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Havward, Raymond W/The dynamics of field * QC100 .U556 V154;1976 C.I NBS-PUB-C 1976 NBS MONOGRAPH 154

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

The Dynamics of Fields

of Higher Spin

Vqc

100

.U556

NU.15^

C.2

1976
NATIONAL BUREAU OF STANDARDS

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The Dynamics of Fields of Higher Spin

Raymond W. Hayward

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Elliot L Richardson, Secretary

Edward 0. Vetter, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued August 1976
Library of Congress Cataloging in Publication Data

Hayward. Raymond W. 1921- The dynamics of fields of higher spin.

(National Bureau of Standards monograph; 154) Includes bibliographical references. Supt. of Docs, no.: C 13.44:154

1. Quantum field theory. 2. Nuclear spin. 1. litle. II. Series: United States. National Bureau of Standards. Monograph: 154. QC100.U556 no. 154 [QC174.45 1 602'.ls [530.r43] 76-608089

National Bureau of Standards Monograph 154

Nat. Bur. Stand. (U.S.), Monogr. 154, 99 pages (Aug. 1976) CODEN: NBSMA6

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1976

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Stock No. 003-003-01612-8 Price 11.70 (Add 25 percent additional for other than U.S. maiUng).
Contents

page 1. Introduction 1

2. Lorentz Transformations 4

3. Representation of Operators and Eigenvectors 14

4. Lagrangian Formalism 31

5. Free Field Solutions and the Quantization of the Field 45

6. Discrete Symmetries 58

7. References 68

List of Tables

page Table I The matrix operators 5/*', 5/'', B, and / corresponding to fields of spin |, 1, f

,

and 2 70

Table II Rotation and boost operators for fields of spin ^, 1, f, and 2. Included are the

hyperbolic functions involving the parameter in terms of p, E, and m. In- cluded also are the projection operators for the polarization component | cr |, denoted by "iP^ 72

Table III The matrix operators S/^'>', 5,^'^', B', and /' corresponding to a field of spin 1 in

a cartesian representation 74

Table IV Plane wave, positive energy spinors u''"Hp) for fields of spin |, 1, f, and 2. with

a nonzero mass 75

Table V Plane wave, positive energy, charge conjugate spinors v'-'^p) for fields of spin

^, 1, f, and 2 and their relation to the negative energy and momentum spin- ors ( —p) with a nonzero mass 77

Table VI Plane wave, positive energy spinors w'-'\p) for fields of spin |, 1, f, and 2 with

a nonzero mass 79

Table VII Plane wave, positive energy, charge conjugate spinors j^'''(p) for fields of spin

^, 1, f, and 2 and their relation to the negative energy and momentum spin- ors p) with a nonzero mass 81

Table VIII Completeness and orthogonality relations involving spinors of the spin ^ field with a nonzero mass 83

Table IX Completeness and orthogonality relations involving spinors of the spin 1 field with a nonzero mass 84

Table X Completeness and orthogonality relations involving spinors of the spin f field with

a nonzero mass 85

Table XI Completeness and orthogonality relations involving spinors of the spin 2 field with a nonzero mass 86

Table XII Plane wave positive energy spinors z''''(p) for the gauge fields corresponding to

fields of spin 1, f , and 2 with a nonzero mass 87

Table XIII Plane wave positive energy spinors m^''Hp) and t<;^''^(p) for zero-mass fields of spin

I, 1, f, and 2 in the "physical gauges." 88

Table XIV Plane wave positive energy spinors for the zero-mass spin | parity nonconserving

field 91

Table XV Plane wave positive energy spinors u^'''(p) for the zero-mass spin 1 field in the

"nonphysical" covariant gauge 92

Table XVI Some relationships between spinors in the indefinite metric. 93

Table XVII Transformation properties of bilinear covariants and the energy-momentum

tensor under space inversion, time reversal, and charge conjugation and their product 94

iii

The Dynamics of Fields of Higher Spin

Raymond W. Hayward

Institute for Basic Standards, National Bureau of Standards, Washington, D. C. 20234

There are several difficulties that plague all existing relativistic equations of motion describing

elementary fields having an intrinsic spin greater than one. While the free field equations can be shown

to be explicitly covariant, the introduction of interactions gives rise to a phenomenon of noncausality. In the presence of interactions, the retarded solutions spread beyond the light cone and the influence

travels faster than light. Furthermore, the solutions in certain simple potentials do not have a finite norm, violating the probabilistic requirements of quantum mechanics.

This paper develops a relativistic theory that is free of the aforementioned difficulties. This

Lagrangian theory describes fields and particles with arbitrary mass and charge and having any dis- crete spin, integer or half integer. Apart from gauge conditions there are no subsidiary conditions. A matrix formulation is used. The generators of the inhomogeneous Lorentz group for a field of any intrinsic spin and mass are defined in terms of Wigner operators of the group SU(2) and a metric operator. A maximal Abelian set of invariants is formed which defines two completely re- ducible representation bases of the inhomogeneous Lorentz group having distinct structures. A set of 7

matrices, obeying a Cliff^ord algebra, is also defined in terms of the Wigner operators and the metric

operator. State vectors having different structures and Lorentz transformation properties can be related

to one another by operators involving the 7 matrices. The equations of motion can be obtained from the Lagrangian by variational methods, and

certain aspects of the canonical formalism can be used to quantize the fields. Invariance of the La- grangian under infinitesimal displacements and rotations yield conservation laws and constants of the motion for pertinent physical observables. The metric of the Hilbert space of the states is uniquely

defined for any spin field, assuring positive definite four momenta and charge.

The Dirac formulation for the spin one-half field and the Maxwell-Lorentz formulation for the electromagnetic field are special cases of this theory.

Key words: Causality; high spin fields; inhomogeneous Lorentz group; relativistic fields; wave

equations.

I. Introduction

Elementary particles of finite mass and with spin greater than one half have assumed a role of

increasing importance in physics in recent years. It is ironical, however, that no "true" dynamical

theory of such particles has emerged that enjoys the privilege of existing as an entity like the Dirac

theory of the spin one-half particle. There are a host of composite theories that are explicitly covariant

but possess certain intrinsic difficulties in application. These theories make explicit or implicit use of the invariance properties of the inhomogeneous Lorentz group and often of the one-to-two homomorphism between the homogeneous Lorentz group and the group SL(2c). There have been proposed

two classes of theories that yield first order equations of motion for the particles. The first of these is the Dirac-Fierz-Pauli type [1-3] ^ involving the usual four-dimensional Dirac matrices obeying a Clifford algebra. A set of Dirac matrices is used for each constituent of the composite state vector and each set

is orthogonal to all other sets. Specifically, for example, the formulation of Bargmann and Wigner

[4] contains extraneous components that must be constrained or eliminated to yield the required

number of degrees of freedom. The prescription for accomplishing this feat is not always unique, but

depends rather on the choice of irreducible combinations that are to be considered on the basis of

a preconceived model. Because of these auxiliary conditions the Bargmann-Wigner equations are some- what intractable, and when interactions are present they become both formidable and questionable. In addition there are some difficulties in formulating a Lagrangian theory.

The requirement that all differential equations of motion result from a variational principle in- volving an action integral requires the introduction of auxiliary fields into the Lagrangian. Only with

the requirement that these auxiliary fields vanish in the absence of interactions is an explicit Lagrangian

obtained. Fierz and Pauli [3] long ago recognized the difficulties arising from inconsistencies in this formulation.

1 Figures in brackets indicate the literature references at the end of this paper.

1
Free particles of spin values 0, 1/2, and 1 described by the Bargmann-Wigner equations are equivalently described by the Klein-Gordon, Dirac and Proca equations, respectively. Free particles having spins greater than one half are described only by a set of coupled differential

first order equations [5] . The number of equations may be reduced, however, by application of conditions

for a lower spin field. For example, a spin 3/2 particle may be described by a set of three coupled

equations involving a three-spinor. Conditions applicable to a spin-one field allow this three-spinor to be described in terms of a vector-spinor and an antisymmetrical tensor-spinor. This formalism proposed

by Rarita and Schwinger [6] allows the set of three coupled equations to be reduced to two. A second class of theories are characterized as the Bhabba-type [7]. In these, attempts are made

to avoid the auxiliary conditions by inclusion of only those irreducible composites that are necessary

to maintain Lorentz covariance together with the requirement that all components of the state vector obey a second order wave equation involving the same mass. This procedure requires matrices of higher

dimension than the Dirac matrices. These matrices obey a rather complicated algebra. For particles of

spin 0 and 1 these matrices are of dimension five and ten, respectively, and obey the Duffin-KemmerPetiau algebra [8]. The Bhabba-type equations of motion may be readily obtained from a Lagrangian,

however the representation of the Lagrangian is by no means unique. In fact, the Lagrangian used by

Kemmer might be questionable from the point of view of having an excessive number of powers of the

four-momenta needed to describe the fields. In any event, for spins greater than one there are profound

difficulties in a theory with interactions. Because of the attractiveness of the possibility for the elimination of subsidiary conditions there is still lively attention [9] being given to the first order Bhabba-type

theories. The problem remains to get a theory with interactions that describes a particle of unique mass

and charge without subsidiary conditions and without extraneous components. Both classes suffer from an inherent nonuniqueness in the selection of irreducible combinations required to express the

dynamics of a field of a particular spin.

Theories that lead to second order equations of motion, apart from the familiar Proca [10] and

Stuekelberg [11] formalisms tor spin one fields, have received less attention. These are usually tensor formulations [12] and, like the first order theories, usually have supplementary conditions invoked to limit the number of degrees of freedom. These theories also become formidable when interactions are introduced.

Velo and Zwanziger [13] have looked into the origin of some of the difficulties. Wave propagation

is usually associated with hyperbolic systems of partial differential equations. Such equations allow an

initial-value problem to be posed on a class of surfaces, called "space-like" with respect to the equations,

and they possess solutions with wave fronts that travel along rays at finite velocities. The rays through

any point form a ray cone that is entirely determined by the coefficients of the highest derivatives. Thus

for hyperbolic systems when coupling occurs only in lower derivatives, the ray cone is the same in the

interacting and free case. The free Klein-Gordon and Dirac equations are familiar examples of hyperbolic

systems, and so, when they are coupled through lower order derivatives, the ray cone remains the

light cone. On the other hand, for spins greater than one half, the free Lagrangian equations are not hyperbolic

but constitute instead a degenerate system because they imply constraints. However, it may be shown

that they are equivalent to a system of hyperbolic equations which describe the wave propagation,

supplemented by constraints that are conserved in time. But, if any low or nonderivative coupling is added to the free higher-spin Lagrangian, the resulting equations do not remain equivalent to a hyperbolic system with the light cone as the ray cone supplemented by the same number of constraints. There is at present no known example of a satisfactory equation with interaction for spin greater than

one. The case of spin one is marginal; some interactions appear to lead to satisfactory equations, but

others are unacceptable. Similar doubts were expressed long ago by J. W. Weinberg [14]. Note that the requirements of special relativity are not automatically satisfied by equations that transform covariantly.

This paper will develop a relativistic theory of higher-spin fields employing the variational methods

of classical Lagrangian field theory. The chief aim is to present an unambiguous method for constructing

a dynamical description of a field having any discrete spin, integer or half integer and including zero,

2
and having an arbitrary, but unique, mass including zero. The formahsm will be similar for all spins

while, at the same time, will have no supplementary conditions or inherent nonuniqueness except

those resulting from internal symmetries arising from variation of the Lagrangian.

One of the main considerations in elaborating relativistic quantum mechanics for particles comes from the fact that the law of conservation of the number of particles ceases, in general, to be true. To

produce a complete theory we must encompass in a single scheme dynamical states specified, not only

by the quantum state, but also by the number and the nature of the elementary particles of which they

are composed. This requires that the theory, to be useful, must be quantizable. If it can be a Lagrangian

type theory where the usual canonical methods apply, so much the better, for then the quantization

methods may be straighforward.

Another consideration in relativistic theories is concerned with positive definiteness of the probability density and the energy. In the early days of quantum mechanics, there was a belief that any

effective theory describing a particle with spin should yield a first order equation of motion in such

a way that the probability density should be positive definite at all times. Viewed in hindsight, the arguments for the superiority of first order equations over second order

equations interpreted as single particle theories are not convincing. Of course, at the time the Dirac

equation had the obvious advantage that it described most experimental facts involving spin one-half

particles. A quantized theory removes these apparent differences between first and second order equations

of motion, however neither is exempt from all difficulties, or even of contradictions.

One of the characteristics of any quantized relativistic theory describing particles of spin one or greater is the appearance of a negative metric for some of the components of the Hilbert space [15]. This leads to a negative probability of a different sort than that previously discussed. Dirac suggested

that, in a relativistic quantum theory, use should be made of an indefinite metric so that the negative

probabilities could be eliminated in order to maintain the probabilistic interpretation of the formalism

of quantum theory.

A more fundamental reason for the indefinite metric may be seen in the fact that, disregarding the

translations, the Lorentz group is a noncompact group, since there is no transformation that corresponds

to the limiting velocity, c. Any finite representation of a non-compact group requires a space with an

indefinite metric, although infinite representations with a definite metric may exist [16]. Although much of the contents of this paper may be based on arguments of group theory which

could be used extensively to make the exposition more economical, much of this paper is written in terms of conventional matrix theory. The use of matrix theory in specific representations makes the physical arguments clearer and closer to the import of Dirac theory and also makes them more easily

understood by people not well versed in spinor calculus and abstract algebra. In particular, we shall develop specific matrix representations of the inhomogeneous Lorentz group with the homogeneous

subgroup employing the nonunitary four dimensional orthogonal group 0(4) in a complex Minkowski

space in which = (x,it) where three of the six generators are regarded as specific representations

of the three dimensional rotations about the spatial axes and the remaining three generators are specific

representations of Lorentz transformations (boosts) along the three spatial axes. The use of 0(4) in a Minkowski metric allows all matrix operators in spin space to be hermitian in our treatment^.

The plan of the paper is as follows: In section 2 we establish and summarize the main features of

the inhomogeneous Lorentz group. We introduce a set of invariants that supplement those customarily

employed in order to obtain a set of commuting operators whose expectation values will completely

characterize a physical state of a particular spin and mass. In section 3 we first develop certain matrix representations of hermitian operators in a subspace

of the Hilbert space of the physical states in terms of Wigner operators of the rotation group SU(2)

2 Alternatively, we could have used in our formalism the group 0(3,1) to describe the homogeneous Lorentz subgroup in a real space in which = ((,x).

It should be pointed out that the apparently trivial differences in sign in the real four-dimensional Euclidian group 0(4) and the group 0(3,1) result in very important differences between the respective covering groups, SU(2) ® SU(2) which is homomorphic to real 0(4) and SL(2c) which is homomorphic to 0(3,1). On the other hand, the group 0(4) with the Minkowski metric has SL(2c) as its covering group, the same covering group as for 0(3,1). There is no funda- mental reason in special relativity, apart from personal taste, to choose 0(3,1) with a real metric over 0(4) with a Minkowski metric. The latter is that which

is used in the original Dirac-Pauli description of spin one-half fields.

3
and a metric operator. Using these hermitian submatrices we develop general rules to construct a set of 7 matrices obeying a Clifford algebra, and a set of spin operators S^y that form a six parameter Lie

algebra. We establish Lorentz transformation properties of the physical states with arbitrary spin using

the generators of the inhomogeneous Lorentz group. In section 4 we establish a uniform Lagrangian formalism for classical fields of any spin and mass. From this Lagrangian formalism, the equations of motion are obtained and conservation laws established

by means of variational methods and Noether's theorem.

In section 5 we obtain the plane-wave solutions for fields having several specific spin values and

determine the completeness and orthogonality properties of these solutions. The Hamiltonian formalism

is developed and the conditions for quantization of these fields is established in the Heisenberg picture. In section 6 we subject the quantized fields to the discrete symmetries of space inversion, time

reversal, and charge conjugation and find the transformation properties of these fields and bilinear combinations thereof. Furthermore we develop the commutation relations between the discrete symmetry operators and the generators of the inhomogeneous Lorentz group.

2. Lorentz Transformations

Of the many important invariance principles of relativistic quantum mechanics, the most fundamental is that which arises from the group of transformations denoted as the inhomogeneous Lorentz

group. In our canonical approach, it is necessary to consider the transformation properties of the

Hilbert space of the physical states, as well as those of the coordinates, in order to discuss the symmetries

of the Langrangian and the equations of motion obtained therefrom. The properties of the inhomogeneous group and its subgroups have been established and understood for many years since the classic paper of Wigner [17]. This understanding, however, has not led to a totally satisfactory dynamical

theory for particles and fields of higher spin. This section is devoted to establishing and summarizing

the main facts concerning the Lorentz group, and to introduce a supplement to the customary method of

construction of invariants of this group in order to classify the representations employed in this presentation.

The inhomogeneous Lorentz transformation, {o,A), is a linear transformation of the coordinates

conserving the norm of the intervals between different points of space-time. The new coordinates xj

are obtained from the old coordinates by the relation^

Xj = Aa$X0 + aa (2.1)

The quantity aa represents the translation of the space-time coordinates, Xa- The condition of

invariance of the norm required that Ka^Aay = 5/37, from which it follows that det A = itl. The set of transformations in which the translations are omitted (ca = 0) is denoted as the full homogeneous Lorentz group, which in turn can be divided into four subsets:

(1) The subset with det A = +1 and A44 > 1 is called the group of proper homogeneous Lorentz

transformations. It is a six-parameter continuous group, containing the identity operator.

(2) The subset with det A = — 1 and A44 > 1 is called space inversion.

(3) The subset with det A = — 1 and A44 < — 1 is called time inversion.

(4) The subset with det A = 1 and A44 < — 1 is called space-time inversion.

The latter three subsets are disjoint and not continuously connected. Those subsets having A44 > 1 may be classified as orthochronous transformations, transforming a time-like vector into a time-like vector.

3 We employ the following notation; All boldface letters. A, Pi x, J, etc.. denote three vectors. The fourth components of the coordinates and momenta, = it and pi = iE, are pure imaginaries. All Greek subscripts a, V, • • vary from 1 to 4 and all Roman subscripts i, y", k vary from 1 to 3, except when specifically indicated otherwise. Repeated indices are to be summed over. Scalar products of four vectors are written asp-x— PfiX^i. The operator d Idxfi is often written as dfi. The D*Alemhertian operator = dud^. Units are chosen so that h and c are equal to unity.

4
To each a and A there corresponds an operator L{a,K) which acts on the Hilbert space of the physical states. A sequence of Lorentz transformations is again a Lorentz transformation according

to the law,

I(a',A')i(a,A) = L{a' + A'a,A'A) (2.2)

A physical state of a free particle of spin, s, and mass, m, may be characterized by a four momentum,

p = (p,pi), and the component of spin along p, the latter quantity is designated the helicity, \. For an

irreducible representation there are at least 25+1 linearly independent basis states in a complete

orthogonal set, one for each value of X = s, s — 1, • • •, —s when the mass is nonzero. If the mass is zero there are only two linearly independent states corresponding to X = ±s. We denote the state vector describing a physical state by \p,s,\).

The Hilbert space of the physical states is endowed with a metric such that the norm is determined

by the Lorentz scalar product of the state vector and its adjoint. The adjoint state vector is denoted as {p,s,\ I /3tj. The hermitian unitary operator /3 assures that the scalar product is independent of the Lorentz frame. The operator tj introduces an indefinite metric into the Hilbert space so that the norm is positive definite for all state vectors. In an unquantized theory, the tj operates only upon the state vectors and has eigenvalues plus or, minus unity.

The covariant norm of this Hilbert space is given by

{p'sX' \l3r,\ps\) = 5(p - p')axx' (2.3)

Any operator, involving the space-time symmetry of the system, that preserves the norm of the

Hilbert space need not be unitary but must obey the relation

0-1 = 00^0 (2.4)

Of course, all observables will correspond to unitary operators, "U, that do not depend upon the metric

of Hilbert space so that they obey the relation

[^,/3] = 0 (2.5)

Any representation of the group is specified by the infinitesimd transformation. A translation in the direction is generated by P^. The unitary operator which represents a infinitesimal translation in the direction is

L{a,l) = 1 - iaj'^ (2.6)

Any finite transformation of the proper Lorentz group may be considered as the product of suc- cessive infinitesimal transformations. The finite translation by the amount in the direction of x^ is obtained by exponentiation of (2.6), thus

L{a,l) = e-»°Mf. (2.7)

Clearly the are the four-momenta of the system.

A rotation in the x^Xy plane is generated by the operator

Aaff^"'' = + a;„/'-'> (2.8)

where the coa^'""' is an infinitesimal Lorentz "six vector," an antisymmetric tensor where

Wag = —Uffo (2.9)

5
co^^f"-') =

^

(£,.Z,„)„^ (2.10)

where the generator, Z^i^, is a hermitian antisymmetric tensor having the structure constants

{Z^y)aff = —i(0a,i5ffy — havO^y) (2.11)

The constant, z^v, represents the magnitude of "rotation" in the x^Xp plane and is of the form

ty^v = i^23,'^31,^12,i^U,i^2i,i^u) (2.12)

i.e., the angles in the spac^-like planes are real, while the "angles" in the space-time planes are imaginary.

The operator in the Hilbert space of the physical states which represents an infinitesimal rotation

is denoted as Z,(0,A) and is defined by

I(0,A) = 1 + ~ (2.13)

with the nonhermitian operator, J^t,, obeying the relations

/,yt = /,,. = (2.14)

(37,4^/3 = -Ju - J4k (2.15)

The J23, /31, and Jn are the three components of the total angular momentum J. The finite rotation

operator L(0,A) is also found by exponentiation

L(0,A) = ei/2(c,./,„) (2.16)

The operator L(0,A) is not unitary, however its inverse can be found from the relation (2.4)

L-i(0,A) = ^Lt(o,A)/3 (2.17)

The following commutation relations hold among J^y and P\

[JtLy,J\p\ = ii^nxJyp + ^fpJ^X — 0^/Jy\ — 6y\Jyp) (2.18)

[J,M = j(5,xP. - 5.xP,) (2.19)

[PM = 0 (2.20)

The generators for translations form an Abelian group, while the generators for "rotations" form

a non-Abelian group obeying a Lie algebra. The problem of finding this representation of the Lorentz group is equivalent to finding all of the representations of the commutation relations (2.18-2.20).

The operator J^^, can be expressed as the sum of two parts; L^^ which operates on the variables x^

or and 5^^, which operates on the intrinsic spin variables.

Jiiv — Li^y -\- S^y (2.21)

where

L.p = XnPy — XyPfj. (2.22)

6
The three spatial operators Liz, L31, and L12 act on the orbital variables alone; L is the orbital angular

momentum operator. Likewise 523, 53i, and 5i2 act on the internal spin variables alone; S is the spin vector of the particle. The total angular momentum is

J = L + S (2.23)

Once a representation is found, the task is to find all of the invariants of the group in order to test for irreducibility. Clearly, only scalar or pseudoscalar operators can be invariants, and one must construct these invariant quantities from the generators.

One invariant is the Lorentz scalar

P-P = P^P^ (2.24)

This invariant obviously commutes with each of the generators P^ and

Another invariant has as one of its constituents the Pauli-Lubanski spin operator, a pseudovector,

i wx = — - e\^yJ^vPp (2.25)

The orbital term, Lf,^, in iv\ drops out identically so that

i wx = —

2

t^y-ypSf.yPp (2.26)

The commutation relations involving w\ and the generators, J^^, and P^, are

[A^,i<.'x] = ii^^xtVy — 6,xttv) (2.27)

[P„wx] = 0 (2.28)

[wi„Wy] = e^yXpW\Pp (2.29)

and also w^P^ = 0 (2.30)

This second invariant is the quantity [4, 18]

W^W^ = — - PxP\S^yS^y + P^S^xPpSyX (2.31)

Although WfiW^i commutes with the generators /^.^ and P^, a requirement that it be a Lorentz scalar puts rather strong requirements on the properties of 5^^. If we designate

M = (523,531,5x2) (2.32)

N = (5i4,524,534) (2.33)

then w^w^ will be a scalar provided that M = 0N; N = ©M; 0^ = 1 (2.34)

where 0 is some unitary hermitian operatoi that commutes with M and N.

7
This may be readily seen if we express in vector form

w = -iP X N + PoM (2.35)

Wi = iV'M (2.36)

Then

w,iv, = -(P-P)(N.N) + (P-N)(P.N) - (P.M)(P-M) + PoKM-M) - iPoP-(N X M - M X N)

(2.37)

This quantity is a Lorentz scalar only when conditions (2.34) are obeyed, then

w^w^ = -P,P,M^ = -P,P^N2 = - ^P^P^(M2 + W) (2.38)

We shall consider subsequently two distinct representations of the operators 5^^^, one representation

obeying the condition (2.34), and the other not. The basis states in a Hilbert space are, of course, defined

as the eigenvectors of a complete set of commuting operators. In order to get unique eigenvalues in the case of a nonzero mass we evaluate the operators P^P^, Wy,w^, P,, and wz in a rest frame where = 0, 0, 0, im). We have

= m(MuM2,Mz,0) (2.39)

iv^w^ = mm^ (2.40)

The eigenvalues of the operators in the rest system for an irreducible representation are given by

P^P^ \ps\) = -m^\ psX) (2.41)

iv^u'i^ \ps\} = m~s{s + 1) I ps\) (2.42)

ws I psX) = mX I psX) (2.43)

P^ I ps\) = ±im I ps\) (2.44)

This set of eigenvalues gives a physical characterization of the eigenstate by specification of the mass,

the spin, the component of spin along the z axis and the sign of the energy. Unfortunately, this Abelian

set of operators does not suffice to characterize the differences between distinct irreducible representations of S^^ describing the same spin in a way that is independent of the Lorentz frame of reference. There are no other invariant quantities that may be formed from functions of the ten generators

of the inhomogeneous Lorentz group. It should be noted that none of the invariants P^P^, w^w^, P^, or W3 contain the coordinates x^. Group theory is stating succinctly that under translations there is no

invariant with a coordinate dependence.

We may describe a physical state in a particular way that does not depend on a specific relativistic equation of motion. Following Jacob and Wick [19] we may define the nonzero mass state | psX) in terms

of the state | psX) in the rest system by the sequence of operations

I psX) = i)(-^>-i(</),^,-<^)e"34iV3I PsX) (2.45)

8
The rest state | ps\) has its spin component X along the z axis. The boost operation e^'^^\ depending

upon N3 (since Lu is zero in the rest system), yields a state with momentum p along the z direction. The particular spatial rotation

D'-'^-\<t>,e,~<l>) = e-'>-^3e-iej-2et«/3 = g-^n-J (2.46)

yields a state with momentum p = p(cos</) sin9, sin</) sinS, cos0) without changing the helicity component, X. We denote this particular sequence of homogeneous Lorentz transformations and the

corresponding operator as h{p) and L(h{p)) respectively, i.e.,

L{h{p)) = i)")-i(<^,e,-<^)e«34iV3 (2.47)

The general Lorentz transformation and the corresponding operator are still denoted x' = Ax + a and L{a,A) respectively.

According to the definitions (2.45, 2.47) we also have

Furthermore, we have

hence

\p's\) = L{hip')) \ps\) (2.48)

p' = Kp = Ahip)p = h{p')p (2.49)

h-'iAp)Ah{p)p = p (2.50)

The sequence of transformations h ^{Ap)Ah{p) corresponds to an ordinary spatial rotation of p. Thus we can write the corresponding operator relation

I(0,A) \ps\) = L{0,Ah(p)) \ps\) = L(0,h{^p))L{0,h-'{Ap)Ahip)) 1 ps\) (2.51)

where we have employed (2.2) for the composition of successive Lorentz transformations. The nature

of the operator L{0,h~^(Ap)Ah(p)) is that of a Wigner rotation operator which we denote as D^-^^(h-^(Ap)Ah(p)). Combining (2.47 and 2.51) we obtain

I(0,A) \psX) = I Ap,s,<T)D,x^^>ih-\Ap)Ahip)) (2.52)

where the momentum states are related by (2.49).

This well established, but still surprising to many, Lorentz transformation has nothing to do with

the representations of the complete operator /^^ but rather involves only the familiar representations

of the spatial rotation group depending only on Jij. The transformation properties of the state vector under translations are more straightforward.

We have

i(a,l) \ps\} = e-'-.P.

I ps\) (2.53)

We may represent the combined proper inhomogeneous transformation x' = Ax + a on the

physical states as

L{a,A) \ps\) = e-'O/.-P-'

| Ap,s,(r)D,x^-'^{h-\Ap)Ahip)) (2.54.)

9
The adjoint of the above is

{ps\ I L\a,k)^y] = {ps\ I /377l-i(a,A) (2.55)

The norm is preserved with the rotation matrix obeying (2.4)

D^J^~^{h-'{Kp)Kh{p)) = ^D^-'^^{h-\Kp)Kh{p))^ (2.56)

We may factor out of L{a,A) that part which operates on the intrinsic spin variables which we

designate as D^^^A) so that

L(a,A) = ei/2(€,^^„)-ia^p,D(«)(A) (2.57)

where

DM (A) = e'/2(S^«M.) (2.58)

The operator Z)^*'(A) is not unitary but obeys (2.4) also

2)w-i(A) = /3Z)Wt(A)^ (2.59)

The projection of a physical state vector onto a four dimensional coordinate space defines a wavefunction

^{x) = {x\ps\) (2.60)

and

(p'(x) = {x I L{a,A) \ ps\) = {A-'{x - a) I D^iA) \ps\) = DM{A)<p(A-'(x - a)) (2.61)

or relabelling the coordinates x —y x'

<p'{x') = D^iAMx) (2.62)

The operator, Z)'*'(A), effects the Lorentz transformation from the coordinate system xto x' = Ax -\- a. The group representation of D^"\A) is completely determined in its infinitesimal form, since the

hermitian operators, 5^„, in the generators obey a Lie algebra similar to that of (2.14).

Using the relations (2.32, 2.33), we can write the commutation relations involving the components

of S^„ as

[M„Mj] = UiiMk (2.63)

[Mi,/Vy] = iei^,N, (2.64)

[/V„/Vy] = UiiM, (2.65)

When (2.58) is expressed in terms of M, and N, the operators M generate spatial rotations and the operators N generate boosts in the basis (2.62).

10
Two invariants that commute with each of the Mi and Ni may be specified and will serve to classify

the representations of this algebra. These invariants are given by

F = |(M2 + W) (2.66)

G = M-N (2.67)

While the F and G are not among the set of operators PyPy., w^w^, P^,, and wz that characterize

representations of this inhomogeneous Lorentz group, they commute with each of the set. Furthermore

the state vectors \ psX) are eigenvectors of all of the foregoing operators. We may use F and G to supplement the set iVy.iv^, Pf^Pn, Pu., and W3 to obtain a maximal Abelian set of operators whose eigenvalues completely specify a particular representation of the inhomogeneous Lorentz group whether or not condition (2.34) is obeyed.

The commutation relations (2.63-2.65) obey the same algebra as that for the group 0(4) which is used to describe the subgroup of homogeneous Lorentz transformations.

Although we intend in this paper to use the representation having the algebra of 0(4) in a Minkowski

space, it is helpful to use the classification scheme of the covering group, SL(2c), even if we do not employ the representations of the latter group. The commutation relations (2.63-2.65) may be decoupled by the introduction of two new hermitian

quantities

S(+) = i(M + N) (2.68)

S(-) = i(M - N) (2.69)

with the commutation relations

[5/+',S/+>] = ie.,,5,(+) (2.70)

[5/-',Sy(-'] = jeo-.5.(-> (2.71)

[5i(+',5/-'] = 0 (2.72)

The operator Z)'^''(A) defined in (2.58) becomes

D(»)(A) = e'(*+'"''S'^^e''*-'"''S'~' (2.73)

The operator i)'''(A) looks formally like the product of two independent rotations through a complex

angle.

The S*+' and S*~^ are independently 2s'+^ + 1 and 2s'~' + 1 dimensional representations of the algebra of the rotation group 0(3), denoted by {S<+>^53^+'} and }S'~>^53'~' 1^ respectively. Taken

together they form a (25^+^ + l)(2s'~' + 1) dimensional irreducible representation, (5'+',s'~'), defined

for any value of s'"""* and s'^' integer or half integer. The representation (s'+^s'"') exhaust all finite dimensional irreducible representation of the group 0(3) ® 0(3) which is homomorphic to SL(2c).

Only the (0,0) representation is unitary.

It will be shown that under the operation of space inversion the operator N changes sign while M

does not. As a consequence, under space inversion S*"^^ becomes S^~^ and vice versa. Any irreducible

representation of well defined parity must be given by (s^+^s'"') ® (5<~>,s'+') and will have a dimension

2(2s<+) + l)(25(-> + 1) whenever s^+^ differs from s^'K

11
The invariants F and G expressed in terms of the new operators S*"*"^ and S^~^ are F = S^+>2 + S^-'2 (2.74)

G = S(+'2 - S(->2 (2.75)

In what follows we shall use the labels (s'+^s*"') © (s'^^s'"''') to characterize a particular representation

but will use the basis vectors of the operators obeying the representation (2.63-2.67).

The eigenvalues of F and G for a specific irreducible representation (s'+',s(~') are given by

F |p5(+>X(^-)5(->X(->) = [5(+'(5<+> + 1) + 5<-)(5(-) + 1)] |j9s<+)XWs<-)X(-') (2.76)

G lps(+'XW5<->X(->) = [5(+'(s<+> + 1) - 5<-Hs<-^ + 1)] |psWX(+>s(-'X(->) (2.77)

The irreducible or completely reducible representations of the Lorentz group can be classified according to whether is time-like, light-like, space-like, or if all components of jr?^ are equal to zero. The last case will not be considered further since the only physical system of interest corresponds to the vacuum state where the representation is the trivial identity that is one dimensional.

We have already treated in some detail the case in which is time-like corresponding to —p^p^ = > 0. There are two representations for each value P^P^, w^Wu,, + N^, and M-N; one for each sign of the energy po = ±\/m^ -\- p^. The spectrum of eigenvalues of are most conveniently

obtained in the rest system according to (2.37). This is true only for the case of nonzero mass in which we are able to transform to the rest frame. In other frames in which = {p,p4) we can use the helicity

operator (J'P)/| p | to characterize the basis state. This is compatible with the description of a frame

with motion along the 2 axis since (J 'P)/ 1 p | commutes with J and hence the operator i)'-^'~i(0,9, —0).

Here,

JP

f— \ps\) = X \ps\) (2.78)

I P I

The subgroup of homogeneous transformations which keep one of the four components of p^

unchanged is called by Wigner a "little group." The little group that leaves pi invariant is the three dimensional rotation group. For any fixed momentum component the operators are generators of the little group. Since if^P^ = 0, only three of the generators are linearly independent. The operators

L(A') (where A'p^ = p^) form an irreducible representation of the little group and determine the

irreducible representation L{a,K) of the inhomogeneous Lorentz group. There is a second class of representations which are of physical interest, namely those corresponding to the light-like or zero rest mass case in which p^p^ = 0. Here the invariants P^P^, and WfiW^ do

not suffice to characterize the representation since they are equal to zero. The invariants +

and M'N are independent of the mass. For a massless particle, there does not exist any coordinate frame in which all but one component

of Pf^ vanish. There is a frame, however, in which p^ = {0,0,p,ip). In this frame the components of

are obtained using (2.35, 2.36)

wi = PiMr + iN2) (2.79)

W2 = PiM2 - iNi) (2.80)

W3 = PMs (2.81)

w, = iPMz (2.82)

12
We can put these in the form of raising and lowering operators

w+ = wi + iw2 = P(M+ + N+) (2.83)

w^ = wi- iw2 = P(M_ - 7V_) (2.84)

where M^. = Mi ± iM2, etc. We also have = XP^ from w^P^ = 0, so that

X = ^ = Ms (2.85)

We can write

w^W/i = w+w^ (2.86)

since in this particular Lorentz frame 103^ = — ^4^, and [ti;i,i{;2] = 0. The w^. and X obey the commutation relations

[X,w+] = w+ (2.87)

[X,m;_] = -m;_ (2.88)

[w+,w_] = 0 (2.89)

The basis states may be identified by finding a representation of this algebra. This algebra is not

semi-simple because the elements and form an Abelian sub algebra. In order that the basis states form a finite set it is necessary that the expectation values of w+ and w_ be zero. This latter requirement

estabhshes the expectation values of X, the helicity operator, to be dbs for a particle of spin 5, either integer or half integer. The matrix element of w+w^ is also zero, and hence that of w^w^ are both

zero for this zero mass case. We may denote such a basis state still as | p,s,\) so that

w^lpsX) = p^\\ps\) (2.90)

Only one eigenvalue of X, either +s or — s, survives for each irreducible representation after taking

the expectation value, and hence X is an invariant. Particular illustrations of the basis states will be

given after a specific representation of the operators is developed. Here the maximal Abelian set of

conamuting operators Pn,w^, + N^, and M-N serves to define the basis states. The nonsingular case, in which the matrix elements of w+w^ or w^w^ do not vanish, corresponds

to the zero-mass, continuous-spin representation [17]. This representation does not seem to have any

physical significance and is not considered further. We are limited to only one physically permissible irreducible representation of the two dimensional

little group, a rotation in the plane perpendicular to the direction of propagation. The vector p^ =

{0,0,p,ip) is unchanged by this rotation. The basis states for a zero mass particle of any momentum p' and helicity X, propagating in a

specified direction, may be obtained by a Lorentz transformation L(a,A) from the particular frame

p^ = {0,0,p,ip) that we have considered. Here

p,' = A,.p, (2.91)

13
We can conveniently choose A to be a "boost" of j9 along its initial direction of propagation to a value

p' followed by a spatial rotation that takes the initial direction into the fined direction of propagation

along p' where p' = p'{cos(t> sind, sm<j) sin d, cosd). So that A is given by

A,„ = I>,p<-^>-K0,9,-<^)[e"^^^»^]p. (2.92)

and where the boost operator e"'^^'^ is given by

6^34^34 = [/ - Zu' + Z342(cOshfi34 + Z34 sinhfi34)] (2.93)

fi34 = In 7^ (2.94)

IP I

The quantity Z34 is the hermitian antisymmetric tensor defined in (2.11).

The spatial rotation operator D^p'--^^"'^ is the usual rotation operator of the little group where pi

is unchanged in the rotation. The representation of a zero mass state in any Lorentz frame is characterized by the eigenvalue

p^\ in (2.90). No proper Lorentz transformation will relate states having opposite sign eigenvalues of X. There is a third class of representations where is space-like, —p^p^ = < 0. Here, the

particles would have imaginary mass and would propagate outside the light cone. Although these tachyonical representations are admissable, there is no evidence that particles corresponding to these representations exist physically.

We shall defer until section 6 the discussion of the improper Lorentz transformations, space and

time inversion, after specific representations are developed. These transformations, along with charge

conjugation, are treated in a manner similar to those treated in textbooks on field theory. The group

theoretical treatment of associated conceptual problems and phase questions is nontrivial [20]. Time inversion is actually the product of two distinct operations: Wigner time reversal or simply

time reversal, an operation in which all velocities are reversed, and charge conjugation, an operation in which all particles are changed into their antiparticles. Charge conjugation is a relativistic quantummechanical concept only. Both time reversal and charge conjugation are discontinuous operations like time inversion.

3. Representation of Operators and Eigenvectors

Our task is to develop a covariant description of particles and fields of discrete spin and arbitrary mass. We would expect that there would exist 2s + 1 independent states for a particle of spin s and

non-zero mass in the rest system. On the other hand, if the mass of the particle is zero, there will exist only two independent states for a particle of spin s corresponding to the two states of maximum helicity.

We want to construct an explicit representation of these state vectors in a Hilbert space and a set of linear operators that will directly account for all observable properties of the particle in a straightforward way. Our ansatz is to postulate that for a particle of spin 5 the state vector has as elements,

25 + 1 space-like components and 25—1 time-like components, the latter being required to give the proper number of degrees of freedom for an observable in any Lorentz frame. Further requirements of

relativistic covariance with components of unique parity may double the number of components so

that the state vector in our Hilbert space in the general case will have 2(25 + 1 + 2s — 1) = 8s components. We can designate the subset of 25 + 1 space-like components as | s>,(r) and the subset

of 2s — 1 time-like components as | 5<,o-). The time-like components have some characteristics of a spin s — 1.

14
We may represent the total state vector as a column matrix of 8s components.

2s + 1 space-like components

\p,s,a) -

2s — 1 time-like and "small" components (3.1)

4s "small" components

The 8s dimension of the state vector in the Hilbert space implies that the linear operator Z)*^''(A) introduced in (2.58) will be represented by an 8s by 8s dimensional matrix. This operator is also a "Lorentz six vector"; it is the generator for the three space-like rotations and the three space-time

rotations or "boosts." This implies that S^p and any irreducible submatrix of S^^ shall behave as a special tensor operator of rank one. We introduce, as constituents of the spin operator, 5^^;,, and other operators to be defined subse- quently, two operators of tensor rank one. These quantities we shall designate as 5/^' and Si^". The

superscript (s) designates an operator that permutes space-like components with one another and timelike components with one another. The superscript (t) designates an operator that permutes space-like

components with time-like components.

The operators Si**' and S,*^" may be defined in terms of Wigner operators [21,22] of rank one. A Wigner operator of SU(2) is designated by the notation

/ k + K V (2k O)

^ k + q

'

and is an irreducible tensor operator of type T{k,q) [23], where q = k,k — 1, • • •, —A:. It is an operator

which effects the shift k on the label s of a generic state vector | s,a) as well as the shift q on the component <r. More precisely, it is defined by

/ k + K . (2k 0 > I 5,0-) = [ s + /c,o- + q){s + K,(T -\- q \ sakq) (3.2) ^ k + q ^

In particular, k may take any of the values A, A: — 1, • • •, —fc, so that there are {2k +1)^ Wigner

operators of rank k, one for each value of k and q. Note that because the vector coupling coefficients are defined to be zero if the triangle conditions on s, k, and s + k are not satisfied certain irreducible

representation spaces of |s,a-) will be annihilated by a given Wigner operator, i.e., each Wigner operator

has an associated null space.

15
We restrict our representation subspace of dimension 4s to those of s and s + k, both of which

can have values 5 and s — 1 which we may denote as

>

(3.3)

The operator S^**' that operates in this subspace of the eigenvectors is defined as

5

0>

<2 o)

^ 1 + g 'k

(3.4)

The subscript K on the Wigner operator indicates a "renormahzed" Wigner operator from that defined in (3.2) in such a way that the denominators of the vector couphng coefficients are eUminated

and the sign of the ^ = +1 components are reversed. This estabhshes a generahzation of the usual

angular momentum operators in our representation subspace that includes time-like components and

is in accordance with the established Condon and Shortley phase convention [23]. With these definitions we can reexpress (3.2) in this generalized and renormalized form.

5+(^) I 5^,<t) = [(s + <T + i(l + 6))(5 - (T - i(l - 6))]l/2

I 5^,<7 + 1) (3.5)

5J^> |5>,<t) = [(5 - + i(l + &))(5 + <T - 1(1 - 6))]1/M5^,'^ - 1) (3.6)

S,^^^\s>,a) = a\s>,a) (3.7)

where the nonhermitian raising and lowering operators, 5+*'' and may be defined in terms of

hermitian operators Si^*' and S2^*', by

5 (a) = 5j(.) ± (3.8)

and where h is the eigenvalue of a diagonal, hermitian, and unitary operator B on the states | s>,(7) or I s<,(t) such that

B I 5>,o-) = I 5>,a->

B I 5<,o-) = — I S<,cr)

(3.9)

(3.10)

We will denote this operator B as the metric operator.

Successive operations on | 5>,(t) with S^''^ for various values of ^ = +, 0, — establishes that

,(s)2 5>,tr) = 5(s + h) I s>,o-) (3.11)

and that

(3.12)

16
Furthermore

[Si^^\B] = 0

We will have occasion to use a shorthand notation

The operator 5g"^ that operates in the subspace of the eigenvectors | s^,a) is defined as

2

(3.13)

(3.14)

= B

0

<. 0>

(3.15)

The conditions of renormalization are the same as those for Sg*^', including the Condon and Shortley

phase convention. The operator B is introduced in order to make 5g<'^ have hermiticity and reality

conditions similar to S^''^. Using this definition we have

S+(« I = -h\{s + 6(7 + i(l + 6))(5 + 6er - |(1 - \ s^,a + 1) (3.16)

5J» I 5^,cr) = b[{s -ba + 1(1 + b)){s -ba- |(1 - bW \ s^,a - 1) (3.17)

5o<'' I 5^,<r) = [(s + a){s- \ s^,<r) (3.18)

where, similar to (3.8),

Note that the operation of Sg^" on | s>,a) always changes 5> into s< and vice versa. Furthermore

{Si^'\B} = 0 (3.20)

Again we can use the shorthand notation to indicate the product of B and 5i''^ = 55/'> = -5/"5 . (3.21)

By successive appUcation of 5g''* we establish that

S(«2 I s>,a) = s{2s - b) I s^,a) (3.22)

We note that in this representation subspace only one of the six operators 5g^'' and 5g**' is diagonal,

namely 5o'''. Successive application of 5g**' and Sq^^^ establishes the additional commutation relations from straightforward algebraic manipulations

[5/'>,5/"] = ieij,(2sB - 1)S,<»)

(3.23)

(3.24)

17
The latter commutation relation is rather unexpected. The six operators 5/'^ and do not obey

a Lie algebra, a fact that will have interesting consequences.

Certain anticommutation relations may also be established in the same way as the commutation

relations were established

{5i<^',5/»M + i5.<",5/«) = 2sHij (3.25)

|5/«>,S/'>) - {5/'\5/^M = 2516, yA'" (3.26)

We note that neither S'^^^ nor S^"^ commute with all of the components 5/'^ and 5/'\ but that the sum S<^*'^ + S''^^ does, however. Adding (3.11) and (3.22) we obtain

(S«2 + S(«2) I s>,^) = 3s2 I 5>,(t) (3.27)

and recalling

5o(*' 1 s>,a) = (T

I s>,a) (3.7)

We have here defined a basis for this representation: the simultaneous eigenvectors of (S'^'^ + S*^''^) and So^^K The values of 5, or equivalently s>, are subject to the restriction that s> is fixed for a given representation and may take on one of the values 0, 1/2, 1, 3/2, 2, • • • The value of s< equals s> — 1 except

in the case when s = 0, 1/2 where the components corresponding to 5< are absent. The 4s-dimensional representation whose basis is given by the eigenvectors | s>,(j), where a =

s^, — 1, — 2, • • •, — s^, is in a group-theoretical sense irreducible. It is clear that, by successive use of the operators Sq^"'' and Sg^", with g = +, 0, and — , we may transform any vector in this set into any other. We shall denote this representation by !5> © s<). This irreducible representation can

be regarded, generally speaking, as a reducible representation of the spatial rotation group. Here

{5> © s< I regarded as a representation of the spatial rotation group transforms as Drot**' («?)? a relativistic operator, which decomposes into the representation, e.g.,

Aot'''(^3i) -> D(^KO,m e D(^-'KO,m (3.28)

where the operators on the right hand side are the usual nonrelativistic D functions. In nonrelativistic quantum mechanics we are concerned only with the Z)'*' representation since all time-like states that transform under the representation are neglected.

For a given eigenvector ] s^,cr), there is a limit to the number of successive raising or lowering

operations S^'-"'' and 5±**'. This limit follows from the associated null spaces of the Wigner operators.

Inspection of (3.5-3.7) and (3.16-3.18) allows us to write

1 S>,5) = (); 5_ 1 5>,-s) = 0 (3.29)

S+« h<,5 - 1) = 0; SJ^^ 1 5<,-(5 - 1)) = 0 (3.30)

1 s>,s) = (); S_ 1 5>,-s) = 0 (3.31)

|5>,5 - 1) = 0; |5>,-(S - 1)) = 0 (3.32)

|s<,s - 1) = - [2s(2s - i)]''M5>,5) (3.33)

18
5J« I s<-{s - 1)) = [25(25 - I 5>,-5) (3.34)

5o(« I s>,±5) = 0 (3.35)

Thus, we are able to generate only the 45 eigenvectors in our representation basis by successive use of the operators 5,''' and 5,*".

It should be strongly reiterated that the Sq^^'> and Sj*" are not in themselves spin operators, but

rather should be regarded as operators having certain group properties in a subspace of other higherdimensional operators which well may have diiferent group properties.

The invariants of the group involving Sg^'' and S^'" are given by

SM.SM + S(«-S('> = 352/ (3.36)

SW-S(« = 0 (3.37)

where / is the unit matrix of dimension 4s by 4$.

The matrix elements of the operators Si'-^\ 5/'\ B, and I in the representation {s> ® s<) define the matrix representation of these quantities which are listed in table I for values of 5> corresponding

to 1/2, 1, 3/2, and 2.

It is sometimes convenient to use the quantities

o-w = -S(»'; a^" = -S(« (3.38)

5 s

The commutation and anticommutation relations written in terms of these quantities are

[<r/'V;W] = j-eoW') (3.39)

5

ki(«\(7/'>] = i - ei,*(7*(« (3.40)

s

[(r,'",cr/«] = i€ij,(2B - s-'W^ (3.41)

+ = 25.y (3.42) _ {,r/»,(T/'M = 2/e,y.ff*(« (3.43)

[<Tt'",c7/"] + [cr/",(r/«] = 2ieij,&,^'^ (3.44)

[(r/«',<r/»] - [<r.(«,<7/»'] = 0 (3.45)

[<r.('\B] = 0 (3.46)

{<Ti(«,51 = 0 (3.47)

These relations are generalizations of the famihar Pauli algebra for the spin 1/2 a matrices.*

* For spin 1 /2 there are no time-like components, furthermore St- and (Ti^ are null operators. Such results as (3.36) are not contradictory since

3«2 = s{s + 1) when s = 0, 1 /2.

19
Combining some of these commutation relations allows us to find additional relations that are often useful

(t)

If we introduce two arbitrary vectors A and C it is straightforward to show that

(a(»>-A)(a(»>-C) + ((r'«-A)(o-("-C) = A-C + ia^-A X C

((r(^>-A)(<r<«-C) - (a">-A)(<r<''-C) = ia^-A X C

(3.48)

(3.49)

(3.50)

(3.51)

Again, these are generalizations of some familiar relations of the spin 1/2 Pauli algebra.

We now form several anticommuting unitary operators of dimension 8s by 85 from the components

(r/'>, o-/'\ and B. We define:

(3.52)

&i (s)

B 0

0 -B

(3.53)

These operators obey the anticommutation relations

which may be proved straighforwardly using (3.42-3.47).

Indeed, there are a number of familiar relations that can be developed. If we define

(3.54)

(3.55)

ji = —Ilia,

la

i(Ti

74 = ^

.(«) -i<Ji

.(() (3.56)

(3.57)

We have the generalization to any spin of expressions for the Dirac 7 matrices in the Dirac-Pauli

representation. These generalized matrices obey the anticommutation and commutation relations characteristic of a Clifford algebra [24]

:

(3.58)

(3.59)

We should emphasize strongly that this a^^ is not, in general, proportional to the spin operator of

the particle. The spin operator is yet to be determined.

20
We may also define a symmetric operator:

75 = 71727374

0 -/

-/ 0

(3.60)

There are additional relations that follow immediately:

[o'^v,7x] = 2j§^x7^ — 2i5„x7^

!cr^„,7x} = 2ie^,xp7p75

W^i',yi] = 0

{<^li>',yi\ = 2€j.^XpO-pX

[o-^»,7x75] = 2i5f.x7.75 — 2f5.,x7M75

{o-M.,7x75l = 2i£^,xp7p

<^ij = — «Ofc75«fc

(3.61)

(3.62)

(3.63)

(3.64)

(3.65)

(3.66)

(3.67)

(3.68)

(3.69)

AU of the 7^ and 75 are unitary hermitian operators in this Pauli-Dirac representation; 71 and 73 are imaginary and antisyimnetric while 72, 74, and 75 are real and symmetric. All have trace zero [25]

.

We now introduce an 85 component wave function <p{x) which is the projection of the state vector

(3.1) onto coordinate space. For the moment we consider the 8s component column matrix to describe

a one-particle state at coordinate x = {\,it). If we operate on this wave function with the scalar product of the 7^ operator with the four- gradient operator 6^ we obtain another function x(x).

x{x) = y^d^(pix) (3.70)

The function x(^) will also be an 8s component column matrix but it does not necessarily have the same Lorentz transformation characteristics as <p(x). The adjoint wave function is defined as the projection of the ajoint state vector, {ps\ \ /8»j, onto

coordinate space. The definition is similar to usual Pauli-Dirac adjoint but modified by an operator r) that reverses the sign of those states that are subject to gauge conditions or, equivalently, those states that have time like components present in the rest system for fields with nonzero mass. This operator 17 introduces an indefinite metric into the Hilbert space in such a way that the norm and the expectation

value of the four-momenta for each orthogonal quantum state is positive definite. This adjoint wave

function is given in terms of <p^{x) by

ip{x) = <p^{x)yi-n (3.71)

and the adjoint to x(^) is

x(^) = xKx)yin (3.72)

21
The properties of the operator 7^3^, under hermitian conjugation together with the commutation

relations yield

x{x) = —^{x)d^y^ (3.73)

Placing 5^ to the left of the 7^ implies that it operates towards the left. We now consider a proper imhomogeneous orthochronous infinitesimal Lorentz transformation as defined by (2.62) of the basis states (p{x) and x(^) so that

^'(x') = Z>(^.^>(A)<p(r*;) = (1 + i/2 £,,5,/^')^(;c) (3.74)

and

x'{x') = D^^.^KA-)x{x) = (1 + i/2 eA'-^)x{x) (3.75)

where

PC^

~~ Xfx I Z fiv^C 1/ I €Lfx

The spin operators S^^^*"' and 5^^'"' are as yet undefined other than that they are the appropriate

spin operators for <p(x) and x(^)? respectively.

The relation (3.70) under this Lorentz transformation becomes

x'(x') = Z)(».x)(A)7xaxD<^""-i(A)D(»'^>(A)^(x)

^'ix')

= y.d,Vix') (3.76)

7xax + ^£..(5,.W7x - 7x5,/^')ax

where

The functional relationship between x(^) and (p{x) is the same in the primed and unprimed system

provided that

5^.<'<>7x - 7x5^.'^> = iS^T. - i5,x7. (3.77)

This requirement (3.77) is the condition for covariance in the Lagrangian formulation to be discussed

shortly. Note that (3.77) is not a commutation relation unless there is an equality between 5^^^"' and

S^^'-'^'K The commutation relations of 5^^^'"' or 5^/*"' with 7x when S^^'-^'' 9^ S^/'^^ are rather complicated

and do not usually occur in practice. The 5^/"' and 5^;,'*"' always satisfy, individually, the Lie algebra

of relations (2.18)

[5,/^',Sxp<-'] = j[5,x5„p(^) + §.p5,x<^' - 5,,S.x<^> - 6.x5,p(^'] (3.78)

[5,/x),5,/x)] = i[6,xS,/x) + _ 6^^5^,(x) _ 6,,5,pW] (3.79)

22
Inspection of the commutation relations (3.78, 3.79) suggests that one of the possibiHties for the

representation of S^, is given by

Sij €{

5.(') 0

0 5i(^'

0 5i<^'

Si(»> 0

(3.80)

(3.81)

We may associate (3.80, 3.81) with the spin operator for the x wave function, i.e., 5^;,^"^ We can also in certain instances make the opposite association, i.e., with the <p wave function, but we will not give

the arguments here. The condition (3.77) then determines uniquely S^,^*"' since both 5^,'^^' and 7x are already determined

in terms of the operators 5,^'', 5/", and B. We find

5i(«' 0

0 S,^^^

(«

(3.82)

(3.83)

These expressions for 5;^/''* satisfy the commutation relations (3.78), as they must. We see that the space components 5,^^"^ = 5jj'*°' so that x and tp wave functions always have the same spatial

rotation properties. Relation (3.77) is a commutation relation for these spatial components for any

spin; it is also a cormnutation relation for all components in the case of spin 1/2. Here o-*^*' = 25^*' while <r''^ is a null matrix so that 5;^,^*"' = 5^,*^^='. For spin 1/2 alone do the x and ip wave functions

have the same Lorentz transformation properties.

Other relations important in establishing covariance conditions are given by the commutation

relations

[5,/>^),(7xp] = [5,/-',<rxp]

and, by taking the hennitian conjugate of (3.77)

(3.84)

(3.85)

(3.86)

We may apply the criteria of (2.66, 2.67) to test these representations for irreducibility. We can

define

M(^) = (S23<^\531<^\5l2<^0 (2.280

(2.29')

and

23
(2.28")

(2.29")

Considering, for the moment, particles of nonzero mass, we have for the tp wave function, from

(2.66, 2.67),

/ 0

0 /

(3.87)

CM = Mf^'-N(^) = (s2 - 1)75 (3.88)

Both F^*'^ and G*'^' commute with all of the generators of the inhomogeneous Lorentz group

and P^, and F^^^ is a multiple of the unit matrix.

The corresponding quantities f-^^ and C-^^ are

s(s + 5) 0

0 sis + 5)

(3.89)

(3.90)

The invariants F^^^ and C-^^ also commute with all of the generators of the inhomogeneous Lorentz

group. While F^^^ is not a multiple of the unit matrix it is completely decomposable into two independent

multiples of the unit matrix.

We can introduce a change of representation to display the structure of the invariants F'*"', G'*'\

F'-^\ and G*"' in a more familiar way by the introduction of the hermitian quantities S'+' and S'-"^ defined in (2.68, 2.69). The eigenvalues of the operators F and G in this new representation are given

by (2.67, 2.77)

F('p) s(+>(5W + 1) + /-'(s'-> + 1)

G(^) + 1) - s<->(s(-> + 1)

(3.91)

(3.92)

Equating F'*"' and G^''^ in this representation with the corresponding F'*"' and G**"' in the 0(4) representation (3.87, 3.88) allows us to solve for and s'~' in terms of s. Retaining only the positive values, we obtain s^+' = s — 1/2 and s*"' = 1/2. The space of the wave functions involves a (s — 1/2,1/2) ©

(1/2, s — 1/2) representation. The <p transforms according to a

£)(s-i/2,i/2)('j\^-) 0£)(i/2,s-i/2)(|^) j.gp. resentation and has 8s components except for the special case when s = 1 where the irreducible

representation has four components. A polar vector or an axial vector transforms as Z)'^'^'^'^)(A). In a similar way we can compare the invariants for the x wave function

Px) + 1) + 5(-)(sf-> + 1)

G(X) + 1) - 5(-)(5(-) + 1)

(3.91')

(3.92')

with the corresponding invariants in the 0(4) representation. We obtain the values s^+^ = s and

s'~' = 0 when 6 = +1 and s*^"*"' = s — 1 and 5'"^ = 0 when b = —1. The space of the x wave function

or state vector involves a completely reducible (s,0) © (0,s) + (s — 1,0) © (0,5 — 1) representation.

The X state transforms according to a D(«'<"(A) © Z)<*''^>(A) + D^'-'^ ^'> (A) © D'O'^-i^A) representation

24
and has 85 components that can be divided into two irreducible groups of 4s + 2 and 4s — 2 components. The special case when s = 1 has three independent groups a Lorentz six vector of six components, a scalar of one component and a pseudoscalar of one component.

The operation of A/j.^"', Mo'-^\ N±^^\ and TVo^"' transforms any vector in a particular irreducible

subspace into another vector in the same subspace.

It is the linear independence of these two irreducible subspaces of the x state vector that provides

a gauge freedom in the description of fields with spin greater than one half. This gauge freedom will be

further elaborated upon in section 4. Equation (3.70) may be given a group theoretical classification. We recognize that a four-gradient

operator is a four vector represented by the classification (1/2,1/2) and the structure of the 7

matrices couples this four-gradient operator to the (p state vector in such a way that the resultant quantity transforms as the x state vector. Then

x(x) = y;.d,<p(x) (3.70)

is represented by the symbolic relation

(s,o) © (o,s) + (s - 1,0) e (o,s - 1) = (i,i) ® [{s - i,i) e (is - ^)] (3.93)

There are two special cases of interest. Where s = 1/2 the representation (s — 1,0 ® (0,s — 1) does not appear and (3.93) becomes

(iO) 0 (0,i) = (ii) ®

[(0,i) © (i 0)] (3.94)

We see that the (p{x) and x(^) are basis states of representations having the same group structure and

hence have the same transformation characteristics and that F'*"' = F'"' and G'*"' = G^^^ whenever

s = 1/2

When 5=1 (3.93) becomes

(1,0) ® (0,1) + (0,0) + (0,0) = (ii) (g) [(ii) + (ii)] (3.95)

We have three independent irreducible quantities on the left hand side, a Lorentz six-vector, a scalar,

and a pseudoscalar, while on the right hand side we have the independent quantities, a polar four- vector and an axial four-vector. We are not limited to situations where the basic field is of the (s — |,^) © (|.s — ^) representation. We also have the converse relation where the roles of the x(*) and (p(x) fields are interchanged.

We may have, independently, the symbolic relationships.

is - hh) © (is - i) = (i,i) (g) [(s,0) © (0,s)] (3.96)

and

is - hh) © {h,s -h) = (1,1) ® [is - 1,0) © (0,s - 1)] (3.97)

Finite rotations and boosts of the (fix) and x(^) functions representing a particular spin can be

calculated from (2.12) using the expressions for 5^„<*'' and 5^^*"' and the explicit matrices in table I. Since M'*"^ = M'^^ the rotation operator is the same for either wave function.

We label the argument of the operator Z)**^(A) differently when making a specific rotation or

boost. For a rotation we write Z)rot'''(«?iy) and for a boost we write i^boost*-'' (^m) . The rotation

operator is

D.ot^^'^\^ij) = D,J^'^\§i,) = ex^[i^iMk^^n (3.98)

25
The operator for a boost along the k axis for the x function is given by

fboost^^'''Hfi=;) = exp[-^ijNk^^^ (3.99)

and a similar operator applies for the <p function. Each of these operators may be calculated by writing

the exponential as an infinite power series and looking for algebraic combinations involving even powers

of Si/^^ or of 5*4^''^ that become projection operators for states of a given spin component | X |. This

direct procedure becomes laborious as the value of s increases. An alternative method is to make use

of homomorphism between the homogeneous Lorentz group and the group of unitary 2X2 matrices

of SL(2c). The rotation and boost operator in the space of the x function for any spin may be calculated from recursion formulae involving the matrices of spin 1/2. The reader is referred to works on group

theory for details [26, 271- The boost operator for the <p function may be conveniently calculated by noting that (3.83) allows us to write D(^'*''(^m) as

i>boost^^''^HflM) = [coshflw - <Ti4Sinhflt4][I>boost'^''''-HM] (3.100)

Specific rotation and boost operators for spins 1/2, 1, 3/2, and 2 calculated by the direct method

are listed in table II. That irreducible portion of the x field with the classification (s,0) ® (0,s) is related to fields dis- cussed by Joos [28] and by Weinberg [29] who have developed extensively the transformation properties

of fields with the SL(2c) group classification. While the properties of these fields are well understood

within the framework of the inhomogeneous Lorentz group, no Lagrangian formulation can be achieved

with this Joos-Weinberg representation alone, other than for the spin one-half case. The rotation operators for a rotation about the y axis by the amount tt will be of particular use. We denote this operator simply as D.

D s D,J^\&n = w) =

rf(^'(7r) 0

0 d^'^iw)

(3.101)

From table II we obtain

-1

(3.102)

(3.103)

26
-1

-1

(3.104)

-1

-1

-1

(3.105)

-1

spin.

The S''>{Tr), and consequently D, are symmetric for integer spin and antisymmetric for half integer

This opecator D has the property^

(3.106)

In order to obtain a physical picture of the operational meaning of the y operators, it is instructive

to look specifically at the spin one case. Here, we can also consider a cartesian as well as a spherical

basis for the components.

Consider a four vector field A{x). We can write the components in either basis:

A{x) = A^{x)ey, = Ai{x)ei + Aiix)^ + Az{x)e3 + V{x)ei (3.107)

6 This D is the operator that Pauli calk B; however, we have already usurped B for the designation of the metric operator. Arguments by Pauli require

i) to be antisymmetric for any representation of the y matrices. This reasoning does not apply to matrices of dimension higher than the 4X4 Dirac matrices

for spin 1/2 fields. For the spin 1 /2 case in the Dirac-Pauli representation, D — — y-fyKyh = icn27
or

(3.108)

where the d are unit vectors along the cartesian axes and the |m are unit vectors in a spherical rep- resentation; 64 = i and ^4=1.

1+1 = - ^ (ei + tea)

1-1

e3

1

(3.109)

(ei — 162)

ip(x)

(3.110)

Consider a wave function (^(x) in a cartesian and in a spherical basis where the components of <p{x) are the components of a polar four vector, = {A,iV) and an axial four-vector, = {C,iW) in the

cartesian basis. We can write in a cartesian basis

A^ix)

Azix)

<Pc^Tt{x) = iV{x) s j?(x)

iCi(x)

iCiix)

iCsix)

-W{x)

The phase difference of and is chosen for convenience.

In a spherical basis^

- (Mx) - iA2ix))

A^ix)

^ (^i(^) + iA^ix))

V{x)

i - ^ (C'i(^) - iCiix))

iCz{x)

^ {C,{,x) + iC,{x))

iW{x)

(3.111)

' There is some confusion in the literature as to the designation of spherical components. Those components directed along the unit vectors are

V2

respectively^ according to the definition (3.108).

28
u =

u 0

0 u

(3.113)

The elements of u are defined by the matrix elements of the unit vectors in the two representations

u^v = e^-|„ (3.114)

giving

i i (3.115)

The operators S^^'*"', 5^,^"^ in the cartesian basis are all given by relations similar to

(3.116)

The S^/"^^', S;,/")'^ and y/ are defined in terms of the matrices 5i<''', S,^''', and B' Hsted in table III using (3.56, 3.57, 3.80-3.83).

The matrices Si'"'' and 5i'''' are identical to the matrices which are the generators involved

in the Lorentz transformation of coordinates and momenta, the components of a four vector. The

correspondence is

5/"' = Zu

(3.117)

(3.118)

They are the generators of the four-dimensional orthogonal group 0(4) and obey the commutation

relations (2.14).

The 7^ in a cartesian representation are all real and symmetric'. It is instructive to write out the

' It is possible to interchange in each -y-matrix the 4^'^ with the 8'^ row and the 4"* with the 8'*^ column by a unitary transformation. The dimension

of these transformed 8 by 8 matrices may be reduced to 4 by 4 matrices by making use of the homomorphism of the 2 by 2 submatrices:

1 0 0 -1

1 0 1 1 0

The resultant 4 by 4 matrices correspond to the complex Dirac-Pault 'yni^trices for the spin one-half field. The real and symmetric 8 by 8 7-matrices are a

real irreducible representation of the complex irreducible 4 by 4 matrices.

29
components of (3.70) to indicate the operational meaning of Ji^d^

In a cartesian representation

is)/. V V -

62^3 - ^3^2

A2 dzAi - diAz

As diAi — diAi

iV 0

(3.119)

(3.120)

Ai

A,

Az

iV

-diiV

-dziV

dlAl + ^2^2 + ^3^3

(3.121)

The function Xcart(^)» related to ipcart(^) by (3.70), contains a Lorentz six-vector (i£,(B), a scalar

80, and a pseudoscalar (Bq, so that we can write (3.70) in terms of components, where again we have

chosen phases for convenience

Xcart(a;) =

181 ^1

182 A2

183 A,

-So iV

-(Bi iCi -(B2 iCi -(B3 iCz -i(Bo -W

(3.122)

Carrying out the matrix multiplication in (3.122) we obtain the following relations:

£ = - vr - a^A + V X c

80 = V-A + = d^A^

(B = V X A + vr + a«c

(Bo = - V-C - dJV = -d^C^

(3.123)

(3.124)

(3.125)

(3.126)

We will see in section 6 that, while these relations are invariant under proper Lorentz trans- formations, they change under the product of space inversion and charge conjugation. Since in this spin-one case, the four axial-vector components are independent of the polar-vector components, we may set = 0 and obtain familiar relations that are invariant under all symmetry operations.

80 ~ d^Afi

(3.127)

(3.128)

30
« = V X A (3.129)

©0 = 0 (3.130)

The four divergence, d/^A^, is not required to be zero. Relations (3.127-3.130) are a specific

example of the symbolic relation (3.95) for a spin-one polar vector field.

4. Lagrangian Formalism

A Lagrangian formalism provides a systematic way for obtaining the equations of motion of a

field -vs-ith an infinite numb^ of degrees of freedom and for identifying and extracting the constants

of the motion in a classical field theory. The equations of motion are obtained from Hamilton's action

principle, which requires the action integral to be an extremum for arbitrary variations of the field function. There is also a conservation theorem and a constant of the motion corresponding to each

continuous synmaetry transformation that leaves the action integral unchanged and the equations of

motion invariant in form. This conservation theorem permits observed selection rules in nature to be

described directly in terms of the symmetry requirements of the Lagrangian.

These symmetries follow directly from the Lorentz transformations discussed in section 2. The

proper transformations such as translations, rotations and boosts lead to familiar conservation laws. In addition, the improper transformations of space reflection, time reversal, and charge conjugation

lead to important selection rules in interactions. There are other invariance principles that arise from internal symmetries of the Lagrangian.

These are based on an arbitrariness in the Lagrangian which manifests itself as the concept of gauge

invariance. As was mentioned earlier in section 3 there are certain representations of the states in Hilbert space that are the direct sum of certain irreducible representations. The existence of these

representations leads to a gauge invariance of the second kind. There is also an arbitrariness in the

phase of the states in Hilbert space, where they occur in bilinear combinations. This phase arbitrariness leads to a gauge invariance of the first kind. The coexistence of these two types of gauge invariance

further implies a particular form for interactions between fields. The Lagrangian density describing a field of n components will depend, in general, on n field functions and their adjoints, if the latter are independent^, and on the first order derivatives, with

respect to the space-time coordinates, of the n field functions and their adjoints. This Lagrangian

density is denoted as £{<Pa,<Pa,d^(pa,d^(pa) where a = I'-'ii. The dependence of £ on the space-time

cordinates is only through the fields and their derivatives, and x,, cannot appear explicitly in £.

The action integral

/ = j Ldt = j £d^x (4.1)

does not depend on the special choice of any reference frame, thus it must be a Lorentz scalar. Since

the term d'^x = dx\ dxi dxz dt is a Lorentz invariant, the Lagrangian density, £, transforms as a scalar under the transformations of the proper Lorentz group. This condition makes it much easier to find the functional form of the Lagrangian density. In section 3 we demonstrated that the wave functions ip{x)

and x(^) each spanned a particular representation space from which we can construct invariant bilinear combinations with the adjoints ip{x) and x(^), respectively. Furthermore, we want to have equations

8 We shall always assume, unless we specifically indicate otherwise, that the adjoint field is a quantity that is linearly independent of the field, such

that separate equations of motion can be obtained from the Lagrangian for a field and its adjoint. There are certain self adjoint fields, e.g., the photon, the

neutral pion, or the neutral rho meson where ip = or, in a quantized theory, where the wave functions are considered to be operators, ip = ip. \n this self adjoint case, we have only one set of equations of motion for the n fields. The Lagrangian will have an additional factor of 1 /2 to avoid double counting for these self adjoint fields.

31
of motion linear in the field and exclude therefrom derivatives of higher than second order. These

requirements shall be met if we demand that £ shall contain bilinear quantities x{x)xix) and (p(x)(p{x)

and not powers of these bilinear quantities. Two other bilinear quantities occur in the case of spin 1/2

fields where x(^) and (p(x) span the same representation space, namely, x(x)(p(x) and (p(x)xix).

Variation of the action integral with respect to the field functions, (p{x) or ip{x), independently,

while requiring the action integral to be stationary, yields the Euler-Lagrange equations of motion

d£ ^ d£

d(p{x) " d d^tp{x) (4.2)

d£ ^ d£ _ ^ (4 3)

dip{x) d{(p(x) d^)

One can construct a canonically conjugate momentum, irix), for each variable <p(x)

. s d£ , ^ t(^) = —-r (4.4) d^{x)

and similarly

Hx) = -1^ (4.5)

From these we may calculate the Hamiltonian density

3C = Tr{x)<p{x) + ^{x)tt{x) — £ (4.6)

In (4.6) it is necessary to express <p(x) and ip{x) as a function of Tr{x) and irix) by the use of

(3.70, 3.73). Thus, the Hamiltonian density (ira,<Pa,dk(Pa,Ti'a,'Pa,dk(Pa) is understood to be a function of

the canonical momenta, the field variables, and their spatial derivatives. The time derivatives are elimi- nated. In a quantized theory we will require the field operators and their canonical momenta to obey

the Heisenberg relations:

i[H,<p(x)] = ^{x) (4.7)

i[HMx)] = 7r{x) (4.8)

where H = j X, (Px.

It is natural to write a free Langrangian utilizing the lowest possible number of first order derivatives of the field. For spin 1/2, such a Lagrangian is

£ = -hW{x)x{x) + x{x)^{x)] - m^{x)ip{x) (4.9)

Writing our x and x in terms of (p and tp, we have

£ = M^(^m7m — — <pi'Ynd^ + m)(p] (4.10)

32
The Euler-Lagrange equations yield the famiUar Dirac equations of motion for the spin 1/2 field

(p(x) and its adjoint ^{x):

(tA + mMx) = 0 (4.11)

Hx){d^y^ - m) = 0 (4.12)

The description of particles of fields with spins greater than one half requires a Lagrangian that is at least quadratic in the first order derivatives. We shall reserve the notation and ip for the wave

functions of these fields with spins greater than 1/2 and denote the wave functions of the above first order equations (4.11, 4.12) by and xp. For any spin greater than 1/2 we can write the Lagrangian as

£ = xix)x{x) - m?ip{x)^{x) (4.13)

or when it is expressed only in terms of and ip

£ = —<p{x)d^y^ypd^{x) — m^(p{x)<p{x) (4.14)

The Euler-Lagrange equations give

(y^y.d^d, - m^)ipix) = 0 (4.15)

ip{x)id ^d.y^y, - m2) = 0 (4.16)

We can use (3.58, 3.59) to write the relation

y^yy = + iff^y (4.17)

Substituting this expression for y^y^ in (4.15) will simplify the expression since the term containing

a,,, is zero, for we can interchange summation indices and coimnute the operators and dy and find that the term is equal to its negative, due to the antisymmetry a^y = —ay^.

This yields the Klein-Gordon equation for a free field in which there is no coupling between the

various spin components

( - m'Mx) = 0 (4.18)

Hx)in -m') =0 (4.19)

It is more instructive to consider a charged particle with a spin greater than one half interacting

with a massive polar vector field, A^{x). We use a gauge invariant type coupling with coupling constant g. This is the usual minimal coupling which will be discussed more fully later. Here

-> - igA^ix) (4.20)

The equation (3.70) becomes

x(x) = y^(d^ - igA^(x))<p{x) (4.21)

33
and equation (3.73) becomes

(422)

We can use this formalism to describe the electromagnetic field by considering the mass of the vector boson field to be zero. In this case the neutral polar vector field denoted by ??(x) is best represented in a cartesian basis with components A^{x) since these are the components that occur in the interaction (4.21, 4.22) so that

t?(x)

A,{x) A(:^)

As(x) Ai{x)

Ai{x) 0

0 0

0

0

0

(4.23)

The space-like components Ak{x) are hermitian and the time-like component Aiix) = iV{x) is antihermitian, i.e., V(x) is hermitian. We occasionally use a shorthand notation for vector fields in which we contract the space components 1,2,3 and 5,6,7 of ??(x) into a vector A(:*;) and a null pseudovector.The Lagrangian can be written as

£ = —^{d^ + igA^)y^y,{dp — igA^)ip — m^^iptp

(4.24)

The field z? is a neutral field and is self adjoint, i.e., = so in essence we have three independent

fields (p, (p, and Variation of the action integral with respect to <p gives, from the Euler-Lagrange

equation,

(d^ — igA^)y^y,{d^ — igAy)(p — mfy = 0 (4.25)

Again from the relation 7^71. = + 1(7,,^, the term containing the ct^^^ operator can be written in an

asymmetrical form by changing summation labels

i(r^v{d^ — igA^){dy — igA^) = ^a^yF^, (4.26)

where

Ffji V ^ liA V d yA fi (4.27)

Thus we obtain

(d^ — igA^y —

I

a^yF^y + (4.28)

and a similar equation for the adjoint

g — {d^ + igA^y + - (TiipF^y + m^M = 0 (4.29)

34
These are equations of motion for <p and ^ in a massive vector boson field with couphng

constant g. It was stated that these equations of motion appHed to particles or fields having spin greater

than 1/2. They also may apply to particles of spin 1/2 since the solution of the Dirac equation,

[y^d, - igA,) +m]xl, = 0 (4.30)

is also a solution of (4.28). Indeed, particles of spin 1/2 can be treated by a second order theory with

alteration in the normalization conditions and with certain redundancies [30, 31].

The Lagrangian (4.24), when ip represents a charged vector boson, is equivalent to the formulation

of Lee and Yang [32] with their parameter ^ equal to unity.

Some caution should be exercised in the application of the equations of motion (4.28, 4.29) since

the equations contain only the components corresponding to those used in the variation of the action

integral. This is of importance primarily in the case of the spin-one field where only the upper four

components of ip{x) are involved for a polar vector field and the lower four for an axial vector field. Here, only part of the matrix operator a^^ is effective and we can make the association = S^^'*''. In general, for fields having spins greater than unity, all of the operator a^p contributes, and it contains terms not contained in the corresponding spin operator 5^/*"^ We may consider a field of spin s to have

an anomalous magnetic moment of 1/s by consideration of the nonrelativistic limit of these equations

of motion^.

The equations of motion (4.28, 4.29) obey all of the Velo-Zwanziger criteria that the solutions

are of a causal nature for any spin field. The equations remain hyperbolic and the characteristic surfaces which determine the maximum propagation velocity assure this propagation on or within the light cone. When the variation of £ is performed with respect to the field we obtain

[—yxy.dxd,d{x) + niA'^^ix)]^ = — ig[ipy ^y v{d , — igAy)(p — ip{d, + igA,)y,y^(p] (4.31)

We must specify the n component of the quantity on the left side since the variation was performed

with respect to that component. The quantity on the right hand side is the four-vector current of the

tp field and can be written as

j\ = -igi^y^x + xyM (4.32)

thus we have for the ju component of t}(x)

[-n + rriA'U.ix) = j,ix) (4.33)

This is the equation of motion for the massive vector field A^ coupled to a vector current source, ]\(x).

There is other information to be obtained from (4.31), however. Remembering the example (3.70)

there is another quantity xw that is related to t?, a four vector field, by

X(M = y,d,Hx) (4.34)

9 The nonrelativistic limit may be achieved by letting E —> W -\- M and neglecting higher order terms where gV <^ m and W <^ m. We then obtain

the Schrodingcr equation after dividing by 2m and dropping the rest energy term as well as including only the space-like components in (p(x) and neglecting the time-like and small components. We have effectively — (P - gA(x))2 - (W - eV(x)) -

2m

<f(x)^^ = 0

where p = —iV and W = —di

35
The components of X(<»(^) are those of a Lorentz six-vector (iS,(B) and of a scalar So

i8i(x)

i&2ix) - 8o(x) - (&i{x) - (&2{x) - (BsW

0

iS{x) — &o{x) — (Rix)

0

(4.35)

Rewriting (4.31) as •

[— y\d),xm{x) + TUA^^ix)]^ = ]^{x)

we obtain the component equations

V X (B - - V8o + niA^A = j V'S + dt&o + thaW = p

(4.36)

(4.37)

(4.38)

where ip = Although these are the only equations that are obtained by a variational principle, it is of interest to calculate the lower four quantities on the left hand side of (4.36) where /x = 5,6,7,8. The

right hand side is identically zero. We get

V X S + dt(R = 0

V-(B = 0

(4.39)

(4.40)

These are not equations of motion but rather identities which can be shown to follow from relations (3.129, 3.130).

The four equations (4.37-4.40) are a generalization of Maxwell's equations for a massive vector

field which is applicable to the description of an iosbaric singlet neutral vector meson. In a following

paper we shall discuss a generalization to an octet of non-Abelian massive vector meson fields of the

Yang-Mills type [33]. The equations (4.37-4.40) bear a resemblance to the component equations of the Proca equation

in the limit 8o = d^A^ —> 0. However we shall see in section 5 that 8o vanishes only in one particular gauge. Its presence is required in a gauge invariant formalism. The Proca equation is not gauge invariant. Moreover, if m = 0 these equations are not identical to the classical Maxwell equations. Much of the

difficulty in quantizing the Maxwell field is avoided if it is recognized that the Maxwell equations are

a classical limit of a more general quantum mechanical zero-mass equation that contains the term d^Ap^.

It should be made clear that the free Lagrangian for this neutral boson field.

is exactly equivalent to the Lagrangian,

£ = —\Fy.,F^, - \{dy^A^){dyA,) " ^Hl A y.A

(4.41)

(4.42)

which includes the term involving the four-divergence of the field, first introduced, ad hoc, by Fermi [34] in order to achieve a separation of the longitudinal and time-like components for the photon. The above

Lagrangian (4.42) without the Fermi term, — \/2{d^Ai^{dvAp), is the Proca Lagrangian. It will be shown

36
in a following paper that solutions of this Proca equation do not make physical sense in simple potential

problems.

To show that the Lagrangian (4.24) is gauge invariant we must, for the moment, assume that the

fields and are distinguishable and commute with one another, i.e., they are Abelian fields. We

consider a gauge transformation of the first kind on the (p field. This transformation is a unitary trans- formation

(4.43)

where \{x) is a hermitian scalar field with the same physical mass as the (p field. A change in gauge means a change in phase, a change that is, at first sight, devoid of any physical consequences for a bilinear quantity involving ip and ^. Also we have

<p{x) = G-Vix) = e-i^^M^'ix) (4.44)

In order that the action integral be invariant under this gauge change of <p{x) we must simultaneously

make a gauge change of the second kind on ??(x), such that

i?(x) = t?'(x) - y^d^x)

where A(x) is a scalar field function of the form

(4.45)

(4.46)

Mx) =

0

0

0

-Hx)

0

0

0

0

0

-Hx)

0

0

(4.47)

and where \{x) is the same scalar quantity as in (4.43). Then ^'{x) is of the form

A(x) + V\(x)

Ai{x) + dik{x)

0

0

(4.48)

It is more convenient to use an equivalent form for the Lagrangian where the components of the

I? field are explicitly labelled.

£ = —^{dy. + igAy)y^y„{du — igAy)(p — m^'^^<p — IF^.F^^y — |8oSo — \mA^A^Ay, (4,49)

The gauge change equivalent to (4.45) is

A^{x) A/(x) = A,(x) + dMx) (4.50)

The simultaneous gauge transformation has the invariance properties

G-i(a, - igA/)<p'ix) = {d, - igA.Mx) (4.51)

37
The gauge transformed Lagrangian expressed in terms of the original fields ^, 93, and is

£' = —^{d^ + igA^)y^y,{dy — igA^)<p — m^^^<p - Hd^A, - d.A^y - \{d^A^{dA.) - \mA'^A^A^ - {dA.){d^dMx)) - \{d^d^\{x)y - mA^A^{d,\ix)) - hmA^d^x))^ (4.52)

If we impose the condition that the gauge field obey the equation of motion,

i-n + rnA')Hx) = 0 (4.53)

then

£' = £ - mA^d,[AMx) + ^\{x){dMx))] (4.54)

This latter term contributes nothing to the action integral since the term is a perfect differential. Thus

J

£' d^x = J

£d^x (4.55)

for any physically meaningful local field^°. The equations of motion of (p'(x) and <p{x) have the same

physical content. The A^ fields are altered in their component structure but no physical characteristics of interaction terms are changed. This point will be elaborated upon in a following paper. We have

here a restricted gauge transformation, since \{x) is not entirely arbitrary but subject to condition (4.53).

To show that the current (4.32) is a conserved quantity, we make use of the symmetry properties

of the Lagrangian (4.49) under this gauge transformation, which is considered to be infinitesimal. This

gauge group is a continuous group, and we can make use of variational methods which will yield a con- servation law resulting from the internal symmetry properties.

From (4.48, 4.50, and 4.54) we have

d(p{x) = ip'(x) — (p{x) = ig\{x)<p(x) (4.56)

d(p{x) = <p'{x) — (p{x) = —ig\(x)<p(x) (4.57)

8A^ix) = A/(x) - A^(x) = d^x) (4.58)

d£ = £' - £ = -&o{x)d^d^\{x) - mA^A^{x)dMx) (4.59)

We can also express the variation of £ as

d£ d£ d£ d£ d£ ^ , d£ ^ ^ 5£ = — + —— dd,<pa + — + -— + —-dA, + —— 5d,A, (4.60)

10 The requirement of constancy of the action integral is all that is required by most symmetry arguments. This is a less restrictive condition than the

often imposed requirement of form invariance of the Lagrangian density

To satisfy this latter condition the masses of the A field and the gauge field X(x) must be zero. The equations of motion are form invariant in either case. Arguments that are not based on the spatial integration of £, such as those made in the application of Noether's theorem, require the retention of the

perfect differential term.

38
Equating (4.59) and (4.60) and making use of the Euler-Langrange equations we obtain

d^{ig\{x)[-^{d, + igA,)y,y^ip + ^7^.7.(5. - igAy)<p]

-F^^dM^) - &odMx)} + &od^d^\{x) + niA^A^d.Hx) = 0 (4.61)

which reduces to

Ugixy^'P + ^7mX) + d,F^, — d^&o + mA^A^]d^\(x) + {d^ig{xy^(p + ^y^x)\Hx) = 0 (4.62)

The coefficient of d^\(x) is zero since it is the component of the generalized "Maxwell" equations

(4.38, 4.39). Since the function X{x) is nonzero we have

df.igixy^'P + ^7mX) = 0 (4.63)

or

d^j^ = 0 (4.64)

This conserved current can be separated into two parts. Writing out the current of (4.32) in terms of ip and tp

;V = —igi^y^y^id, — igAy)<p — ip{dy + igA,)y,d^(p] (4.65)

We can write this as the sum of tw^o terms, a "conduction current," which does not involve the

spin components explicitly and a moment current, which does, by using the familiar relation

= + yV-^ (4.66)

where

j/'^ = -igip[{d^ - igA^)ip] + igiHd^, + igA^)]<p (4.67)

;m""^ = gd.i^cx,.^) (4.68)

The four-divergence of the moment current is zero because of the antisymmetry of c^^;

a,;/"' = gd^.d,{^a^^) = 0 (4.69)

The moment current is conserved separately; therefore the conduction current must be conserved

also. The four "Maxwell" equations (4.38-4.41) are, of course, invariant under the gauge transformation

A^{x) -> A,'{x) = A^{x) + d^\{x) (4.50)

From the definitions of £, (B and So we have £' = £ and (B' = (B while

So' = So + d^d^x) (4.70)

39
Involving the condition (4.53) we see that the Maxwell conditions can be written in the new gauge in an invariant form.

V X (R' - dtS' - VSo' + tua^A' = y (4.38')

+ dt&o' + mA^V = p' (4.39')

V X £' + dt(R' = 0 (4.40')

V-(B' = 0 (4.41')

In section 3 it was shown that we have form invariance of the quantities X = 7iid,i(p and x = —(pd^y,,

and thus we have form invariance of the Lagrangian under Lorentz transformations. The invariance of

the action integral under any continuous symmetry group will induce a conservation law for certain

physical quantities which can be derived for any physical system once the Lagrangian is known. This

conservation law is known as Noether's theorem [35] and is written in terms of the variation of the

coordinates and the fields

:

d£ d£ \ d£ d£

' £5^„ dy(p — <pdp —— I 5xp -| d(p + 5(p —— + 5fl^ = 0 (4.71)

The translation group is a continuous group that leads to the energy-momentum conservation law. The translation

Xp xj = Xv \- (4.72)

corresponds to a variation of the coordinates

hxy = ay (4.73)

Some care must be exercised in the expression of the variation of the functions ^(x) and ^pipc). Expression

(4.71) is applicable for the definitions

h<p{x) = ^\x') - <p{x) (4.74)

d^x) = ^'(jc') - Hx) (4.75)

For pure translations, the relation (2.62) indicates that

d<p{x) = 8ip{x) = 0 (4.76)

Other formulations of Noether's theorem use different definitions for the variations of <p{x) and

^{x) [36]. The fact that two Lagrange density functions lead to the same set of equations of motion if they

differ by a four divergence, £ and £ + d^'^^ may be included in Noether's theorem with the condition

that

j d^x a^Sn^ = 0 (4.77)

40
Inserting (4.74, 4.75) into (4.71) and making use of the arbitrariness of a„, we obtain

d£ _ d£' = 0 (4.78)

The conserved quantity in the square brackets is called the canonical energy-momentum density

tensor 3^„, thus

a^3^, = 0 (4.79)

The free Lagrangian (4.14) yields the specific energy-momentum tensor

3^.- = (XX — m^^^<p)d^, — xy^d,(p + ^d^y^x (4.80)

We denote the quantity P„ as the total four-momentum of the field where

(4.81)

The quantities (P* and (Po = are the momentum and energy densities of the fields and their space

integrals are constants of the motion by virtue of (4.79, 4.81), i.e..

—i J

d^'3^yd^x = —i J

(di'34y + d^'ikv) d^x (4.82)

Using the Gauss theorem, the three dimension volume integral involving 3*3^^ vanishes if the fields are

well behaved at infinity, thus

d

dt P. = 0 (4.83)

The momentum and energy densities are

(4.84)

<?o = Xli^KP — ^dijiX — XX + Tn/<P<P (4.85)

The latter expression is of course equivalent to the Hamiltonian density of (4.6) provided that the time

derivatives of <p are eliminated by the usual procedure.

We may denote an energy current or energy flux density by a generalized Poynting vector where

= —i3ki = i{x7kdi(p — (pdiykx) (4.86)

The remaining portion of the canonical energy momentum tensor is the momentum current density or

stress density tensor, 3ij. From (4.79) we may establish the continuity relations that relate the time

41
rate of change of energy density to the divergence of the energy flux density and the time rate of change

of the momentum density to the divergence of the stress density in the field, i.e.,

where the dyadic 0"^ is defined by

V-( + - (Po = 0

at

V • ffy + - (Py = 0

dt

(4.87)

(4.88)

(4.89)

These relations are the continuity equations for the energy and momentum densities of the field. The six parameter group of Lorentz rotations and boosts is the continuous group that leads to the

angular momentum conservation laws. Here the transformation

corresponds to the variation of the coordinates

3Xfi ZfiyXy

while the variation of (p{x) and ip{x) follow from expressions (2.62, 2.17)

8<p{x) = ip'{x') — (p(x) = - £^v5;,/*'V(^)

8ip{x) = <p'ix') — ip{x) = — - £^,-<p(x)5;j/«'^

Inserting these in the expression for Noether's theorem (4.71), we get

(4.90)

(4.91)

(4.92)

(4.93)

J

d^x ^

d£ 1 d&

2 ddyfp

£„x = 0 (4.94)

which we can write in terms of the canonical energy-momentum tensor after making use of the

arbitrariness of the antisymmetrical quantity e^x

d^[x\:i^y — a;,3^x + ixy^S„),<p + i^S^xJuxl = 0 (4.95)

We can call the term in the square brackets 31I^^x, the angular momentum density tensor

d^mi^.x = 0 (4.96)

The conserved angular momentum is defined as

J^x = i j £^4.x d'x (4.97)

42
From (4.96) and the Gauss theorem we can write

^/.x = 0 (4.98)

Using the Lagrangian (4.14) and the definition (4.92, 4.93) we can write Ax in an obvious form:

The sum of the space integrals of an orbital angular momentum density and a spin angular momentum

density

Ax = y

dH[xy(?x - xx(Pv - XT45.x<P - <p5.x74x] (4.99)

Neither the spin angular momentum nor the orbital angular momentum is separately a constant of the

motion, only the total angular momentum is. This is true of all relativistic fields having a spin other

than zero. The tensor character of under proper Lorentz transformations is a consequence of the

Lagrangian density £ transforming as a scalar. This canonical energy momentum density tensor is not

in general symmetric. It may be symmetrized by proper choice of the in Noether's theorem (4.71)

using a generalization of method due to Belinfante [37]. The symmetrical tensor is given by

= 3,. - d^G^,, (4.100)

where

i Gx^v = - [x7x5^^ + xy^S,\(p + X7v5^x^ + ^j^^.txx + ^^^TmX + 'pS^x-y^xl (4.101)

We are assured

d^0^. = 0 (4.102)

since the tensor G\^y is antisymmetric in the first two indices

a^^xCxM. = 0 (4.103)

Furthermore, the angular momentum tensor defined in terms of 0^^ becomes

an^.x' = ^x0M. - xS^x (4.104)

which has the property that

d^'M.yx' = a^an^.x (4.105)

The ten quantities

= -i jQ,,d'x = -i j

^i-'d'x (4.106)

J^., = i j 3114^/ d^x = i j 3114^. dH (4.107)

43
are conserved as a consequence of the invariance of £ under inhomogeneous Lorentz transformations.

The total four momentum P„ and the operator Jy,y are not changed, however, the density distribution features of the energy and momentum given by 3^1^ are lost. There is another interesting feature that can be obtained from the canonical energy-momentum

tensor for interacting fields. Consider the Lagrangian (4.24) involving a charged field of nonzero mass

with spin greater than one half interacting with a neutral vector boson field. This canonical energy-momentum tensor is

3,. = £5,. - TT- - (^a.) T— - (4.108)

aoyip a<po,i oa^v

We make a separation of this tensor, in a particular way, into parts which we shall denote as '5^^'^^ and 3^^**'. The total 3^^ always obeys the conservation law 5^3^„ = 0. These parts can be written in a gauge invariant form by adding and subtracting a term jiiA^ to the canonical energy momentum

tensor 3^„ yielding the separation:

= (xx - m^'^^,p)d^y - x7m(5^ - igA.)^ + ^(3. + igAy)y^x (4.109)

while

= i-h^dxyxipd,^ - ^mA^^)^^. + 0dxyxy^d.^ + 0d,y^yxdx^ + j^A, (4.110)

where

jf' = —igixy^'P + <py^x)

and

X = y\{d\ - igAy)^

Explicitly calculating 5^,3^/*'' and 5^,3;,/"" and making use of the "Maxwell equations" (4.37-4.40) we obtain

5;,3^/*"> = -ig{xy,a> + 9y^x){dvA^ - d^A,) = i.Fr,=f. (4.111)

and

a;.3^/''' = -igixy^f + ^y^x){d^A, - d,A;) = i,F,. = (4.112)

The quantity f is the relativistic Lorentz force and /o = —ifi is the rate of change of energy density

or work done.

f = ;o£ + j X (B (4.113)

/o = j.£ (4.114)

This result is quite well known in classical electromagnetism. The point made here is that the

inclusion of the term £o = d^A^ in the Lagrangian and in the "Maxwell equations" for the massive

44
vector field plays no role in the forces or rate of change of energy. The dynamics do not depend on the

gauge, only the kinematics do. Similar results are obtained for a charged Dirac field interacting with a neutral boson field.

5. Free Field Solutions and the Quantization of the Field

We consider first the solutions of the wave equation

{ — ynd^yyd, + m/)ip{x) = 0 (4.15)

or, with the substitution = y^d^^x), the equation

-TmSmxW + m^VW = 0 (4.360

The solutions for a field of spin, s, and nonzero mass, m, are most easily obtained by choosing a Lorentz frame that corresponds to the rest system of the field. In this rest frame only the diagonal portion,

74^4, of the operator y^d^ will contribute, and the wave function is of a simple form. We denote the wave function in the rest system as ^^""^'(x') where

0

(5.1)

The quantity tj'""' is a column submatrix of 4s elements in which the only nonzero element is the r^^, which has a unit occupation number. The quantity 0 is a column submatrix with 4s null elements.

The normalization iV depends upon whether the wave function <p(x) satisfies a first order Lagrangian,

of the form (4.11), which is unique to the spin one-half field and which we denote by yp{x), or of the second order from (4.18) applicable to spins greater than one-half. The requirement that the Hamiltonian

for a free particle at rest in any intrinsic state should be equal to the rest energy of the particle allows us to write:

N = -4= spin \ (5.2)

N = ^ spin > I (5.3) V2mV'

We want to transform from the rest system with four-momentum = (0,0,0,tm) to a state of

four-momentum pn = (0,0,p,iE). In the unprimed system we have the space-time dependence of the wave functions determined by the relations

E p

t' - t coshl2 — z sinhfl = t z~ (5.4) m m

v = V coshn = - r (5.5)

m

The latter follows from the space-time contraction of the normalization volume along the direction of

motion.

45
The solutions corresponding to a field in motion with velocity p/E =

by making a Lorentz boost so that

tanhQ may be readily found

^W(^) = I>boo (5.6)

The operator i)boost^'''^^~K^34) for a boost along the z axis can be obtained from table II and (3.100).

The solutions for a plane wave travelling in another direction may be obtained by subsequently applying

the operator Drot^'^K4>-,^^~4>) defined in (2.46) and reference [19]. As examples, we shall list here those

solutions propagating in the z directional in order to exhibit the structure of different spin fields.

The special plane wave solutions can be written as

EV

(5.7)

(x) = —=«W(p)e^-

V2EV

(pz-E t) (5.8)

for the first order and second order wave equations, respectively. The (r) superscript refers to the 2s + 1 space-like states and the 2s — 1 time-like states in the rest system. The spinors «''"'(/?) are listed in

table IV for fields of spin \, 1, f , and 2. In a similar manner, the negative energy solutions may be obtained in the rest system with four- momentum = (0,0,0, — im) and then Lorentz boosted to the system moving with velocity —p/E.

There is an important difference however, the first order wave equation for spin ^,

(7m<5m + m)\l/{x) = 0

requires that the solution in the rest system be of the form

0

^w'(x')

1 Vv ,(r) (5.9)

The ry^''' occupies the lower two components rather than the upper two for the positive energy solutions. Higher spin fields obey a second order wave equation and there is an arbitrariness as to whether we

have the t?^""' in the upper 4s components or the lower 4s components of the wave functions. We know

that certain neutral bosons will be their own antiparticles and furthermore that a boson antiparticle

will have the same parity as the particle, hence we require the rj''"' to be "upstairs" for bosons. On the other hand the parity of fermion antiparticles is opposite so, that according to arguments to be made

subsequently, we require that tj^''' be "downstairs" for fermions. We have solutions in the rest system

of the form

^W'(x') = V2mV'

Jr) bosons (5.10)

^W'(x') = V2mV'

fermions (5.11)

11 We could have been more general by boosting along an arbitrary direction followed by an arbitrary spatial rotation. This would greatly increase the complication without making the physical picture more cogent.

46
We shall not write the solutions in the system with four-momentum = {0,0,p, —iE), however, since these negative energy solutions will later be replaced by antiparticle solutions. Rather, we make a Lorentz transformation to a system of opposite momentum so that the four- momentum is = {0,0,—p, — iE). This is the same transformation as in (5.6) since v = p/E. These

negative energy and momentum solutions are of the form

xP^'\x) = ^^uJ')(-jD)e- (5.12)

<p^-\x) = ^ uJ-)(-p)e-'<p^-^') (5.13) V2EV

and are related to the positive energy and momentum antiparticle solutions by a unitary transformation

which, in turn, can be related to the positive energy particle solutions (5.7, 5.8) by a charge conjugation

transformation to be discussed in the next section. These negative energy and momentum spinors

«_'''(—j9) are hsted in table V expressed in terms of the true antiparticle spinors ?;''''(/?) for spins ^,

1, f , and 2. Free-field solutions in a spherical basis and solutions corresponding to fields in a central potential

will be treated in a following paper. The wave function x^'''(^) that is related to (p'-'''>{x) by the relation x^''H^) = 7i^d ^<p'-'''> (x) may be

written as

X^-^ix) = . M;W(D)e''p^-^" (5.14) V2EV

for free positive energy fields. We have

w^^'ip) = iy.P,u^'\p) (5.15)

or in the rest system

^(''(O) = -m74«^^H0) (5.16)

The spinors w^^^{p) may be obtained in two equivalent ways; by calculating from (5.15) using the

explicit representation of the 7 matrices or by performing a Lorentz boost from the rest system to that with velocity p/E by use of the expressions in table II appropriate for the x function. The spinors

w'-''^{p) obtained are listed in table VI for fields of spin ^, 1, f , and 2. The spinors corresponding to the

negative energy and momentum solutions for x*'^'(^) may be obtained from

wJ^\-p) = -iy,p,uJ^\-p) (5.17)

These are also related to the positive energy and momentum antiparticle spinors y^^^{p) by a unitary

transformation to be discussed in the next section. Here, we have a relation similar to (5.15) for anti- particles

yW(p) = -iy^p^v^r.^p-) (5.18)

These are listed in table VII for fields of spin \, 1, f, and 2, together with the relation between y^^'^'ip) and wJ''^{—p).

47
Our special plane wave solutions were obtained from a complete and orthogonal set of solutions

in the rest frame by making a Lorentz boost to a momentum p along the z axis. It is easy to demonstrate

by direct calculation that the solutions in the boosted Lorentz frame also constitute a complete and

orthogonal set. We have listed these completeness and orthogonality relations for fields of spin \, 1, f

,

and 2 in tables VIII, IX, X, and XI, respectively. It is important to recall the rules for the idenfinite metric of the Hilbert space. Relations (3.71, 3.72) imply, for example, that

The first 2^+1 spinors obey the relation

t^w(p) = u^r)^{p)y, for r = 1, . • •, (2s + 1)

and the latter 2s — 1 spinors obey the relation

uir){p) = _i,wt(p)^^ for r = (2s + 2), • • 4s

Similar adjoints are obtained for td'-''^p), v'^^'^ip), and y'^''^{p) by use of the same rules. All of the norms and completeness relations are positive definite with this indefinite metric of the

Hilbert space. The completeness relations for fields of spin greater than 2 are tedious but straightforward to calculate. For example, we may use the relation (5.6) to obtain

/ 0

0 0

The free field wave function <p'^''\x) has 4s orthogonal solutions each having 8s components in the general case. Of these 4s orthogonal solutions only 2s + 1 represent independent dynamical degrees

of freedom since 2s — 1 depend on gauge conditions and may be arbitrarily removed by a gauge trans- formation of the second kind.

<p'(x) = ^(x) + y.dMx) (4.450

The function A(x) satisfies the wave equation

(- + m^'^)A{x) = 0 (4.53')

and has the Lorentz transformation properties of a field of spin s — 1 if it is the gauge field of spin s. In a parity conserving theory A(x) has the group representation ( (s — 1,0) © (0,s — 1) ) . The Lorentz

transformation operators will be given by those of x vector since that irreducible portion that operates on A{x) corresponds to the same representation. We may define a positive energy spinor z^''''(p) by the

solution

The spinors may be determined in the same manner as u>^''^p) by making a Lorentz boost from the rest system. These spinors are tabulated in table XII. It will be observed that tv'-''''{p) = m^z^''\p) where r labels only those states that correspond to the time like states in the rest system. The negative energy

and momentum gauge spinors may be obtained in a similar manner. We shall denote the gauge where we have the 4s (canonical) degrees of freedom as the Feynman

gauge and that gauge where 2s + 1 degrees of freedom survive as the Lorentz gauge. These are generaliza48
tions of the nomenclature for the spin 1 electromagnetic field. The Lorentz gauge corresponds in that case to d^A^(x) = 0. The plane wave solutions describing zero mass fields require special consideration. There is no

Lorentz transformation corresponding to a boost to the limiting velocity c. This implies the zero mass

solutions cannot be obtained from the solutions for a massive field by taking the limiting value as m ^ 0. Inspection of tables IV and VI indicate that those spinor components having | X | < s become

infinite, i.e., u'-^'^p) —> oo while at the same time w^'''>{p) —» 0 in such a way that certain bilinear combinations such as or u)''''(/')74w'''Kj3) do not depend on the limiting value of the mass. This limiting procedure is clearly not acceptable because the expectation value of the helicity operator

u'3/1 p I defined by (2.78) would be

-x'^'Kx)yJz<P^^Kx) - v''''U)/374X<'K^) = X5xx' (5.20)

for all possible values of X; —s< X < s. The zero mass case is characterized by a X having only the expectation values of +s or —5.

The spinors for the zero mass case in the particular Lorentz frame p^ = {0,0,p,ip) may be ob- tained from the finite mass spinors of table IV by "renormalizing" those components having | X | < s such that they have propagation characteristics similar to those components having | X | = 5. Several

interesting features emerge. The 4s independent and orthogonal solutions in the massive field case are reduced to 25 -f- 1 orthogonal solutions in the zero mass case. The 2s — 1 solutions that corresponded

to a time-like field in the rest system become proportional to the 2s — 1 solutions that corresponded to "longitudinal" fields in the rest system, i.e., those solutions having | X | < s, and hence these sets are no longer orthogonal to each other. Of course, the concept of a rest system is meaningless and

actually the introduction of the renormalization artifice is, in essence, the abandonment of this concept.

The normalization constant is completely arbitrary since it may be changed at will by a gauge transformation of the second kind.

Some zero mass plane wave solutions are given in tables XIII for fields of spin 1, f , and 2. These

solutions will have the same form in another Lorentz frame p/ = (0,0,p',ip') which can be shown by

using the Lorentz boosts of table II and then regauging the fields. In this zero mass case, the parameter n is given by

fi = In- ^ (2.73)

P

For s > I the spinors w^^^(p) = iy^p^u'^^^p) are nonzero only for those components corresponding

to I X I = s. This result is independent of the gauge. We should expect that the helicity should be

+s or —s for a field of any discrete spin s, however we obtain that result in this formulation only for the spins | and 1. A most surprising feature is that for s > 1 massless fields have zero norm. It follows that the

helicity and all of the elements of the energy-momentum tensor for the free fields are zero. The well-established zero-mass particles occurring in nature are those of spin | and 1. The neutrino

has spin | and zero mass but appears in a different guise than that formulated here. The spinors of

table XIII are modified by a factor (2)"i/2(i _(_ [33] go that the four possible solutions are reduced

to two: One helicity state for the neutrino and one for the antineutrino. These are shown in table XIV.

The photon has spin 1 and zero mass and may be described with the spinors of table XIII in several gauges. When the longitudinal and time-like components are eliminated we have the transverse gauge;

when present we have the Lorentz gauge. There is a "nonphysical" gauge in which the longitudinal

and time-like components are treated independently. This gauge, shown in table XV, is called the Feynman or covariant gauge. In order that this not be in violation of certain physical facts, certain formal subsidiary conditions [39, 40] must be applied at the expense of a cogent physical interpretation.

This gauge, in particular, violates the characterizing group properties of a zero-mass field. It is ironical

49
that the name covariant gauge is appUed to a gauge that is not among the set of hasis states in the

Hilbert space of the inhomogeneous Lorentz group. The conjectured quantum of gravitation, the graviton, is a massless boson of spin 2. Its source is the sixteen component energy-momentum tensor. This divergenceless tensor encompasses all physical

quantities except gravity that contribute to the energy content of space. The Einstein field equations

may be linearized in the weak field limit with the result that a wave equation for a field of sixteen

uncoupled components is obtained. A zero mass plane wave travelling in a z direction will have at most

four nonzero components with only two independent. This is just the case for the spin 2 boson in

table XIII when all components but those having X = ±2 have been gauged away. Our sixteen component spinor can be recast in tensor form but it is not necessary to do so here. The interesting feature

is that the zero norm of our spin 2 massless boson leads to zero energy content of the free boson field. In the weak field limit the contribution of the field energy to the gravitational Hamiltonian is zero. A further consequence is that the free zero-mass spin 2 boson carries no energy or momentumi^. This

statement does not imply that there should be no gravitational radiation, since we are considering here

only a linear theory. The experimental detection of gravitational waves is, at this time, still a matter

of controversy.

Having determined a complete orthonormal set of positive and negative energy classical fields of

discrete spin, it is possible to develop a quantized operator field theory obeying, in a systematic way, most of the customary assumptions of Lagrangian field theory. In an operator field theory that is quantized by the canonical formalism, it is customary to require

that the canonical boson commutation relations, or fermion anticommutation relations, among the

field operators and their associated canonical momenta are such that the quantum mechanical equations

of motion in the Heisenberg picture agree formally with those obtained from Hamilton's canonical

equations for continuous classical systems. These relations are

<p^{x) = i[H,^^(x)] = (5.21)

TT^ix) = i[H,T^{x)] = - (5.22)

where

H = j3C(x) d'x (5.23)

and the quantity 83C(x)/ 8ir{x) is the functional derivative defined by

SlTaix) dlTaix) ddkTTaix)

There are similar relations involving the adjoint fields and their canonical momenta. The normalization

is determined such that the Hamiltonian has the value

H = Nc^ (5.25)

where A'^ is the total number of particles or quanta present.

12 This feature is also implicit in the linearized Einstein equation, following directly from the Bianchi identities.

50
It should be emphasized that this requirement does not always lead to the postulated [41] equal

time commutation relations between canonically related fields and momenta in the general case. The

relations usually invoked

[<Pa{x),-K»{x')]i, = iha$b{-X. — x') (5.26)

[ip^{x),Tff{x')]± = i5^ff6{x - x') (5.27)

X0=I0

[;p,{x),Mx')]± = 0 (5.28)

xo=xo

[^„{x)Mx')]± = 0 (5.29)

10=^0

are not always valid. Simultaneous satisfaction of relations (5.21, 5.22) and (5.26-5.29) cannot be

obtained for field operators corresponding to particles having spin 5 ^ f

The failure of the canonical quantization method for 5 ^ f does not imply that these fields cannot

be quantized, but rather that we must give a particle interpretation of the field (p{x) as the basic postulate.

The Hamiltonian (5.23) is a function of one or more fields which are linear combinations of the creation

and annihilation operators. With this particle interpretation along with Lorentz invariance and locality the usual connections between spin and statistics [42, 43] follow with the conventional choice for commutation relations involving the creation and annihilation operators of the boson fields and anti- commutation relations involving the creation and annihilation operators of the fermion fields. Crossing

symmetry follows from the same assumptions.

Care must be exercised in the construction of Hamiltonian density (4.6) to include only the

canonical conjugate momenta corresponding to each component of the field operator ip{x) and to maintain the ordering of the operators in the expressions. Consider the Lagrangian for a free field with

spin greater than one half:

£ = —ipd ,d ytp — m^^<p<p (4.14)

We introduce the generalized coordinates

<Pa{x) = ^Mx) (5.30)

ip,{x) = :pixn^ (5.31)

where is an idempotent projection operator that projects the a component of <p{x). The canonically

conjugate momenta are given by

d£

TTaix) = -—— = iHx)^|,y^,y^'^a = -ix{x)yi% (5.32)

dipa{x)

d£

Tj-a{x) = = i%yiy^d^<p{x) = I'^aJiXix)

0(Pa{x)

(5.33)

then

(5.34)

51
We must write <Pa, and £ in terms of the canonical coordinates so that the Hamihonian does not

exphcitly depend on the time derivatives of the fields. The '^a is a diagonal operator that will commute

with 74 Then

'Pa = T^a -\- '^a(yikdk^ (5.35)

(5.36)

(5.37)

where we have used the relation 7>,7^ = + ^^n" so that the Hamiltonian density is

3C = IT air a + Ka'^a'^ikdkfP + ^dkdki^aT^a + m^'^qxp + ^5t7i(l — ^a)lidj(p (5.38)

Inspection of tables III and V shows us that

/ 0

0 0

0 0

0 /

spin 1, polar vector

spin 1, axial vector

(5.39)

(5.40)

/ 0

0 1

spin > 1 (5.41)

The last term in (5.38) drops out for all but spin one fields.

It is straightforward to show that Hamilton's canonical equations

a3c(x) a3C(x)

^a{x) = 7-7 — dkdiTaix) ddkir{x)(5.21)

TTaix) = — — + dk dd^paix)

(5.22)

are satisfied and are compatible with the equations of motion (4.15).

To establish contact with the particle interpretation we want to Fourier analyze the field operators

(p{x) and Tr{x). We start from the classical fields and decompose them into expansion coefficients involving

the plane wave solutions of momentum p times certain creation and annihilation operators a^ip) and

a(p) for particles and b^{p) and b{p) for antiparticles, respectively.

We will consider, in the main, non-hermitian fields where a mathematical operation of conjugation

will be associated with particles and antiparticles. Only one type of quantum will be associated with

a real hermitian field, while the complex non-hermitian field has twice as many degrees of freedom and

describes particles which carry opposite "charges." Whenever complex or hermitian conjugation is involved, it is convenient to start at once with non-hermitian fields so as not to be confused by irrelevant

identies. On the other hand, zero-mass boson fields which carry no "charge" must be real hermitian

52
fields. In the following we do not wish to join the cult of the arbitrary complex phase factor. We shall write all non-hermitian fields such that the hermitian field may be obtained by letting - ( - 1) W t (p)y; W (p) t (p)u(^) t (p)

and - ( - 1) (p) 'v (p) -> aW (p)m(^) t (p)

This phase choice'^ shall also guarantee crossing symmetry where all particles are replaced by antiparticles.

The usual ascribed commutation relations for bosons and anticommutation relations for fermions

occur among these creation and annihilation operators

[a«(p),a'^)t(p')]^ = [fe"-Hp),^'<^'np')]± = 5„5(p - pO (5.42)

[a«(p),a(«'(p')]± = [a(^'Up),«^^'^(p')]± = 0 (5.43)

[6W(p),fe(»)(p')]± = [6'^'^(p),6(^'Up')]± = 0 (5.44)

Rather than express these quantized fields as infinite series of plane wave solutions of momentum p, we make use of the relationship

1 to 1 ^00 -7= Z - / d^P (5.45)

to express the quantized fields in integral form.

These quantized fields are: For spin \:

'^^''^ ^

(2^0^ /

^

{a(^)(p)^.(^)e'>- + i('-)t(p)t,W(p)e-'>-) (5.46)

ml/2 r (Po

"^^^^ ^

(2^0^^ / £^ !«'^'np)ti^'Hp)e-*- + 6W(p)i;w(p)eiP-x} (5.47)

For spin > 1

<p(x) = T^— f , f

,

ja('-'(p)u(^)e'>- - (-l)2«6(^>t(p)j;(r)(p)e-ip.xj

(2Tr)3/2 y (2£^„)"2 ^ ^ ' (5.48)(27r)3/2; (2^p)

^""^ ^ (2^2/ (Jy^^ i«<^'np)^<^'(p)e--- - (-l)^'6<^>(p)i;«(p)e--! (5.49)

13 For bosons, this phase choice follows from the definition tJ^'''(p) = 7?J3m^'''(p). The properties of these spinors in the indefinite metric give the equivalent description u^'''(p) = (Brf(7r))rr'w^'''^(p). Likewise we have the relationship 6^^^'''(p) = -(S(i(7r))rr"a^'""^^(p)- We have {Bd(jv))rr'{Bd{-K))rT" =

( —l)^*5r'r"» hence the expression for the hermitian field follows directly.

53
^^^^ " '(2^J (2^ {aW(p)u;«(p)e'>- - (- l)^^6Wt(p)yW(p)e-'>x) (5.50)

"^^""^ ^

(2;^2 i

!a«t(p)tZ;W(p)e-p- - (-l)2»&W(p)r«(p)e'>-! (5.51)

For a zero mass field, only the spin ^ form differs by the factor v^/n/f'p

^P{x) = j

d'p{a^^KpW(p)e'^- + b^^^Hv)v''-Kp)e-'^-'} (5.52)

^(x) = f d'p\a^'^Hp)u^'^(p)e-^p-=' + ¥'Hp)d('^(p)e'p-^} (5.53)

(zTr)^'^ J

In all of these solutions the sum over the index (r) extends over the 45 orthogonal spinors, some of

which may be zero, depending on the gauge. The adjoint spinor is now defined as

mW(p) = ,,-iuWt(p)Y4,, (5.54)

J5W(p) = rj-it>Wt(p)^,r7 (5.55)

where r] is the operator that reverses the sign of the last 2s — 1 solutions to achieve the desired idenfinite metric.In a quantized field theory the field amplitudes are operators on the state vectors in a Hilbert

space. The physical observables are the matrix elements of these operators, and the requirements of

symmetry and covariance shall be imposed upon these matrix elements rather than on the state vectors of the unquantized theory. In a Lorentz transformation x —> x' = Ax + a we may expect the matrix element of the field operator, which is a c-number, to transform in the same manner as the unquantized field according

to (2.53). We require

i^'Mx'W) = D{A){^,<p{x)^) (5.56)

There exists an operator L(a,K) that accomplishes the desired transformation of the state vectors

^'{x) = L{a,A)Hx) (5.57)

From this we get the transformation property for the operator

L{a,A)cp{x)L-^ia,A) = D-^{A)<p{Ax + a) (5.58)

We first consider an infinitesimal displacement by the amount

L(a,l) = 1 - ia^P^ (5.59)

where is the total four momentum defined in (4.81). We make a Taylor expansion of the right hand

side of (5.58) and obtain after cancellation

[PMx]] = idMx) (5.60)

54
In a similar way we consider an infinitisimal Lorentz rotation by the amount Zt^v with the J^^ defined

by (4.99)

L(0,A) = 1 + ^

£. J,. (5.61)

D-\K) = 1 -

^

£..5,/-> (5.62)

Expanding the right hand side, as before, and keeping the lowest order terms we obtain

[/mv,^(^)] = [i(M. - ^.^J - S^.^-^^^pix) (5.63)

The matrix elements of certain observables, formed from products of the field operators such as the four-momentum densities or the charge density, may include undesirable singularities resulting

from certain inherent properties of the vacuum. To remove these features of the vacuum we must

either symmetrize or antisymmetrize the factors in the Hamiltonian and Lagrangian or alternatively

postulate a normal ordering. For the free field theory the two methods are completely equivalent. For

the case of interacting fields in the Heisenberg picture the normal product cannot be as straightforwardly

defined, thus we invoke the procedure of symmetrizing the order of boson operators and antisymmetrizing

the order of fermion operators [44]. The notation : : stands for a symmetrized or antisymmetrized product, e.g.,

: xy^d^ : = WxJ^dytp ± '^d,i^\] (5.64)

The difference between xyn^y^ and : xJii^'^ '• is always a c-number, and this is sufficient also to remove

undesired zero-point singularities that appear when the product is not properly symmetrized.

The operator for the four-momentum of a field of any particular spin may be calculated directly

from (4.81) using the solutions (5.48-5.51) and the spinors taken from tables IV to VII. We obtain

= I

d^p p,{a(r^HvWip) + 6«t(p)5W(p)) (5.65)

P, = Id^pp,{N+^^Kp) + NJ^Kp)) (5.66)

The sum over (r) extends over the 2^+1 space-like states and also the 25—1 time-like states,

if they are present in the gauge considered. The positive definite property of this four-momentum

operator follows from the adopted prescription of the indefinite metric of the Hilbert space of these

time-like states. The charge operator can be similarly expressed in this representation by direct calculation from

(4.32)

O = j

d^p : —iji{x) :

= j

d^p ^(aW + (p)aW(p) - fe« + (p)6W(p)) (5-67)

55
or Q = y"(f3p^(/V+M(p) _ 7V_«(p)) (5.68)

The effect of the symmetrization or antisymmetrization is to dispose of an infinite constant, the

total charge of the negative energy states of the vacuum. The equations (5.66, 5.68) show the symmetric

appearance of particles and antiparticles of equal masses and opposite charges in the quantized theory.

The angular momentum operators ^^.^ for fields of discrete spin are not as easily calculated as the operators for the four-momentum and charge, due to the complexities of the non-diagonal nature

of the spin density terms in the angular momentum tensor. The task is not made simpler by using the

angular momentum tensor that employs the symmetrized energy-momentum tensor. Furthermore,

the particular Lorentz frame that we have chosen with the 5i2 component of the spin along the direction

of motion for illustrative purposes (giving the spinors of tables III to VI) is not suitable for calculation

of in the general case.

If one starts from spinors formulated in an arbitrary Lorentz frame, the calculation of the operator

^iiy can be done in a straightforward way, but it is very tedious for fields of spin greater than one and

will not be done here. It is simpler to verify that relation (5.63) is indeed satisfied for all components

of the operator J^i, with ip{x) in the rest frame or for the components J12 and J34 when (p{x) is boosted

from the rest frame along the z axis corresponding to the spinors in table IV or by calculation of J;^^ in the rest system.

Using the identity

[AB,C] = A[B,C] + [A,C\B = A\B,C\ - \A,C}B

we are able to establish the relations

[a(^)t(p)o(r,(p)^^(.,(p/)] = -§^^5<3)(p - p')a«(p) (5.69)

[a«t(p)aW(p),a(s)t(p')] = 5„6<3)(p - pOa^^'^p) (5.70)

These relations hold whether the a'''^^(p) and a^'''(p) obey the commutation relations of bosons

or the anticommutation relations of fermions. Thus one can establish by direct computation the commutation relations of P^, J^^^ and Q for any field ip{x) and ip{x)

[?„<p{x)] = id,^{x) (5.71)

[?,,^{x)] = idMx) (5.72)

[J^„^(a;)] = [i{x^d, — x,d^) — Sy.,^f'>](p{x) (5.73)

[}^.,^{x)] = \iix,dy - x.d,) - S,/^>]<^(x) (5.74)

[Q,<p{x)] = -g<p{x) (5.75)

[QMx)] = gHx) (5.76)

The relations (5.71-5.74) can be considered to express the invariance of the quantized theory

under infinitesimal translations, rotations, and boosts and the operators and i^^ may be considered

to be displacement and Lorentz rotation operators of the quantized theory.

56
All of the commutation relations of the inhomogeneous Lorentz group are reproduced in quantized

operator form

[P.,P.] = 0 (5.77)

[J^.,Px] = i5,xP. - iS.xP, (5.78)

[J^.Jxp] = i{5^\iyp + 5„pJ^x — d^Jvx — 5,xJ^p) (5.79)

The latter tAs'O relations may be readily established by showing that Fx and Jxp transform as a four

vector and a Lorentz six vector, respectively, under the infinitesimal homogeneous Lorentz trans- formation {0,A).

Fx' = I-i(0,A)FxI(0,A) = Fx -^z.AKM

Jxp' = L-i(0,A)JxpI(0,A) = Jxp - ^s,.[J,.,Jxp]

Furthermore, one can estabhsh that

[Q,F,] = 0 (5.80)

[Q,J..] = 0 (5.81)

Relations (5.77, 5.80) establish P;, and Q as constants of the motion. contains the coordinates

explicitly so that we must calculate the time derivative by the usual Heisenberg relation.

= i[i?,J..] + -J,. (5.82)

dt

Using an expUcit representation for J^, we see that the last term on the right cancels the nonzero

value of the commutator so that

j,. = 0 (5.83)

and J^, is also a constant of the motion.

We conclude that the postulated canonical commutation relations (5.26-5.29) are a special case holding only for spin zero and spin one fields'^ and do not necessarily provide the basis for a quantized

theory, but rather, the particle interpretation of the field is the more fundamental postulate for fields of arbitrary spin. Schwinger [45] using a different line of reasoning, also concluded that the conventional canonical quantization approach fails for fields having a spin s > f

We are now in a position to consider the vacuum expectation values of the field operators; of

particular interest is the Feynman propagator which is readily evaluated. We are interested in a transi- tion from vacuum to vacuum in which a particle is created and later absorbed and only certain

" The canonical commutation rules do not hold if the Lagrangian density for a spin one half field is of the form (4.10). Only if one considers a particular Lagrangian density of the form £ — ~^W(7m^** ~i~ '")^(^) which the canonical momentum corresponding to xpix) is undefined, do we have the canonical commutation relations obeyed. Of course all physical phenomena are described by either Lagrangian.

57
combinations of creation and annihilation operators give nonvanishing contributions. The Feynman

propagator involves the vacuum expectation value of the time ordered product of the field functions

<p{x) or xW- For example, the propagator for the ^ field is

vac (5.84)

where

= i-iy^Hx'Mx)

if Xo > Xo'

if Xo < Xo'

The expressions (5.46-5.53) may be used to calculate this propagator by standard methods of field theory (e.g., see reference [41]) to yield the result

where d^p = d^p dpo and e is a small positive number which tends to zero after the integration, such

that the contour of integration of the four variables po, • • •,pz is along the real axis. The values for the quantities ii^'"'(p)M^'''(p), which give the matrix operators involving the spin sums, may be taken

from tables VIII through XI for fields of spins |, 1, f and 2 in some particular gauges. Unfortunately,

the Lorentz gauges for spins f and 2 are particularly complicated and are not listed. Similar arguments

provide the propagator for the x field

The values for the quantities ii;^'''(p)?Z)''''(p) are also taken from tables VIII through XI. If the x field

is considered the basic and not a derivative field we may set —p^p^ = in the expressions for

tf ^''Hp)^^^'''(p)- Here the results agree with the expressions given by Weinberg [29] for the propagators

corresponding to that portion of the x field in the (s,0) ® (0,s) representation. Other invariant A

functions for fields of any spin may be calculated with the procedures developed in this paper, however

these quantities lie outside the scope of the considerations of this present work.

Thus far we have considered only those symmetries and invariance properties that are obtained

from transformations that differ infinitesimally from the identity transformation. Here, we investigate

those improper Lorentz transformations that do not contain the identity, namely space reflection and

time reversal, and an additional transformation, charge conjugation, under which each state is mapped

into one where all particles are replaced by their antiparticles, the other properties of the state being

essentially unchanged. A satisfactory formulation of charge conjugation cannot be made without

quantization of the field such that required commutation relations are obeyed. It is important in this symmetry transformation that all physical observables involving bilinear products of the field operators

be symmetrized or antisymmetrized according to (5.64). This symmetrization or antisymmetrization is implicitly understood and is written explicitly only when required.

The Lagrangian (4.24) for a charged non-hermitian field of any discrete spin interacting with

a neutral hermitian vector boson field provides us with a convenient model to study these discrete transformations. The Lagrangian, the equations of motion, and the equal time Heisenberg commutation,

relations shall remain invariant under these transformations. The form invariance of the relations

(4.21, 4.22) establishes the properties of the operations for space inversion, time reversal, and charge

conjugation.

(5.85)

(5.86)

6. Discrete Symmetries

58
The first of these discrete symmetries to be considered is that of space inversion, or the parity

transformation in which x —^ — x, while X4, X4. This transformation is the subset of Lorentz trans- formations = Aj:withdet A = — 1 and A44 = 1. We may characterize this transformation by defining

a unitary operator 5" that has the property

(?£{x,t)(P-^ = £<-P^(x,t) = £(-x,0 (6.1)

The action of this unitary operator (P on the field operators is- defined as

(P<pix,t)(P-' = <p^''\x,t) = P<p{-x,t) (6.2)

(Px(x,0(P-^ = x^^'(x,0 = Px(-x,0 (6.3)

where P is a unitary matrix of dimension 8s by 85. Applying this transformation to (4.21) we obtain

Px(-x,0 = P[-y,id, + igA,i-x,t)) + 74(^4 - igA,i-x,t))]P-'P<p{-x,t) (6.4) We can write this in the space-inverted system as

x(P)(x,0 = y^id, - igA,^P^{x,tW\x,t) (6.5)

provided that

P7*P-' = -7* (6.6)

P74P-1 = 74 (6.7)

and that

A(-P'(x,0 = (PA(x,0(P-' = -A(-x,0 (6.8)

^4<^'(x,0 = (S>Ai{x,t)(9-' = Ai{-x,t) (6.9)

The matrix P that satisfies (6.6, 6.7) is

P = vpy^ (6.10)

where the constant -qp has modulus one but must be limited to the values

VP = ±1 (6.11)

since two successive space inversions returns the coordinate system to the original such that P^ = 1. The choice of the + or — sign in (6.11) defines what is called the intrinsic parity of the particle described

by this field. It is a specific rule of transformation of the field which creates the particle by operation on

the vacuum state. This sign may be determined only when interactions are present between different

particles. As we shall see the requirement (6.8, 6.9) determines tjp = — 1 for a polar four-vector field. In general rjp may be determined only for boson field operators. Fermion field operators always occur

in bilinear combinations such that only | rjp p occurs. The invariance of (4.21) under the parity transformation, and similarly also for the adjoint relation

(4.22), assures that a scalar Lagrangian and the equations of motion will be invariant.

59
The invariance of the commutation or anticommutation relations may be verified by consideration

of the transformation properties of the expansion coefficients for the quantized field

=/(i|s(d)irJ<'"'(p)^"'"(p).-'—> - (-l)2*fcWt(p)Pt;M(p)ei(p-x+S()} (6.12)

Changing p to — p in the right hand side of (6.12) and making use of the spinor properties

P«W(-p) = VpBrr'U^^'^ip) (6.13)

Pt;W(-p) = i-iy^r,pB„'V^^'^{p) (6.14)

we obtain

(PaW(p)(P-i = vpa^''K-v)Br'r (6.15)

(P6wt(p)(p-i = r?p(-l)2»6('-')t(_p)5^,^

The factor ( — )^* is positive or negative whether bosons or fermions, respectively, are described by the

field, and Brr' is the rr' component of the diagonal matrix B, the metric operator.

It is possible to get explicit representations of the operator (P making use of the commutation

relations of a''''(p), 6'''*^(p) and their hermitian conjugates, however it is not necessary to pursue this here.Similarly one may show for the adjoint field

(Pipix,t)(P-' = = ip^P^{-x,t)P-' (6.17)

and

(PaW^(p)(P-i = vp*a^''^H-p)Br'r (6.18)

(P5('>(p)(P-i = vp*{-iy'b^''\-p)Br" (6.19)

The properties of the single particle states under space inversion follow immediately from the

definitions (6.13, 6.14).

The requirement (6.8, 6.9) for the transformation properties of the components of the field An{x,t) under space inversion anticipate the transformation properties of the neutral vector boson

field described by i}{x,t). The remaining portion of the Lagrangian is also invariant provided that

m{x,t)(p-' = j?<^'(x,0 = ijp<'')74??(-x,0 (6.20)

60
Here r^p*^"" is not arbitrary but has already been determined by the requirements (6.8, 6.9) so that

rjp'-^^ = — 1. All transformation arguments for the ??(x,?) field parallel those for the <p(x,0, if we

recognize the fact that here the hermitian field is described by letting

_(_l)2.5(r)t(p)j;(r)(p) ^a(r)t(p)„(r)t(p) ^nd - ( - 1) ^^fc*'' (p)i; « (p) 0^''^?)^^ ^ (p)

Next we consider the operation of time reversal in which Xi-^ —Xi while x —> x. This transformation is the subset of Lorentz transformations x' = Ax with det A = — 1 and A44 = — 1. We may

characterize this transformation by defining an antiunitary operator 3 that has the property that

3£(x,03-i = £'^Hx,i) = £(x,-0 (6.21)

The action of this antiunitary operator 3, so chosen that the commutation or anticommutation relations of the field operators remain invariant, is defined as

3^5(x,03-i = ^'^>(x,0 = T^{\,-t) (6.22)

3x(x,03-i = x^^'(x,0 = mx-t) (6.23)

where T is an 85 by 85 unitary matrix. Applying this transformation to the transpose of (4.22) we obtain

mx-t) = T[-y,{d, + igA,{x-i)) + 74(^4 - igA,{x-t))]T-^TUx-t) (6.24)

we can write this in the time reversed system as

X<^'(x,0 = 7.(a. - j^^/^'(x,0)^'^'(x,0 (6.25)

provided that

Ty,T-' = -7* (6.26)

7^74^-1 = 74 (6.27)

and that the field Ay!'''"' {x,t) obeys the relations

A(^Kx,0 = 3A(x,03-i = -A(x,-0 (6.28)

A^^'''ix,t) = 3^4(x,03-^ = ^4(x,-0 (6.29)

A matrix T that fulfills the relations (6.26, 6.27) in our "Dirac-Pauli" representation is given by

T = VTr|Dy^ (6.30)

where D is the operator, corresponding to a rotation by the amount tt about the j axis, with the properties

(3.106). The operator T is symmetric or antisymmetric depending upon whether it corresponds to bosons of fermions respectively. The restriction on the value Tjr is that | »?r |^ = 1- Interaction terms determine the value of Tjr for bosons but not fermions. The operator rj is uniquely defined and determines

the metric of the time reversed states.

61
The invariance of the commutation or anticommutation relations may be verified by considering

the transformation properties of the expansion coefficients of the quantized fields

= / -(^,j2k^2

i«'''Up)7^^'^Kp)e-^-(P-+^« - (-l)^^6«(p)r^«(p)e'-(P-+^')) (6.31)

changing p to —p on the right hand side of (6.31) and making use of the spinor properties in table XVI

to give

we obtain

aa('-'(p)3-i = 7,T(-l)^^a<^''^(-p)((f(^'(7r)).'

3&'^' + (p)3-l = r,Ti-ir^b^^'K-p){d'^'M)r'r

(6.32)

(6.33)

(6.34)

(6.35)

The factor (d'**^(7r))r'r is the r'r component of the rotation matrix d^^'>{ir) defined in (3.102-3.105).

As for the case of the parity operator it is possible to obtain a representation for the time-reversal

operator, 3, however it is not necessary to do so since its action is defined by (6.34, 6.35).

Similar considerations yield the time reversal analog of (4.21)

where

(6.36)

(6.37)

The phase rjT*'" is determined uniquely by the interaction term as in (6.25) to be r/r'"'^ = 1. This calls for a closer examination. We have specified a cartesian basis for the ??(x,?) field. If it is described in a spherical basis, D is given by

D = (3.101)

(^W(7r)

where

d^^iir) = (3.103)

62
The relation (6.37) can be rewritten as

t?(^)(x,0 = DK-&{x,-t) (6.38)

where K is the complex conjugation operator. We make a unitary transformation to get to a cartesian

representation using (3.112).

f/??<^)(x,0 = UDKWm{x,-t) (6.39)

t?cart<^Kx,0 = -i^J?cart(x,-?) (6.40)

This transformation results from the action of the complex conjugation operator on W so that KU^ =

UK, with the result

UDtj = Z)eart = "

/ 0

0 /

(6.41)

In a cartesian representation all of the ju, are real and symmetric so that (3.106) is satisfied even though

•Dcart is the negative of the unit matrix.

Arguments similar to those made for the ip{x,t) field hold for the time reversed creation and destruction operators of the quantized !?(x,f) field. The commutation relations involving the quantized fields will be maintained provided the anti- unitary property of the time-reversal operator is observed. The order of all operators is reversed. Charge conjugation, while not a Lorentz transformation, is a discrete symmetry similar to the improper Lorentz transformations, space inversion and time reversal. We may characterize this trans- formation by defining a linear unitary operator Q with the property that

e£(x,i)e-i = £^^\x,t) (6.42)

where in the transformed Lagrangian the same dynamical state occurs but with all particles transformed

into antiparticles and vice versa. The action of this unitary operator, so chosen that the commutation or anticommutation relations of the field operators remain invariant, is defined as

e<p{x,t)e-' = = C5(x,0 (6.43)

ex(x,Oe-i = x'^'(x,0 = ±Cl{x,t) (6.44)

where C is an 8s by 8s unitary matrix. The plus or minus sign represents an arbitrariness to be resolved

depending upon our choice for the matrix C. Applying this transformation to the transpose of (4.22) we obtain

Cl(x,t) = C[-%{d, + igA,(x,t))]C-'CU^,t) (6.45)

The invariance of the Lagrangian is explicit for it is quadratic in (p{x) and its adjoint and in x(^)

and its adjoint. We can have two operators with the properties

Cfj.Cf = -y, (6.46)

63
and

Cbj^Cb = (6.47)

that leave the Lagrangian invariant. We know, however, that certain neutral boson fields are self conjugate with the same parity so that the matrix associated with boson fields is such that

^<c'(x,0 = CB${x,t) (6.48)

x'^Hx,^) = -CBl{x,t) (6.49)

where

Cb = vc^^^vD (6.50)

On the other hand the invariance of the first order Lagrangian and the assurance of the proper

anticommutation relations require for fermions that

.

. ^'^Hx,0 = CF^{x,t) (6.51)

X^^Hx^t) = C,l{x,t) (6.52)

where

Cf = -vc^^'^vDy, (6.53)

As in the case of space inversion and of time reversal, the phase factor rjc*^' cannot be determined

for fermions but is limited to | ??c'^^ |^ = 1- The interaction term involving the electromagnetic field, a vector boson, determines tjc^^' = 1. The metric operator t] is always defined'^.

If we consider Ap,{x) to be an external field not subject to the conjugation operation then our

transformed relation is

X<^'(^) = 7m(<9m + igA,{xW<^^{x) (6.54)

This is similar to (4.21) except for the reversed sign of the coupling coefficient g. On the other hand,

if A^{x) is subjected to the conjugation operation then

^/«(:t) = eA,{x)e-' = -A^(x) (6.55)

so that the transformed relation is

x(C)(^) = ^^(a^ _ j^j/c)(^))^(C)(^) (6.56)

The charge conjugation oi Aii{x), the component of «?(x), implies that

" ??(<^>(x) = 7,D%ix) (6.57)

15 The relative forms and phases of relations (6.48-6.53) are identical to those obtained by the so-called operation "strong reflection" combined with

the CPT theorem. Strong reflection implies a finite proper Lorentz rotation through an angle tt in the xiX2 plane followed by a finite complex proper Lorentz

rotation through an angle tt in the x^x^ plane and a subsequent transposition of the order of all operators. The expressions for the complex rotation in the plane may be obtained from the formulae in table II with the substitution = —tTT.

64
In a cartesian system D is the negative unit matrix and the ??(x) field is self adjoint, iJcartW = catii^) > so that

t?cart'^K^) = -^oUx) (6.58)

for a neutral vector boson field. The transformation properties of the expansion coefficients of the quantized fields are determined

by the relation

We can make use of the spinor properties of bosons in table XVI to give

CbDW(p) = rjc<^'"<^'(p)

or similarly, the spinor properties of fermions in table XVI to give

Cf^(p) = vc^V^ip)

so that by comparison of the two schemes

eaW(p)e-i = -{-iy'vc¥'\p) (6.64)

efeW + (p)e"' = -(-l)2^r;caWt(p) (6.65)

Lagrangians of the type (4.24) are not the only ones that it is possible to write down. For any

process we wish to describe involving several fields, we must consider that the Lagrangian consists of

the individual Lagrangians of the free fields and interaction terms involving two or more fields. These

interaction terms must be invariant under proper Lorentz transformations and at least invariant under

the product of the improper transformations (PCS.

It is worthwhile to study the transformation properties of covariant quantities of several fields. Five bilinear covariants can be formed from the quantities (p{x) and x(*) and their adjoints. They

transform as a scalar S, a polar vector V, an antisymmetric second rank tensor T, an axial vector A,

and a pseudoscalar P. They are

y(S)(x) = ci:ip{x)<p{x): + C2:x{x)x(x): (6.66)

//^Ka;) = :x(x)yM^)- + :^(^)7.x(^): (6.67)

fiiv^'^Kx) = ci:ip(x)a^^{x): + c2:x(^)o"^.'x(^)". (6.68)

f^^^Kx) = :x(x)y^ys<p{x): + -.ipix) j.yMx)-- (6.69)

fP^(x) = ci:;p{x)ys<p(x): + C2:x(^)75x(^): (6.70)

65

(6.59)

(6.60)

(6.61)

(6.62)

(6.63)
where ci and ci are arbitrary constants. We have put these in symmetrized or antisymmetrized form

depending on whether they are covariants of boson or fermion fields, respectively.

It is straighforward to calculate each of these covariants in another Lorentz frame x' by considering

infinitesimal Lorentz transformations such as

<p'{x') = (l +

^

£..5,/^)^ ^{x) (3.74)

x'(^') = (l +

^

£^^5,/-'^ (3.75)

Relations (3.76, 3.77, 3.84, 3.85) are used to get

fis){x) =f(S)'(x') (6.71)

//^>(^) = A,-y;(^''(^') (6.72)

/./"(^) = A,x-iA. -yx/"'(;c') (6.73)

f,(^\x) = A,ry/^>'(x') (6.74)

y(P)(^) = f(P)\x') (6.75)

in which

A^i,""^ = 5^, — e^v (6.76)

We see that the proper Lorentz transformation properties of these bilinear covariants do not depend on the value of the spin of the field, whether integer or half integer. Each of these bilinear covariants has characteristic transformation properties, also independent of the spin value of the field, under the discrete operations (P, Q, and 3. The properties follow in a straightforward way from the

definitions (6.2, 6.22, 6.43). The transformation properties for bilinear covariants and for components of

the energy-momentum tensor 3^^ are listed in table XVII for the operations ff", Q, 3, and (Pe3. We note

that they are identical to the transformation properties of bilinear covariants and the energy-momentum

tensor involving Dirac spinors [46]. It is these properties that determine the intrinsic phase factors

rjp, rjc and 7]t for boson fields in interaction terms where a scalar product of the field and a bilinear covariant occurs.

Finally, we are in a position to extend the proper inhomogeneous Lorentz group, abstractly defined

by the commutation relations (2.18-2.20), by adjoining the generating elements of the improper

operations, space inversion and time reversal and also charge conjugation. We can make use of the

specific commutation relations of the operator D and the matrices 74 and 75 with the spin matrices

'•'^'> and S^y'-^^ to obtain

PSi/^^P-' = Si/^^ (6.77)

PSu^^^P-^ = -5,4<^' (6.78)

TSi/^^T-' = -5i/^) (6.79)

66
TSu^^^T-^ = Su^^^ (6.80)

CS,/^'C-i = -S,/^) (6.81)

where we have used the definitions (6.10, 6.30, 6.50, 6.53). There is a second set of similar relations where the corresponding components of S;^/^' are substituted for those of 5^/*^'. The transformation

properties of coordinates and momenta coupled with the relations (6.77-6.81) then determines the

transformation properties of the generators of the proper inhomogeneous group to be

(P(l - iaj',)(?-' = 1 + ia,P, (6.82)

(P(l - iaiPi)(P-^ = 1 - iaiPi (6.83)

(P ^1 +

^

^ijji^ (P-I = 1 + i (6.84)

(P(l - %Jki)(P-' = 1 + (6.85)

3(1 - iaj'*)3-i = 1 - ia^Pk (6.86)

3(1 - 104^4)3-1 = 1 + ia4P4 (6.87)

(1 + ^ ^^J.^ 3-1 = 1 + ^

^ijJij (6.88)

3(1 - 0^47*4)3-1 = 1 + ^kjki (6.89)

e(l - ia^P^)e-^ = 1 - ia^P^ (6.90)

(6.91)

These relations hold whether the generators are in the Hilbert space of the <p field or of the x field. Thus we may effectively write the commutators involving the space inversion operator (P as

{(P,Pi] = 0 (6.92)

[(P,P4] = 0 (6.93)

[<P,Jii] = 0 (6.94)

\(?Ju] = 0 (6.95)

The commutators involving 3, an antiunitary operator, will appear to be different than those for (P because of the transposition involved. They are:

= 0 (6.96)

[3,^^41 = 0 (6.97)

= 0 (6.98)

[3,7.4] = 0 (6.99)

67
The commutators involving Q are:

[e,P,] = 0 (6.100)

[ej,.] = 0 (6.101)

The remaining commutators among (P, 6, and 3 will not be considered here, since phase relations are rather delicate and are treated in detail elsewhere for the spin one-half case [47] which corresponds,

when expressed in the Dirac-Pauli representation, to our representation.

7. References

[I] P. A. M. Dirac, Proc. R. Soc. A 155, 447 (1936).

[2] M. Fierz, Helv. Phys. Acta. 12, 3 (1939).

[3] M. Fierz, and W. Pauli, Proc. R. Soc. A 173, 211 (1939).

[4] V. Bargmann, and E. P. Wigner, Proc. Nat. Acad. Sci. 34, 211 (1946).

[5] The wide scope of diflferent approaches to the treatment of relativistic equations of motion and the prodigious number

of investigators associated with these various avenues of development makes it hopeless to list all of the principal references. We, therefore, give those references which should be useful to the reader who wishes to see some of the technical developments and the difficulties which are not discussed in our treatment.

[5a] S. N. Gupta, Phys. Rev. 95, 1334 (1954).

[5b] P. A. Moldauer, and K. M. Case, Phys. Rev. 102, 279 (1956).

[5c] L. L. Foldy, Phys. Rev. 102, 568 (1956).

[5d] G. S. Guralnik, and T. W. B. Kibble, Phys. Rev. 139B, 712 (1965), Phys. Rev. 150, 1406(E) (1966).

[5e] S. J. Chang, Phys. Rev. 161, 1308 (1967).

[5f] S. N. Gupta, and W. W. Repko, Phys. Rev. 177, 1921 (1969).

[5g] A. Aurilia, and H. Umezawa, Phys Rev. 182, 1682 (1969), Nuovo Cimento 51A, 14 (1967).

[5h] L. M. Nath, B. Etemadi, and J. D. Kimel, Phys. Rev. D3, 2153 (1971).

[5i] C. R. Hagen, Phys. Rev. D4, 2204 (1971).

[5j] J. D. Kimel, and L. M. Nath, Phys. Rev. D6, 2132 (1972).

[6] W. Rarita, and J. Schwinger, Phys. Rev. 60, 61 (1941).

[6a] For a more detailed treatment of the Rarita-Schwinger method, see P. A. Carruthers, Spin and Isospin in Particle Physics (Gordon and Breach, New York, N.Y., 1971).

[6b] L. S. P. Singh, and C. R. Hagen, Phys. Rev. 9D, 898 (1974), Phys. Rev. 9D, 910 (1974).

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[8b] N. Kemmer, Proc. R. Soc. 173, 91 (1939).

[8c] G. Petiau, Thesis, University of Paris (1936).

[9] The same statement as in Reference 5 applies.

[9a] N. Kemmer, loc cit. [9b] Harish-Chandra, Phys. Rev. 71, 793 (1947).

[9c] S. N. Gupta, Phys. Rev. 95, 1334 (1954).

[9d] K. Johnson, and E. G. C. Sudershan, Ann. Phys. (N.Y.) 13, 126 (1961).

[9e] Wu-Ki Tung, Phys. Rev. Lett. 16, 763 (1966).

[9f] J. S. Dowker, Proc. R. Soc. A297, 351 (1967).

[9g] A. Z. Capri, Phys. Rev. 178, 2427 (1969).

[9h] A. Shamaly, and A. Z. Capri, Nuovo Cimento 2B, 236 (1971).

[9i] W. J. Hurley, Phys. Rev. D4, 3605 (1971).

[10a] A. Proca, C. R. Acad. Sci. (Paris), 202, 1420 (1936).

[10b] See also P. M. Morse, and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, New York, N.Y., 1953).

[II] E. G. C. Stueckelberg, Helv. Phys. Acta. 11, 225, 299 (1938).

[11a] J. D. Jenkins, J. Phys. A: Math., Nucl., Gen. 6, 1935 (1973).

[12a] S. C. Bhargava, and H. Watenabe, Nucl. Phys. 87, 273 (1966).

[12b] E. Kyriakopoulos, Phys. Rev. 183, 1318 (1969).

[12c] Julian Schwinger, Particles, Sources, and Fields (Addison Wesley, Reading, Mass., 1970).

[12d] Y. Takahashi, and R. Palmer, Phys. Rev. Dl, 2974 (1970).

[12e] A. J. Macfarlane, and W. Tail, Commun. Math. Phys., 24, 211 (1972).

[13a] G. Velo, and D. Zwanziger, Phys. Rev., 186, 1337 (1969), Phys. Rev., 188, 2218 (1969).

[13b] G. Velo, Nucl. Phys. B 43, 389 (1972).

[13c] M. Hortacsu, Phys. Rev., D9, 928 (1974).

[14] J. W. Weinberg, Thesis, University of California, 1943.

[15a] P. A. M. Dirac, Proc. R. Soc. A 180, 1 (1942).

[15b] W. Pauli, Rev. Mod. Phys. 15, 175 (1943).

[16a] W. Heisenberg, Indefinite Metric in Space State, in Aspects of Quantum Theory, A. Salam and E. P. Wigner, Editors (Cambridge University Press, 1972).

[16b] N. Nakanishi, Prog. Theor. Phys. Suppl. 51, 1 (1972).

[17] E. P. Wigner, Ann. Math. 40, 149 (1939).

[18] W. Pauli, Continuous groups in quantum mechanics, in Ergeb. Exakten. Naturwiss. 37, 85 (1965).

[19] M. Jacob, and G. C. Wick, Ann. Phys. (NY) 7, 404 (1959). See also G. C. Wick, Ann. Phys. (NY) 18, 65, Appendix

(1962).

68
[20] E. P. Wigner, in Group Theoretical Concepts and Methods in Elementary Particle Physics, F. Gursey, Ed. (Gordon and Breach, New York, N.Y., 1964). See also P. A. Carruthers, loc. cit. [21] L. C. Biedenharn, On Racah CoeflBcients as Coupling Coefficients for the Vector Space of Wigner Operators, in Spectroscopic and Group Theoretical Methods in Physics (North Holland, Amsterdam, 1968).

[22] J. D. Louck, Am. J. Phys. 38, 3 (1970).

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[24] W. K. Clifford, Am. J. Math. 1, 350 (1878).

[25] W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, in Encyclopedia of Physics (S. Fliigge, Ed.), Vol. 5, Part 1, (Springer-Verlag, Berlin, Gottingen, Heidelberg, 1958).

[26] B. L. van der Waerden, Group Theory and Quantum Mechanics (Springer-Verlag, New York, Heidelberg, Berlin, 1974).

[27] P. A. Carruthers, loc. cit. [28] H. Joos, Fortschr. Phys. 10, 65 (1962).

[29] S. Weinberg, Phys. Rev. 133, B1318 (1964), Phys. Rev. 134, B882 (1964).

[30] H. A. Bethe, and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Academic Press, New York,

N.Y., 1957).

[31] L. M. Brown, Phys. Rev. Ill, 957 (1958).

[32] T. D. Lee, and C. N. Yang, Phys. Rev. 128, 885 (1962).

[33] C. N. Yang, and R. L. Mills, Phys. Rev. 96, 191 (1954).

[34] E. Fermi, Rev. Mod. Phys. 4, 87 (1932).

[35] E. L. Hill, Rev. Mod. Phys. 23, 253 (1951).

[36] See, e.g., D. Lurie, Particles and Fields (Interscience Publishers, New York, London, Sydney, 1968).

[37] F. J. Belinfante, Physica 6, 887 (1939).

[38] T. D. Lee, and C. N. Yang, Phys. Rev. 105, 1671 (1957).

[39] K. Bleuler, Helv. Phvs. Acta. 23, 567 (1950).

[40] S. N. Gupta, Proc. Phys. Soc. A 63, 681 (1950).

[41] G. Kallen, Quantenelectrodynamik, in Encyclopedia of Physics (S. Flugge, Ed), Vol 5, Part 1 (Springer-Verlag,

Berlin, Gottingen, Heidelberg, 1958).

[42] W. Pauli, Phys. Rev. 58, 716 (1940). This proof is based on a first order field equation of motion.

[43] N. Burgoyne, Nuovo Cimento, 8, 604 (1968). This proof makes no reference to the form of the equation of motion

that the field obeys.

[44] See, e.g., S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and Co., Evanston,

111., 1961).

[45] J. Schwinger, Phys. Rev. 130, 800 (1963).

[46] See, e.g., G. Kallen, Elementary Particle Physics (Addison-Wesley, Reading, Mass., 1964).

[47] P. A. Carruthers, loc. cit.

69
Table i The matrix operators S;'^', Si''', B, and I corresponding to fields of spin 5, 1, |, and 2

Spin 1/2:

,(s)

s: ' = -r- 0 1

1 0

As) _

0 0 B =

1 0

I =

0 0 0 1

1 0

0 1

Sp in 1

:

0 10 0 10 10

0 10 0

0 0 0 0

,(s) 1 /2

0 -i 0 0

1 0 -i 0

0 i 0 0

0 0 0 0

,(s) 10 0 0

0 0 0 0

0 0-10

0 0 0 0

1 /2

0 0 0 -1

0 0 0 0

0 0 0 1 -10 10

0 0 0 i

0 0 0 0

0 0 0 i

-i 0 -i 0

,(t)

0 0 0 0

0 0 0 1

0 0 0 0

0 10 0

10 0 0

0 10 0

0 0 10

0 0 0 -1

10 0 0

0 10 0

0 0 10

0 0 0 1

0 /3 0 0 0 0

/"3 0 2 0 0 0

0 2 0 /3 0 0

0 0 /3 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

s(^> =4-

0 0 0 0 0

/3 0 -2 0 0 0

0 2 0 -/3 0 0

0 0 /3 0 0 0

0 0 0 0 0 -1

0 0 0 0 1 0

0 0 0 0 10 0 0 0-100

0 0-30

0 0 0 1

0 0 0 0

0 0 0 0 -/3 0

0 0 0 0 0 -1

0 0 0 0 1 0 00000/3

-/3 0 1 0 0 0 0-1 0/3 0 0

,(t) /2

0 0 0 0 /3 0

0 0 0 0 0 1

0 0 0 0 1 0 00000/3

-/3 0 -1 0 0 0

0 -1 0 -,/3 0 0

,(t) _ /2

0 0

0 0

0 0

0 0

0 1

0 0

0 0

1 0

0 1

0 0 1

0 0 0

0 0

0 0

0 0

0 0

0

•1

0 0

0 0

0 0

0 0

1 0

0 1

0 0

70
Table i The matrix operators Si'*', Si**', B, and I corresponding to fields of spin 5, 1, f, and 2—Continued

(s) _ _i

I ~ /"2

2 ~ /2

0 . i 0 0 0 0 0 0 0 0 0 0 0 0 0

/2 0 /3 0 0 0 0 0 /2 0 /3 0 0 0 0 0

0

At / J 0 /

J

0 0 0 0 0 0 / J 0 0 0 0

0 0 /J 0

ro

I 0 0 0 0 0 Jo 0

- / ^ 0 0 0

0 0 0 / ^ 0 0 0 0

2 /2 0 0 0 / z 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0

0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 -1

0 0 c 0 0 0 1 0 0 0 0 0 0 0 1 0

9 u

A

U

AuAu 0 0 0 A

u

AUAu

AuAu

- /J: 0 0

0 0 0 0 0 Q 0 Q 0 0 0 0 0

u u

AuAuAUAu AuAuAuAU AuAuAu11 n

u u

A

u - i AU

A

u Au

A

u .(t) _ 1 Au

AuAuA

u Au

A

u V J Au

u u

Au n

U

A

u

A

u

A

u

^1 " /2 A

U Au

A

u

AUA

u

A

u Au / b

0 0 0 0 0 1 0 0 -/6 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 V3 0 /3 0 0 0 0

U 0 u U u

AUAU - i 0 0 -11 0 / b 0 0 0

0 0 0 0 0 /b/A 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 /3 0 0 0 0 0 0 /3 0 0

0 0 0 0 0 1 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0

r-x J i 0 0 0 0 0 0 0 0 /3

0 0 0 0 0 0 0 yb 0 0 0 0 0 0 0 0

-/6 0 -1 0 0 0 0 0 0 /3 0 0 0 0 0 0

0 -/3 0 -/3 0 0 0 0 0 0 2 0 0 0 0 0

0 0 -t 0 / D 0 0 0 0 0 0 0 0 0 0

1 u Au

A

U

A

u Au

A

u

A

u I A

u

A

U

AU A

u 0 0 0

0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0

u u

1L nu AUAUAUAU Au U1i AUAuAu uA

u

u Au

A

U i AU AU

A

u

AuAuA

U

A

u i AU n

U

A

u

A

u

I =

nU

AuA

U

AU 1 0 0 0A

u

A

U 0

A

U

1i AU 0 0

0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1
Table ii Rotation and boost operators for fields of spin i 7, §, and 2. Included are the hyperbolic functions involving the parameter Q. in terms of of ^, E, and m. Included also are the projection operators for the polarization component \ a \, denoted by

D^*^ = cos^t?., + i 2 sff^slnW. . (II i) rot ij 2 ij ij 2 ij

ri^^COk*) = coshiq^-2S^sinhini^ (II. 5)

(II. 6)

<o;ft(^) = [l-(4^^)1 + (4^^n coshf^-sL^^sinhf^,] (II. 7)

+ i[(Sk^^r-i]["^h|nk, -fs^tf sinhfq^] (II. 8)

+ ["^^K* (II. 9)

72
Table ii Rotation and boost operators for fields of spin J, 1, f, and 2. Included are the hyperbolic functions involving the parameter U in terms of p, E, and m. Included also are the projection operators for the polarization component | cr |, denoted by ^„—Continued

ki = tanh —

. n lE+m , , n /E+m P

cosh- n = -^S^ slnh- n =«/S^ .2I±E _E_

2 ^ 2m m 2 ^ 2m m E+m

cosh 2 n = i sinh 2 n =

Spin 1/2: Spin 1: Spin 3/2:

Spin 2;
Table hi The matrix operators Si^^', Si'*'', B', and I' corresponding to a field of spin 1 in a cartesian representation

0 0 0 0

0 0 - i 0

0 i 0 0

0 0 0 0

0 0 i 0

0 0 0 0

-i 0 0 0

0 0 0 0

0 - i 0 0

i 0 0 0

0 0 0 0

0 0 0 0

,(t)'

0 0 0 -i

0 0 0 0

0 0 0 0

1 0 0 0

,(t)'

0 0 0 0

0 0 0 -i

0 0 0 0

0 i 0 0

,(t)'

0 0 0 0

0 0 0 0

0 0 0 -i

0 0 i 0

B' = B I' = I

74
Table iv Plane wave, positive energy spinoTS u'-^'' (p) forfields of

spin \, 1, 5, and 2 with a nonzero mass

Spin 1/2:

E+m

0

Spin 1:

n(=^(p)=!

Spin 3/2:

1

0

0

0

0

0

E+m

0

0

0

0

0

(P) M 2ai 3m

0

2E+m

0

0

2/2p

0

0 •(2E-m)p E+m

0

-2/2(E-m)

0

(P)

/ E+m J_

2m 3m

0

2/2(E-m)

0

0

0

1

0

0

0

0

0 _E_

E+m

0

0

2/2 p

0

0

(4E-m)

0

0

-2/2(E-m)

0

0 -(4E+m)p E+m

0

(s), , /E+m 1

0

0

2/2p

0

0

(4E- m)

0

0

2/2(E-m)

0
Table iv Plane wave, positive energy spinors u'-'Mp) forfields of spin §, 1, f, and 2 with a nonzero mas5—Continued

Spin 2:

1

0

0

0

0

0

0

0

v_E

0

0

0

0

0

0

0

0

0

0

0

/3pE

0

0

0

-pE

0

0

0

0

0

(3), . E

u (p) = -

0

0

0

0

1

0

0

0

0

0

0

0

_E.E

0

0

0

0 /3pE

0

0

0

3E^-m^

0

0

0

-/3p=^

0

0

0

- 3pE

0

0

-Hp) = ^

76
Table v Plane wave, positive energy, charge conjugate spinors v'^'Hp) forfields of spin J, 1, f, and 2 and

their relation to the negative energy and momentum spinors uJ^\—p) with a nonzero mass

Spin 1/2:

.Hp) -J^ E+m

0

p E+m

0

1

0

Spin 1:

v^p) - !

" E

0

0

0

0

v«(p)

Spin 3/2:

0

0

0

E+m

0

0

0

0

0

-1

0

0

0 (2E-m)p E+m

0

0

2/2(E-m)

0

0

(2E+m)

0

0

2/2p

(2E-m)p E+m

0

0

2/2 (E-m)

0

0

-(2E+m)

0

0

-2/2p

0

-P

E+m

0

0

0

0

0

1

0

0

0

0

0

0

0

2/2(E-m)

0

0 (AE+m)p E+m

0

0

2/2p

0

0

(4E-m)

0

2/2(E-m)

0

0 (4E+m)p E+m

0

0

-2/2p

0

0

-(4E-m)

0
Table v Plane wave, positive energy, charge conjugate spinors v<'>(p) for fields of spin i 1, f, and 2 and their relation to the negative energy and momentum spinors u_'=>(—p) with a nonzero mass—Continued

Spin 2:

(P) = e (p)

1 1^

0

0

0

E +m

0

0

0

/3pE

0

0

0

pE

0

0

0

/3p=

.(4

(p)

1 2tn2

1

0

0

0

0

0

0

0

pE

0

0

0

0

0

0

0

1 2m2

0

0

0

/3pE

0

0

0

0

0

/3p^

0

0

0

3pE

0

/3pE

0

0

0

3E^-m^

0

0

0

-A3p^

0

0

0

-3pE

0

0

v^'^^p) = (-l)^'(Bd(K))^^, u^P(-p)

78
Table vi Plane wave, positive energy spinors w'-^'Cp) forfields of spin |, 1, f, and 2 with a nonzero mass

Spin 1/2:

Spin 1:

(p) wHp) = -m wHp) = -E w^*\p) = m

Spin 3/2:

2E-m

0

0

0

0

0 (2E-hn)p E+m

0

0

0

0

0

„<^(p)

0

1

0

0

0

0

0

p E+m

0

0

0

0

(a), ^ w -^(p)

0

0

1

0

0

0

0

0_E_

E+m

0

0

0

wW(p) =

0

0

0

2E-m

0

0

0

0

0 -(2E+m)p E+m

0

w^^\p)

_£_

E+m

0

0

0

0

0

1

0

0

0

0

0

p E+m
Table vi Plane

Spin 2:

0

0

0

0

0

0

0

2Ep

0

0

0

0

0

0

0

wave, positive energy spinors w<"->(p) for fields of spin I, 1, f, and 2 with a nonzero mass—Continued

wHp)

2E

0

0

0

0

2 2 -m

0

0

0

0

0

0

0

-2Ep

0

0

0

wHp) =

80
Table vii Plane wave, positive energy, charge conjugate spinors y^^Hp) for fields of spin f, i, ^,.and 2 and

their relation to the negative energy and momentum spinors w_'^^(—p) with a nonzero mass

Spin 1/2:

y^p) = -m v^l\p) y<^=>(p) = -m v^^(p)

Spin 1:

yHp) y^*\p) = m

Spin 3/2:

y^^\p) =1

0

0

0 -(2E+ni)p

E+ra 0

0

0

0

0

(2E-m)

0

0

y<=\p)

0

0

E+m

0

0

0

0

0

1

0

0

0

y^P)

E+m

0

0

0

0

0

1

0

0

0

0

y4p) = -V^

E+m

0

0

0

0

0

(2E-m)

0

0

0

0

0

y^P)

0

0

0

0

0_E_

E+m

0

0

0

0

0

1

0

0

0

0

p E+m

0

0

0

0

0

1

0
Table vii Plane wave, positive energy, charge conjugate spinors f'Hp) forfields of spin h, 1, f, and 2 and their relation to the negative energy and momentum spinors vfj"^ ( —p) with a nonzero mass—Continued

Sp in 2 :

All

(p)

0

0

0

0

0

0

0

0

0

0

0

-2Ep

0

0

0

.(a),'(P) = y^^>(p)

y^^\p) =

0

0

0

0

0

0

0

2Ep

0

0

0

0

0

0

0

y^^>(P)

0

0

0

0

0

0

0

E

0

0

0

0

0

0

0

-P

y^^\p) y^P)

y^'^^P) = (-i)"(Bd(H))^^. w^' Vp)

82
Table viii Completeness and orthogonality relations involving spinors of the spin \ field with a nonzero mass

rs (VIII.l)

— 4 — — 4 — m

(VIII. 2)

(VIII. 3)

r=i

E + n

2m 2m

(VIII. 4)

(r) (r), , UI 0\ + (m^^\ p)

' irp,p^-m

2m

(VIII. 5)

(VIII. 6)

83
Table ix Completeness and orthogonality relations involving spinors of the spin 1 field with a nonzero

(IX.l)

w^^>(p)r^ J%) = -y^%)r^ v^%) = -E6'rs (IX. 2)

(IX. 3)

u^^\p)v^^\p) = v^^\p)u^%) = (Bd(n))

rs (IX.A)

2 u(^>(p)u(^\p) = ^ (r). ._(s) /I 0

(P)V (P) =

q

^w(%)w(^\p) = r=i

I OA /1-fT. ^ r (n(^>> p)^- e(n^^>-p)

(IX.5)

(IX. 6)

(IX. 7)

3 (r) ._(r), , 1 /i+B 0 \ -tA (r), . _(r), . J]w (p)w (p) =

21 0 I+b) ^^^"^

r=

1

(IX. 8)

In Cartesian Coordinates

r=i

(IX. 9)

^ (r), ,_(r), . . _^ PgP/g > u (p)uo (p) = o a H (IX. 10)

84
Table x Completeness and orthogonality relations involving spinors of the spin f field with a nonzero mass

— 4 — — 4 — L£>

-(x), N (s), -(r), . (s), , , '(p)w^ Xp) = -y^'(p)y '(p) = -P|^P^6rs

u^'^\p)v(^>(p) = v(\p)u<^\p) = 0

i (u«(p)aW(p) - v«(p)v«(p)) = (J ;)

r=i^

^ (r), . -(r), . l/I+B 0 \ ^ (r), ,_(r), , E y (p)y (p) = 2\ 0 i+b) E y (p>y

r=i r=i
Table xi Completeness and orthogonality relations involving spinors of the spin 2field with a nonzero mass

u^''\p)u^%) = v^''\p)v^'\p) =

6^^ (XI. 1)

w^''\p)r^u('\p) = -y^%)r^ ^^%) =-£6^3 (xi.2)

w^''\p)w^^\p) = y ^''\p) y^^\p) = -P^P^Srs (XI. 3)

u^''\p)v^^\p) = v^''\p)u^^\p) = (Bd(n))^g (XI. 4)

tr=i

f; v(^\p)v^'^\p) = g u('^\p)u%) (XI. 6)

,l((ii<»'--j°-/)(fe'^'-£)'^'(g'^'-s)^)]} <xi.7,

E

y^''\p)y^''\p) = E w('^\p)w%) (XI. 8) r=i r=i

r=i r=l

86
Table xii Plane wavepositive energy spinorsz'-^^p) for the gaugefields corresponding tofields ofspin 1, |, and 2 with a nonzero mass Spin 1: Spin 3/2;

z (p) = -

0

0

0

1

0

0

0

0

.Hp,

P E+m

(s), , /E+m

Spin 2:

0

0

0

0

0

1

0

0

0

0

0

0

0

pE

0

0

z^p) = i

m

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

E

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

p

87
Table xiii Plane wave positive energy spinors u''>(p) and w<''(p) for zero-mass fields of spin i, 1, f, and 2 in the "physical

gauges"

Spin 1/2:

u (P) - 7^

1

0

+1

0

(2), - 1 V (p) = 7^

Sp in 1 ;

u(^>(p) u^p) = uHp) .^4p)

u^^\p) = u^*\p) as above, Lorentz gauge

u (p) = u (p) = 0 Transverse gauge

w^^\p) = w4p)

0

0

1

0

0

0

-1

0

w^^\p) = w^*\p) =0 in either gauge

88
Table xiii Plane wave positive energy spinors u''^p) and w''^p) for zero-massfields of spin 5, 1, f, and 2 in the "physical gauges' Continued

Spin 3/2:

1

0

0

0

0

0

-1

0

0

0

0

0

0

1

0

0

/2

0

0

-1

0

0 -n

0

u^p)

0

0

1

0

0A

0

0

1

0

0

/2

uHp) =

w^^\p) -2E w^p) -2E

0

0

0

1

0

0

0

0

0

-1

0

0

u^p) = uHp)

u^p) = u(^\p)

w<2\p) = w<^\p) = 0

w^=\p) = w(^\p) = 0

may be gauged awav

89
Table xni Plane wave positive energy spinors u^'^p) and •w^'^Hp) for zero-massfields of spin |, 7, |, and 2 in the "physical gauges^

Continued

Spin 2:

u<^>(p) -

(2), , u (p)

0

1

0

0

0

/3

0

0

0

-1

0

0

0

-/3

0

0

u4p) =

0

0

0

1

0

0

0

/3

0

0

0

1

0

0

0

/3

uHp)

/^\p) = 2E wHp) = -2E

u^p) = u(^>(p)

.(3), (7). u^-^'(p) = u^"(p)

w^p) = w^p) = 0

w^=\p) = w^'^^p) = 0

wHp) = v,Hp) = 0

may be gauged away

90
Table XIV Plane wave positive energy spinors for the zero-mass spin 5 parity nonconservingfield

0

(l+r5)u^'^(p) = 0 (i-hr3)u^p)

-j- (Hr,)v^p) = -

0

-1

0

1

-j^ (i-trg)vHp) = 0

-j^ (i-hr3)t<^\p) = 72 (1^5>^^'^(P> = °

-J-

(i+r^)v^i{p) = 0
Table xv Plane wave positive energy spinors u'l'Hp) for the zero-mass spin 1 field in the "nonphysical" covariant gauge

u (P) u4p)

w^^\p) = -E w4p) = -E w^p) = wHp) = E

92
Table XVI Some relatiouships between spinors in the indefinite metric

(XVI. 1)

r r 4 — (XVI. 2)

(XVI. 3)

(XVI. 4)

DTi^(^\p) = Drv^''^ (p) = u^^\p)

for

bosons

only

(XVI. 5)

(XVI. 6)

-r Tl DT3-^(p) = -r D r u^^^^Cp) = v^^\p) 5 '4

-r. Tl d¥\) = -r^ D v^^^ (p) = u^^\p)

for

f ermions

only

(XVI. 7)

(XVI. 8)

where, e.g.

— 4 — 4 r r

2

Tl = 1

I B tl - 6 , ,, r r r r r r

93
Table xvii Transformation properties of bilinear covariants and the energy-momentum tensor under space inversion, time reversal, and

charge conjugation and their product

Covariant ^ Q. ^ ^'^^

f^'\x,t) f^'^-x,t) f(^>(x,t) f<^)(x,-t) f^'\-x,-t)

ff^^i.t) -4^>(-x.t) -4^\x,t) -4^>(x,-t) -f^^-x.-O '

f^^\x,t) fi'\-2i.t) -f^^x.t) f^^\x,-t) -ff>(-x,-t)

<f (2..t) f(J>(.,,t) (x.t) -f<f (x.-t) 4f(-^,-t)

ff?(x,t) -ff?(-2<.t) -ffPc^.t) ffj\x.-t) ff?(-i.-t)

f^'^^x.t) 4^\-x,t) 4^^x.t) -f^^\x,-t) .4A>(.x,-t)

ff\^.t) ' -ff\-.,t) ff>(x,t) ff>(x,-t) .ff>(-^.-t) 4 4 4 4 4

f^'^x.t) \ -f^'\-^,t) f<^>(x,t) -f^^>(x.-t) f^'\-.,-t)

S'.jCx.t) C.jC-x.t) 3-^j(x,t) JT.jCx.-t) ?.j(-x,-t)

3". (x.t) -J. (-x,t) 3 (x,t) -J. (x,-t)

14 14 14 14 14

^^,^(x,t) ? (-x,t) ^-^Cx.t) 3:, (x,-t) 3: (-x,-t)

44 44 44 44 44

94

U.S. GOVERNMENT PRINTING OFFICE: 1976 O 205-165
NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PURLICATION OR REPORT NO. 2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

The Dynamics of Fields of Higher Spin

5. Publication Date August 1976

6. Performing Organization Code

7. AUTHOR(S)

Raymond W. Hayward

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 76-6O8O89

16. ABSTRACT (A 200-word or less (actual summary of most si^iticar\t information. If document includes a significant

bibliography or literature survey, mention it here.)

There are several difficulties that plague all existing relativistic equations of motion describing

elementary fields having an intrinsic spin greater than one. While the free field equations can be shown

to be explicitly covariant, the introduction of interactions gives rise to a phenomenon of noncausality.

In the presence of interactions, the retarded solutions spread beyond the light cone and the influence

travels faster than light. Furthermore, the solutions in certain simple potentials do not have a finite norm, violating the probabilistic requirements of quantum mechanics.

This paper develops a relativistic theory that is free of the aforementioned difficulties. This

Lagrangian theory describes fields and particles vvith arbitrary mass and charge and having any dis- crete spin, integer or half integer, .f^part from gauge conditions there are no subsidiary conditions. A matrix formulation is used. The generators of the inhomogeneous Lorentz group for a field of any intrinsic spin and mass are defined in terms of Wigner operators of the group SU(2) and a metric operator. A maximal Abelian set of invariants is formed which defines two completely re- ducible representation bases of the inhomogeneous Lorentz group having distinct structures. A set of y

matrices, obeying a Clifford algebra, is also defined in terms of the Wigner operators and the metric

operator. State vectors having difTerent structures and Lorentz transformation properties can be related

to one another by operators involving the 7 matrices. The equations of motion can be obtained from the Lagrangian by variational methods, and

certain aspects of the canonical formalism can be used to quantize the fields. Invariance of the La- grangian under infinitesimal displacements and rotations yield conservation laws and constants of the motion for pertinent physical observables. The metric of the Hilbert space of the states is uniquely

defined for any spin field, assuring positive definite four momenta and charge. The Dirac formulation for the spin one-half field and the Maxwell-Lorentz formulation for the electromagnetic field are special cases of this theory.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

Causality; high spin fields; inhomogeneous Lorentz group; relativistic fields; wave equations.

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