

I{Fs)

=

I

s] I

[Fs• H

In the Schrodinger picture we call, by definition,

ff), will then be time-independent, i.e., dt \ 4>//) = 0. The operator 7(f) being unitary, the expectation value of the operator Fs in terms of |U/) is given by

Ik]

9

SCHRODINGER AND HEISENBERG PICTURES

(F) =

Fs*s(t)) = (V(t) *s(t), V{t) Fs*s(t))

= ($h, V(t) Fs7_1(i)

(35)

The unitary transformation function (q' | p') which permits us to trans¬ form from the q representation to the p representation is obtained by tak¬ ing the scalar product of (32) with the bra (q'|

p' = S‘*’(q' - q")

(38)

The wave function T(q') = (q' | T) in configuration space is thus related to the momentum space wave function (p') = (p' | T) by the familiar Fourier transformation

'b(q') =

—ihv

(50) (51)

lc]

13

NONRELATIYISTIC FREE-PARTICLE EQUATION

and the resulting expression is to operate on ^(q; t) the wave function describing the particle. The solution of (47) is given by | ]t) =

| .q

(52)

Thus the time displacement operator U(t, t0) is here given by exp —

(t — t0) .

In the q representation, we may write

(q I ; 0 = / dqo (q I U(t, to) I q0) (qo | J Q

(53)

^(q; t) = I dq0 K(qt; q0to) ^(q0; t0)

(54)

or equivalently

where K(qt] q0t0) = (q | U(t, t0) | q0) 1

(2irh)3

dpe*

' p ■ (q -

qo)e_7 2m

{t~k)

(55)

From the fact that U(t, t) = 1 it follows that K(qt; q0t) 5(3)(q — q0), which is clearly satisfied by (55), as required in order that (54) be an iden¬ tity for t — t0. Now Eq. (52) is defined only for t ^ to, so that K is sim¬ ilarly only defined for t ^ t0. It is convenient to require that K = 0 for t < to. We can incorporate this boundary condition by writing K(qt) qoto) = e(t — to) (q | U(t, t0) \ q0)

(56)

where 6(t) is a step function defined as follows: 0(0 = 1

if t > 0

= 0

if t < 0

(57)

so that m.,«

(58)

The differential equation obeyed by K is now easily derived from Eq. (56) ihdt K(qt] qo x + a

t —> t T r

(60a) (60b)

Experiments have also yielded the fact that space is isotropic so that the orientation in space of an event is an irrelevant initial condition and this principle can be translated into the statement that the laws of motion are invariant under spatial rotations. Newton’s lawT of motion further indi¬ cated that the state of motion, as long as it is uniform with constant veloc¬ ity, is likewise an irrelevant initial condition. This is the principle of Galilean invariance which asserts that the laws of nature are independent of the velocity of the observer, and more precisely, that the laws of motion of classical mechanics are invariant with respect to Galilean transforma¬ tions. These symmetry principles are usually stated in terms of two ob¬ servers, 0 and O', who are in a definite relation to each other. For ex¬ ample, observer 0 may be moving with constant velocity relative to O' in such a way that the relation of the labels of the points of space and the reading of the clocks in their respective co-ordinate systems is given by the following equations: x' = x — vf

(61a)

t' = t

(61b)

The principle of Galilean invariance then asserts that the “laws of nature” are the same for the two observers, i.e., that the form of the equations of motion is the same for both observers. The equations of motion must therefore be covariant with respect to the transformations (61a) and (61b). Two observers using inertial co-ordinate systems (i.e., one in which the laws of motion are the same) are said to be equivalent. The aforementioned invariance principles were experimentally estab¬ lished and may have limited applicability. Thus, Lorentz invariance

1^]

SYMMETRY AND QUANTUM MECHANICS

15

has replaced the principle of Galilean invariance and the discovery of the nonconservation of parity in weak interactions has re-emphasized that an invariance principle and its consequences must be experimentally verified. At the macroscopic level, the notion of an invariance principle can be made precise and explicit with the help of the concept of the complete description of a physical system. By the latter is meant a specification of the trajectories of all particles together with a full description of all fields at all points of space for all time. The equations of motion then allow one to determine whether the system could in fact have evolved in the way specified by the complete description. As stated by Haag [un¬ published, but quoted in Wigner (1956)], an invariance principle then requires that the following three postulates be satisfied: 1. It should be possible to translate a complete description of a physical system from one co-ordinate system into every equivalent co-ordinate system. 2. The translation of a dynamically possible description should again be dynamically possible. 3. The criteria for the dynamic possibility of a complete description should be identical for equivalent observers. Postulate 2 is equivalent to the statement that a possible motion to one observer must also appear possible to any other observer, and postulate 3 to the statement of the form invariance of the equation of motion. In a quantum mechanical framework, postulate 1 remains as stated. It implies that there exists a well-defined connection and correspondence between the labels attributed to the space-tiihe points by each observer, between the vectors each observer attributes to a given physical system, and betvreen observables of the system. Postulate 2 is usually formulated in terms of transition probabilities, and states that the transition probabil¬ ity is independent of the frame of reference. In other words, different equivalent observers make the same prediction as to the outcome of an experiment carried out on a system. Note that this system will be in a different relation to each of the observers. Observer 0 will attribute the vector | To) to the state of the system, whereas observer O' will describe the state of this same system by a vector |T0'). We shall, however, assume that given two systems So and So' which are in the same relation to each of the two observers (i.e., the values of the observable of system So as measured by observer 0 are the same as the values of the observables of S0' as measured by observer O'), the observers will describe the state of their respective systems by the same vector. We shall call the vector |To') the translation of the vector |T0). Stated mathematically, postu¬ late 2 asserts that if |T0) and |o) are two states and |T0') and |0') their translations, then

16

[Id

QUANTUM MECHANICS AND SYMMETRY PRINCIPLES

K'J'o, $o)|2 = 1(^0', $0')|2

(62)

If all rays in Hilbert space are distinguishable, it then follows from Eq. (62) as a mathematical theorem [Wigner (1959)] that the correspondence I’J'o) —> |^o') is effected by a unitary or an antiunitary2 operator, U{0', 0), |*o'> =

U(Of, 0) | *0>

(63)

where U depends on the co-ordinate systems between which it affects the correspondence and U(0', 0) = I if O' = 0. Postulate 3 now asserts that U can only depend on the relation of the two co-ordinate systems and not on the intrinsic properties of either one. For example, for Lorentz transformations, U{0', 0) must be identical with U(0"', 0") if observer O'" is in the same relation to 0" as observer O' is to 0, i.e., if O'" arises from 0" by the same Lorentz transformation, L, by which O' arises from 0. If this were not so there would be an intrinsic difference between the frames O', 0 and O'", 0". The operator U is completely determined up to a factor of modulus unity by the transformation, L, which carries 0 into O'. We write |*o'> =

U(L) | tf0)

(64)

with U(L) = I if L is the identity transformation, i.e., if 0 and O' are the same co-ordinate systems. If we consider three equivalent frames, then we must obtain the same state by going from the first frame 0 to the second O' — L\0 and then to the third 0" — L20', and by going di¬ rectly from the first to the third frame 0" = L^O, Z

/3

L2L\

—

(65)

Hence |4>o") = =

U(L2) UiLx) | ¥o> U (L ) | 'Fo) 3

(

)

66

from which it follows

U(L3) = co(L2, U) U(L2) U{Lx) U(l) = I

(67a) (67b)

where w is a number of modulus one which can depend on Li and L and arises because of the indeterminate factor of modulus one in the state vectors. A set of Us which satisfy (67a) and (67b) are said to form a 2

2 Recall that an operator U is said to be unitary if for every pair of vectors P, x in the Hilbert space, 3C, (t/'P, Ux) = ('P, x) and every vector in 3C can be written in the form U] = at/|p) +HU\x). Note that the product of two antiunitary operators is unitary.

Id]

17

SYMMETRY AND QUANTUM MECHANICS

(unitary or antiunitary) “representation up to a factor” of the group of transformations under which the observers are equivalent. For special relativity, for example, this group is the group of inhomogeneous Lorentz transformations. One is thus led to the mathematical problem of deter¬ mining all the representations up to a factor of the group of interest. It now follows from postulate 2 and from the fact that all the frames which can be reached by the symmetry transformation are equivalent for the description of the system, that together with |T0), U(L) | To) must also be a possible state of the system as described by observer 0. Thus, a relativity invariance requires the vector space describing the possible states of a quantum mechanical system to be invariant under all relativity transformations, i.e., it must contain together with every |T) all trans¬ forms U{L) | T) where L is any relativity transformation. This is the active view of formulating relativistic invariance [Bargmann (1953)] and it deals only with the transformed states of a single observer. Note that for the symmetry transformations which can be obtained continuously from the identity (i.e., no inversions), the transformed states can always be obtained from the original state by an actual physical operation on the system. Consider for example a Lorentz transformation along the x-axis with velocity v. The transformed state, which arises from the state |To) is given by Uiy) | To). This is the state of the system as seen by observer O'. It is, however, also a possible state of the system as seen by 0 and which can be realized by giving the system a velocity —v along the z-axis. If one deals with an inversion, e.g., time inversion, no such oper¬ ation is in general possible. The invariance of the theory under this sym¬ metry operation then essentially postulates the existence of this trans¬ formed state without necessarily giving a procedure for its realization. For quantum mechanical applications, the importance of determining all unitary representations of a relativity group comes from the fact that the knowledge of such a unitary representation can in effect replace the wave equation for the system. For if in the above discussion in the frame 0, we used a description of our system in the Heisenberg picture in which the Heisenberg state |T)# coincides with |T(0))s, the Schrodinger state at time t = 0, then the Schrodinger state vector at time t0 can be obtained by transforming to a frame O' for which t! == t — to while all other co¬ ordinates remain unchanged. If L is this transformation then |T(io))s =

U(L) | T(0))s

(68)

Thus a determination of all unitary representations of the inhomogeneous Lorentz group [Wigner (1939), Bargmann (1948), Shirokov (1958a, b)] is equivalent to a determination of all possible relativistic wave equations. To clarify these concepts further, we consider in the next section the representations of the three- and four-dimensional rotation group.

18

QUANTUM MECHANICS AND SYMMETRY PRINCIPLES

[le

le. Rotations and Intrinsic Degrees of Freedom The relation between the labels of the points of three-dimensional space for two observers whose co-ordinate systems are rotated with respect to one another about a common origin, is given by x' = Rx

(69a)

or 3 X %

ikXk

—

r ikXk

(69b)

k= 1

(We use the summation convention over repeated indices.) We call R a rotation. The length of a vector and the angle between vectors are pre¬ served under rotations, i.e., x'iy'i = Xiyi (70) therefore

RRT = RTR = I

(71)

and rotations are represented by orthogonal matrices.

It follows from

(71) that det RRT = det RT det R = (det R)2 = 1 so that det R = ±1. A rotation for which det R = +1 is called a proper rotation, one for which det R = -1 an improper one. An example of the latter is an inversion of the co-ordinate system about the origin repre¬ sented by /-I

=( \

0

o

0

0

-1

0

o-i

with (R-)2 = +1. R- corresponds to a transition from a right-handed to a left-handed co-ordinate system. Every improper rotation R' with det R' = — 1 can be written in the form (R'R_) E_, i.e., as the inversion R- followed by a proper rotation, since det R'R_ = det R' det /?_ = = ( — l)2 = +1. The set of all proper rotations in Euclidean three-space forms a group: the rotation group. The group of all rotations together with reflections is called the orthogonal group. Since each element of the group can be specified by three continuously varying parameters (e.g., the direction cosines of the axis about which the rotation takes place and the angle of rotation), the rotation group is a continuous three-parameter group. The number of parameters of a group is called the dimension of the group. We wish to determine all the representations of the rotation group. In general, a representation of a group G is a mapping (correspondence) which associates to every element g of G a linear operator Tg in a certain vector space V, such that group multiplication is preserved and the iden-

le]

19

ROTATIONS AND INTRINSIC DEGREES OF FREEDOM

tity e of G is mapped into the identity I in V.3 That is, if e, gi, g2, gf3, etc., are the elements of G and if to these elements are associated the linear operators Te, Tffi, T0i, • • • etc. in V, these operators are said to form a representation of the group G if , and

Te = I TgTQi

=

T0iB2

(73a)

(73b)

If Tg is represented by a matrix one speaks of a matrix representation. In quantum mechanics one is actually interested in a ray correspondence in which case Tg and exp (iag) ■ Tg, with ag an arbitrary real constant, represent the same correspondence. In this case Eq. (73b) is replaced by TgTg^ = co(gv g2) T^. It has, however, been shown by Wigner (1959) that one can determine from a ray correspondence an essentially unique vector correspondence by a suitable normalization. Bargmann (1954) has furthermore shown that for the groups of interest for physical applications (rotation, Galilean, Lorentz group) with a suitable choice of Tg (recall that T0 and Tg exp ia represent the same correspondence), « is either equal to ±1 (restricted Lorentz group, rotation group) or it can be ex¬ pressed by a fairly simple expression (Galilean group). A subspace Vi of V is said to be invariant under the representation T„ if all vectors, v, in Vi are transformed by Tg into vectors, v', again in Fx, and this for all Tg. If the only subspaces of V which are invariant under the representation g —> Tg consist of the entire space and the subspace consisting of the null vector alone, we say that the representation is ir¬ reducible. It is a theorem, which we state without proof [see Gel’fand (1956) or Wigner (1959)], that it is always possible to define a scalar product in V such that the representations of the rotation group in V are unitary,4 i.e., such that the operators Tg are all unitary: Tg* = Tg~l = Tg-i. Fur¬ thermore, the study of such unitary representations for compact groups can be reduced to the study of irreducible representations. For if there exists'a subspace Fi of V invariant under Tg, then the orthogonal comple¬ ment of Vi, Vp, i.e., the set of all vectors orthogonal to Vh is also invariant under Tg. Proof: If v\ is an element of Fx and w an element of Vp, since Tg is unitary we have 0 = (vh w) = (7>x, Tgw). Now by assumption, Tgv is again an element vf of Fx, therefore (vf, Tgw) = 0 for arbitrary Tg, 1

3 We shall always only consider continuous representations, i.e., representations T„ such that (v, T0w) is a continuous function of g for every pair of vectors v, w in V, now assumed to have a scalar product defined in it. 4 The possibility of introducing this scalar product depends in an essential manner on the finiteness of the group volume, i.e., on the compactness of the group, for it in¬ volves an integration over the group manifold. Loosely speaking, a matrix group is said to be compact if the matrix elements of every group element (i.e., of the matrix representing this group element) are bounded. 1 his is clearly the case for the rotation group but is not the case for the Lorentz group. [Compare Lq. (1.78) and Eq. (2.10).]

20

[le

QUANTUM MECHANICS AND SYMMETRY PRINCIPLES

and therefore for all Tg. Hence the set of vectors Tgw for all re in 1 are elements of TV, and TV is therefore invariant under Tg. Thus V has been split into two invariant subspaces. In many cases, this process can be continued until one deals with only irreducible representations. For compact groups (and therefore for the rotation group in particular) it is known [see, e.g., Pontrjagin (1946)] that this inductive process of decomposing invariant subspaces into invariant subspaces terminates: The irreducible representations are all finite dimensional and every repre¬ sentation is a direct sum5 of irreducible finite dimensional representations. Finally it should be noted that one is only interested in inequivalent ir¬ reducible representations. Two representations T and T' are said to be equivalent if there exists a one-to-one correspondence, v v', between the vectors of the representation spaces such that if v corresponds to v' the vector Tgv corresponds to Tgv’ for all g and all pairs of vectors v, v'. This one-to-one correspondence can be represented by a (unitary) operator M, i.e., v' = Mv and v = M~lv'. For equivalent representations MTgv = = Tg'v' = MTgM~lv' for all v. Two representations are thus equivalent if there exists an M such that Tg' = MTgM~l. Two equivalent representa¬ tions can be considered as the realizations of the same representation in terms of two different bases in the vector space. Now every rotation is a rotation about some axis so that a rotation can be specified by giving the axis of rotation about which the rotation is made and the magnitude of the angle of rotation. A rotation can thus be represented by a vector X, where the direction of the vector specifies the direction of the axis of rotation and the length of the vector the magnitude of the angle of rotation. A rotation about the I-axis is thus represented by a vector (X, 0, 0), a rotation about the 0-axis by (0, X, 0), etc. It is evident that if X = (Xi, X2, X3) is a rotation vector then |X| ^ n and that the set of all rotations fill a sphere of radius 7r. Distinct points in the interior of this sphere correspond to distinct rotations, whereas points diametrically opposed on the surface of the sphere correspond to the same rotation and must be identified. A group element can thus be considered a function of X, g = g(k) and similarly for a representation, Tg = T(\). Now X = 0 corresponds to the identity operation so that 1

n 0) = I

(74)

Infinitesimal rotations about an axis will play a fundamental role in the following. Their importance derives from the fact they generate oneparameter subgroups and that any finite rotation can be constructed out 6 If

D, Du

Z>2 are three square matrices,

and one writes

D

=

Dx

® D2.

D

is said to be the direct sum of

Dx

and D2 if

le]

21

ROTATIONS AND INTRINSIC DEGREES OF FREEDOM

of a succession of infinitesimal ones. It is to be noted that infinitesimal rotations commute with one another whereas finite rotations in general do not. Let E(3)(0) be a rotation through the angle 0 about the 3-axis, and let us define

A'

“

IeR's'm 0

(75) =

0

One calls A3 the generator for an infinitesimal rotation about the S-axis. Note that for e infinitesimal we may write i2(3)(t) = 1 + A3e + terms of order e2

(76)

Now a rotation through the angle d about the 3-axis, R(3)(0), can be considered to occur in n steps, each step consisting of a rotation through an angle 6/n. We may therefore write

e6M

(77)

We can define the generators for infinitesimal rotations about the 1- and #-axes in a similar fashion. Explicitly, since

(

cos 6 — sin 6

0

sin 6 cos 6 0

°\

(78)

° 1/

/ 0 l-i

1 0

°\ 0

0

0

0

(79a)

,

and similarly,

A\ =

0

-1\

0 0

0 0

(79b)

/

One verifies that the generators A* (t = 1, 2, 3) satisfy the following com¬ mutation rules among themselves:

[Ai, Aj] = —eijkAk

(80)

where etjk is the totally antisymmetric tensor of rank three which is equal to +1 if Ijk is an even permutation of 123, -1 if Ijk is an odd permutation of 123, and zero otherwise. It should be noted that the reflection oper¬ ator E_, Eq. (72), commutes with all rotations

[R-, Ad = 0

for i = 1, 2, 3

(81)

Rotations about an axis form a commutative one-parameter subgroup of the group of rotations. In general, a one-parameter subgroup, a(t), of a group G, is a “curve” in the group (i.e., a continuous function from the real line into G) such that

22

QUANTUM MECHANICS AND SYMMETRY PRINCIPLES

ait) o(s) = a(t + s)

[le

(82)

Clearly a(0) = e, the identity, a( — A (a), which associates to each element a of the algebra a linear operatoi A (a) in a vector space V, such that A (a + /3) = A (a) + A(0)

(90a)

A(ca) = cA (a)

(90b)

A ([a/3]) = [A (a), A 03)] = A (a) A(/3) - A(/3) A (a)

(90c)

i.e., the bracket operation is mapped into commutator which automatically satisfies Eq. (85c). A representation of the Lie algebra of a group will uniquely determine a representation of the group. Let us illustrate these remarks with the rotation group. The Lie algebra of the rotation group is generated by the three linearly independent operators A1} A2, A3 satisfying Eq. (80) and these operators generate the one-parameter subgroups of rotations about the three spatial axes. An infinitesimal rotation about e through the angle |e| can be rep¬ resented by #(e) = 1 + eiAi + e2A2 + e3A3 + 0(e2) = 1 +

+ • • •

(91)

For a representation we shall write

T(e) = T{e 1,

€2, €3)

= I + eiMi + e2M2 + e3M3 + 0(«2)

(92)

where the M& constitute a representation of the generators of the Lie algebra and satisfy the commutation rules

[Mi,Mj] = -eijkMk

(93)

Let us next show that T(X) for arbitrary X is completely determined by

24

QUANTUM MECHANICS AND SYMMETRY PRINCIPLES

[le

the generators Mi, Mi, M3 and by A, and is given in terms of these quanti¬ ties by T{\ 1, X2> X8) = eXlMl+X2M2+X3M3

(94)

Proof: Since two rotations about the same axis commute R(s\) R(tX) = i?((s + t) A)

(95)

T(sA) T(t\) = T((s + 0 A)

(96)

and therefore, similarly

Upon differentiating both sides of this last equation with respect to s, replacing on the right side the differentiation with respect to s by one with respect to t, and setting s = 0 thereafter we obtain using Eq. (92) | T(t\) = j- T(s\) • T«A) at as s=0

= (M1A1 T Mfki T M3K3) T(t\)

(90

Equation (97) is a differential equation determining T(t\). The solu¬ tion of this differential equation which satisfies the boundary condition T(0) = I, Eq. (74), is precisely given by Eq. (94). For unitary representations, the requirement that the Ts be unitary implies that the M3 are skew-Hermitian, that is

Mj* = —Mj

(98)

The operators J, = —iMi are thus Hermitian and satisfy the familiar commutation rules of angular momenta: \rJ l, Jrri\

'Itlmn'J n

(99)

Now the problem of finding all irreducible representations of the rota¬ tion group is equivalent to finding all the possible sets of matrices J\, J2, J3 which satisfy the commutation rules (99). Clearly every irreducible rep¬ resentation of a continuous group will also be a representation in the neigh¬ borhood of the identity (infinitesimal transformations) although the con¬ verse is not necessarily true. In general, if we find all the irreducible representations of the group G in the neighborhood of the identity, i.e., find all the representations of the infinitesimal generators, then we can obtain all the irreducible representations of the entire group by exponentia¬ tion, Eq. (94). However, it is possible that some of the irreducible repre¬ sentations of G obtained in this manner are not continuous over the whole group but are continuous only in the neighborhood of the identity. These discontinuous representations must then be discarded. In the theory of group representations by complex matrices, Schur’s lemma [see Wigner (1959)] is of fundamental importance. It asserts that the necessary and sufficient condition for a representation to be irreducible is that the only operators which commute with all the matrices of the rep-

le]

ROTATIONS AND INTRINSIC DEGREES OF FREEDOM

25

resentation be multiples of the identity operator. Suppose that the Lie algebra of a group G contains an element A which commutes with all other elements of the Lie algebra. Let g —> T(g) be a representation of G in a vector space V. The operators (dT(g(s))/ds)a=o = a form a representa¬ tion of the Lie algebra of G. The operator which corresponds to A in this representation commutes with all other operators a, and consequently commutes with all operators T(g) (such commuting elements will be called invariants of the group). Because of Schur’s lemma, then, a representa¬ tion is irreducible if and only if the vector space on which the representa¬ tion is defined is spanned by a manifold of eigenfunctions belonging to a single eigenvalue of this commuting operator. Conversely, if we find all the independent invariants of the group and construct a representation whose representation space is spanned by eigenfunctions belonging to the same eigenvalue of each of the invariants, then this representation will be irreducible, since each of the invariants is a multiple of the identity in this representation and by definition there are no other operators which com¬ mute with all the elements of the group. To each set of eigenvalues of all the invariants there thus corresponds one and only one irreducible rep¬ resentation. The problem of classifying the irreducible representations of the group is therefore reduced to finding the eigenvalue spectra of the invariants of the group. For the proper rotation group, J2 = J\ + Jt + Jz1 commutes with each of the generators and it therefore is an invariant of the group. Its eigenvalues, as is well known from the theory of angular momenta, are j(j + l) where j = 0, i, 1, -f, 2, • • • . Every irreducible representation is thus characterized by a positive integer or half-integer value including 0, the dimension of the representation being 2j T 1 and for each j, integer or half-integer, there is an irreducible representation. In order to classify the irreducible representations of the orthogonal group we note that T_, the linear operator corresponding to the inversion operation E_, commutes with all rotations. By Schur’s lemma, in every irreducible representation it must be a constant multiple of the identity. An irreducible representa¬ tion of the orthogonal group is thus classified by a pair of indices (j, t) where the second index is the eigenvalue of T_ in that representation. For integer j, one has t = ±1 (since TJ = I) and there exist two different irreducible representations of the orthogonal group for each integer j. For one of these T_ = +/ and for the other T— — —I. For j _ o the representation is one dimensional, every group element is mapped into the identity and the infinitesimal generators are identically zero. We call the representation for which T_ = +/ the scalar repre¬ sentation, that for which T_ = -I the pseudoscalar. For j = i the representation of the rotation group is two dimensional and the infinitesimal generators Af,