The Dirac Algebra for Any Spin∗

David N. Williams

Eidgenossiche Technische HochschuleZurich

Contents

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1II. Spinor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

III. The 2(2j+1)-Component Formalism . . . . . . . . . . . . . . 8A. Lorentz Covariance . . . . . . . . . . . . . . . . . . . . . . . . . 10B. Plane-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . 13C. Conserved Currents and the Scalar Products . . . . . . . . . . . 14

IV. The Generalized Dirac Algebra . . . . . . . . . . . . . . . . . 16A. Construction of a Complete Set . . . . . . . . . . . . . . . . . . 17B. Infinitesimal Generators . . . . . . . . . . . . . . . . . . . . . . 20C. Representations of 2(2j′+1)×2(2j+1) Matrices . . . . . . . . . 20D. γ-Algebra: Threefold Traces . . . . . . . . . . . . . . . . . . . . 22E. Abstract Characterization of the Algebras . . . . . . . . . . . . 24

V. Plane-Wave Matrix Elements . . . . . . . . . . . . . . . . . . . 25Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 29References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

I. Introduction

The proliferation of resonances in elementary particle physics has broughtabout a renewed interest in formalisms for describing processes involving parti-cles of higher spin. Of course a number of formalisms, based on higher spin waveequations, have existed for many years, for example, those of Dirac [1], Fierz [2],Fierz and Pauli [3], Bargmann and Wigner [4], and Rarita and Schwinger [5].All of these theories use first-order wave equations, and for spins higher thanone-half require subsidiary conditions in order to restrict the number of distinctspins to one.

∗Presented at the Lorentz Group Symposium, Institute for Theoretical Physics, Univer-sity of Colorado, Summer, 1964. Published in Lectures in Theoretical Physics, vol. VIIa,University of Colorado Press, Boulder, 1965, pp. 139-172. Permission for internet publica-tion granted by University Press of Colorado, April, 2003. Except for the table of contentsadded in March, 2008, this version has only cosmetic changes from the original. Some typoshave been fixed; footnotes have been separated from references; unnecessary commas havebeen removed from the spin notation; and a few gaps in equation numbers have been filled,while leaving the original numeration unchanged.

1

For several reasons, there have not been any successful Lagrangian fieldtheories for higher spin based on these equations. First of all, as emphasizedfor example by Fierz [2] and Fierz and Pauli [3], it is difficult to maintainconsistency between the wave equations and the subsidiary conditions wheninteractions are included. Secondly, there is the nonrenormalizability of suchinteractions.1

In recent years, the situation has changed somewhat. One no longer in-sists on the Lagrangian formalism (which is not to say that it is no longerinteresting); but instead one tries to discover as much as possible from generalprinciples such as symmetry, causality, and analyticity. On this view, waveequations are restricted to a kinematic role, when they have any role at all,being applied only to the description of free particles. Their main value is thattheir solutions form Hilbert spaces on which representations of the Lorentzgroup are concretely realized.

With this view, none of the objections mentioned before is applicable. More-over, there is no longer any reason to restrict the wave equations to be firstorder, especially if by doing so one can eliminate the subsidiary conditions. Infact, the emphasis is just the other way. One decides what representation of thegroup one wants to consider, and only then asks what is the most convenientwave equation corresponding to it.

In this lecture, we describe free, massive particles of spin j by means of a2(2j+1)-component formalism, based on the representation (j, 0)⊕ (0, j) of thehomogeneous Lorentz group. Here (j′, j), with j′ and j half integers, refers toan irreducible, finite-dimensional representation of the connected componentof the homogeneous Lorentz group, L↑+; and the direct sum (j, 0) ⊕ (0, j) cor-responds in the well-known way to an irreducible representation of the groupL+, which is L↑+ plus space reflection.

The Dirac equation corresponds to the representation ( 12 , 0) ⊕ (0, 12 ); and

there is a very direct analog of the Dirac equation for the representation(j, 0) ⊕ (0, j), as well as a direct analog of the Dirac algebra. In fact, themain advantage of the 2(2j+1)-component formalism is that all analogies withthe Dirac theory are very close; and one does not really have to learn anythingnew in order to use it.

The 2(2j+1)-component formalism, with the generalized Dirac equation, iscertainly not an essentially new development. The ideas are to a large extentanticipated in the work of Fierz [2]. In the form given here, the formalismhas occurred to a number of people. The first explicit discussion known tome is contained in a paper by H. Joos [7], which also gives the connectionwith the theory of Fierz. Subsequently, S. Weinberg [8] has published a similarconstruction, arrived at in the course of developing Feynman rules for any spin.These lectures are a result of Weinberg’s encouragement to revive and improvean earlier unpublished study on the generalized Dirac algebra.

What I propose to do is first to review the spinor calculus and the construc-tion of generalized Pauli matrices for any spin, and then to list a few properties

1Cf. H. Umezawa [6].

2

of the wave equation and its solutions. Next, the main business of these lec-tures is to show how to compute and classify the analogs of the matrices inthe Dirac algebra, I, γµ, σµν , γ5γµ, and γ5, and to derive the relations amongthem and their traces. This turns out to be quite easy, and it all follows fromthe simple observation that the classification of matrices in the Dirac algebracorresponds to a Clebsch-Gordan analysis.

Finally, I intend to mention very briefly the representation of scattering am-plitudes. In that respect, it is pertinent to remark that the 2(2j+1)-componentformalism offers no substantial advantage over the nonredundant, (2j+ 1)-component formalism, even when discrete symmetries are included, and that,on the other hand, it is also not particularly more difficult to manipulate. Thus,for such applications it seems to be primarily a matter of taste which formalismone should choose.

II. Spinor Calculus

Although I shall try to keep it as painless as possible, it is necessary to use afew detailed properties of spinors and Clebsch-Gordan coefficients in order toget the basic results across. The more important notions are reviewed here.

Recall that L↑+ is the set of real, orthochronous, unimodular, 4×4 matrices,Λ, satisfying

ΛTGΛ = G , G =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

, (1)

where “T” means transpose. I shall use the notation “∗” for complex conjugate,and “†” for Hermitean conjugate. The spinor claculus is based on the well-known two-to-one homorphism between SL(2, C), the group of 2×2 unimodular

matrices, and L↑+.This correspondence can be expressed explicitly with the help of the 2×2

Pauli matrices, written as a four-vector σµ with

σ0 = I , σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

). (2)

We use a special symbol for σµ with the space components multiplied by −1,namely,

σµ ≡ σµ . (3)

Because the σµ are a complete set, we have a 1-1 correspondence betweenfour-vectors vµ and 2×2 matrices V expressed by

V = vµσµ ≡ v ·σ , detV = v ·v , (4)

3

where the summation convention is used. The reality of v implies that V isHermitean. Any A ∈ SL(2, C) induces a Lorentz transformation Λ, accordingto

Av ·σA† = (Λv)·σ , (5)

from which one finds

AσµA† = Λνµ σν , (6a)

Λµν =1

2Tr(σµAσνA

†) . (6b)

For σµ one finds that

A†−1 σµA−1 = Λµ

ν σν . (7)

According to (5), three-dimensional, proper rotations (the group O+(3))are represented by unitary matrices U ∈ SL(2, C), and Lorentz transformationsfrom the rest-frame vector (m, 0, 0, 0) to k, where k ·k = m2 > 0, by

A(k) =√k ·σ/mU , (8)

where the Hermitean, positive-definite square root is intended, and where U isan arbitrary unitary element of SL(2, C).

In the spinor calculus,2 one writes indices according to the following scheme.Let ζ be a two-component spinor. If it transforms according to A, i.e.,

ζ ′ = Aζ ,

then we write ζα, with a lower undotted index,3 taking the values ± 12 . If it

transforms according to A∗ we write ζα; and if it transforms according to therespective contragredient transformations, A−1T and A−1†, we write ζα andζα. The summation convention applies to repeated upper and lower indices ofthe same type. Thus ζαη

α and ζαηα are invariant forms.

Raising and lowering of indices is effected by the metric symbol

ε =

(0 1−1 0

)= −ε−1, (9)

which has the property, for any 2×2 matrix M ,

εMε−1 = M−1T detM . (10)

Thus ε takes any A ∈ SL(2, C) into its contragredient form, A−1T. We writethe spinor indices of ε as

εαβ = εαβ , ε−1αβ = ε−1αβ ,

2Cf. Bade and Jehle [9] and Corson [10], and for the general spin notation, References [7]and [13].

3We use letters from the first part of the Greek alphabet for spinor indices, and lettersfrom the latter part for tensor indices.

4

and we follow the rule of always contracting with the right index of ε for raisingand with the right index of ε−1 for lowering of indices.

With these conventions, the Pauli matrices have indices σµαβ and σµαβ ,

from (6a) and (7); and they satisfy orthogonality relations

12 σµαβ σ

µβ′α′= δαα′δβ

β′ , (11a)

12 σµαβ σ

µβα = gµν . (11b)

The whole apparatus carries over for all finite-dimensional representationsof L↑+ (i.e., of SL(2, C)) [9, 10]. We define a set of matrices Dj(A), wherej is a half-integer, inductively by means of Clebsch-Gordan (CG) coefficientsC(j1j2j3; a1a2a3) (in Rose’s notation [11]):

Dj(A)ab =∑

a1a2b1a2

C(j− 12 ,

12 , j; a1a2a)

× C(j− 12 ,

12 , j; b1b2b) Dj−1

2 (A)a1b1D12 (A)a2b2 ,

(12)

where D12 (A) ≡ A. Equation (12) defines Dj(O) for any 2×2 matrix O, which

may even have operator-valued matrix elements. For A ∈ SL(2, C), the Dj(A)

form a representation of L↑+. They satisfy the identities: Dj(A∗) = Dj(A)∗,Dj(AT) = Dj(A)T, Dj(A†) = Dj(A)†, Dj(A−1) = Dj(A)−1. The general

finite-dimensional, irreducible representation of L↑+, labeled by two half integers

(j′, j), is given by the direct product Dj′(A)⊗Dj(A)−1†.Now we define (2j+1)-component spinors, whose indices take the values

−j,−j+1, . . . , j, with

ζα , ζα , ζα, ζα

transforming respectively according to Dj(A), Dj(A)∗, Dj(A)−1T, and Dj(A)−1†.The metric symbol becomes

[ j ]αβ ≡ Dj(ε)αβ = [ j ]αβ = (−1)j−αδα−β ,

{j}αβ ≡ Dj(ε−1)αβ = {j}αβ = (−1)2j [ j ]αβ .(13)

Then, as before, ζαηα is an invariant form, with

ζαηα = (−1)2j ζαηα .

In order to emphasize that they are invariant spinors [12], we write the CGcoefficients in the form

C(j1j2j3;α1α2α3) ≡ [ j3j1j2 ]α3

α1α2 = [ j3j1j2 ]α3

α1α2

= (−1)2j3 [ j3j1j2 ]α3α1α2

(14)

Our discussion of generalized Dirac matrices is essentially a discussion ofvarious relations betwen spinors and irreducible tensors. The building blocks

5

of the construction are a special class of generalized Pauli matrices, which wereview in the rest of this section. The particular matrices considered herecorrespond to irreducible tensors of the minimum rank, which are thereforetraceless. They are special cases of general constructions given elsewhere [13].

First, for the representation (j, j), we get (2j+1)×(2j+1) matrices, definedinductively by successive application of CG coefficients:

σµ1···µ2j (jj)αβ = [ j, j− 12 ,

12 ]α

α1α2 [ j, j− 12 ,

12 ]β

β1β2

× σµ1···µ2j−1(j− 1

2 , j−12

)α1β1

σµ2jα2β2

,(15)

and from the analogous constructon using σ we get

σµ1···µ2j (jj)αβ = σµ1···µ2j (jj)αβ . (16)

These matrices are Hermitean, and it is straightforward to show [13] that theyare symmetric and traceless, respectively, in the interchange and contractionof any pair of four-vector indices. They satisfy transformation laws preciselyanalogous to (6a) and (7).

We introduce the notation (µ)n = µ1 · · ·µn for the indices of a tensor thatis symmetric under all permutations of indices. When the number of indicesis determined from the context, we usually drop the label n and write (µ) =µ1 · · ·µn.

From the orthogonality relations for the 2×2 Pauli matrices (11a) and forthe CG coefficients, it follows that

σ(µ)(jj)αβ σ(µ)(jj)β

′α′ = 22j δαα′ δβ

β′ . (17)

The “inverse” expression

P(jj)(µ):(ν) ≡ 2−2j σ(µ)(jj)αβ σ(ν)(jj)βα

= 2−2j Tr[σ(µ)(jj) σ(ν)(jj)

] (18)

is the projection operator for the symmetric and traceless part of a 2j-th ranktensor [13].

For representations (j′, j), with j′ 6= j, there are in general several waysto construct tensors of minimum rank. Here, we choose for convenience aparticular symmetry scheme. As in the case j′ = j, we do not prove thesymmetries in question because they are well known, and explicit proofs interms of symmetries of the CG coefficients can be found elsewhere [13].

Consider the representation (1, 0). This corresponds to a second-rank, self-dual tensor:

σµν(1)α ≡ [ 1 1212 ]α

β1β2 [ 01212 ]0

γ1γ2 σµβ1γ1σνβ2γ2

=i

2εµνλρ σ

λρ(1)α , ε0123 = −1 ,(19)

6

where εµνλρ is the alternating symbol. For (0, 1), we have an antiselfdual tensor,

σµν(1)α = σµν(1)α = − i2εµνλρ σ

λρ(1)α . (20)

For (j, 0) and (0, j), respectively, with j an integer, we define inductively

σµ1ν1···µjνj (j)α = [ j, j−1, 1 ]αβγ σµ1ν1···µj−1νj−1(j−1)β σ

µjνj (1)γ

σµ1ν1···µjνj (j)α = σµ1ν1···µjνj (j)α .(21)

We extend our convention for labeling tensor indices as follows: Write(µν)n = µ1ν1 · · ·µnνn for the indices of a tensor that is antisymmetric un-der any interchange µiνi ↔ νiµi, and symmetric under permutations of pairsµiνi ↔ µjνj . Again, we usually write (µν) instead of (µν)n.

We say that a tensor is, respectively, selfdual or antiselfdual (with respectto (µν)) if it is selfdual or antiselfdual in each pair µiνi of (µν).

In particular, σ(µν)(j) and σ(µν)(j) are traceless tensors, with the indicatedsymmetries, that are, respectively, selfdual and antiselfdual with respect to(µν).

Again, we get orthogonality relations,

σ(µν)(j)α σ(µν)(j)β = 22j δα

β ,

σ(µν)(j)α σ(µν)(j)β = 22j δα

β ,

σ(µν)(j)α σ(µν)(j)β = 0 ,

(22)

and projection operators

P(j)(µν):(λρ) ≡ 2−2j σ(µν)(j)α σ(λρ)(j)α

= P(j)(λρ):(µν)

=[P(j)(µν):(λρ)

]∗≡ 2−2j

[σ(µν)(j)α σ

(λρ)(j)α]∗

=[P(j)(µν):(λρ)

]∗.

(23)

To form tensors for any (j′, j), where j′+j is an integer, define

j′ ≥ j : σ(µν)∆(λ)2j (j′j)αβ = [ j′,∆, j ]αα1α2

× σ(µν)(∆)α1σ(λ)(jj)α2β

, (24a)

∆ ≡ |j′−j| ,

j′ ≤ j : σ(µν)∆(λ)2j′ (j′j)αβ =[σ(µν)∆(λ)2j′ (jj′)βα

]∗. (24b)

In the same way as before, we construct (or define)

σ(µν)(λ)(j′j)αβ = σ(µν)(λ)(j′j)αβ , (25)

7

which gives the rectangular matrix relations (square when j′ = j)

σ(µν)(λ)(j′j)† = σ(µν)(λ)(jj′) ,

σ(µν)(λ)(j′j)∗ = {j′} σ(µν)(λ)(j′j) {j}T.(26)

Additional properties of σ(µν)(λ)(j′j) and σ(µν)(λ)(j′j), aside from the sym-metries indicated by the notation, are as follows:

(i) σ(j′j) and σ(j′j) are traceless tensors.

(ii) For j′ > j, σ(j′j) is selfdual with respect to (µν) and σ(j′j) is antiselfdual.For j′ < j, the reverse is true. Moreover, σ(j0) = σ(j) and σ(j0) = σ(j).

(iii) For j′ = j, we get the same matrices defined earlier.

Finally, we have orthogonality relations and projection operators, whichinclude as special cases those given before:

σ(µν)(λ)(j′j)αβ σ(µν)(λ)(jj′)β′α′ = 22M δα

α′ δββ′ ,

M ≡ max[j′, j] ;(27)

P(j′j)(µν)(λ):(ρσ)(τ) ≡ 2−2M Tr[σ(µν)(λ)(j′j) σ(ρσ)(τ)(jj′)

]=[P(j′j)(µν)(λ):(ρσ)(τ)

]∗≡ 2−2M Tr

[σ(µν)(λ)(j′j) σ(ρσ)(τ)(jj′)

]∗=[P(j′j)(µν)(λ):(ρσ)(τ)

]∗= P(jj′)(µν)(λ):(ρσ)(τ)

= P(j′j)(ρσ)(τ):(µν)(λ) ;

(28)

P(j′j)(µν)(λ):(ρσ)(τ) P(j′j)(ρσ)(τ)(µ′ν′)(λ′) = P(j′j)(µν)(λ):(µ

′ν′)(λ′)). (29)

The tensors P(j′j) are invariant, by construction, and thus they are com-binations of the metric symbol, gµν , and the alternating symbol, εµνλρ. Notethat the operation “ ˜ ” on P corresponds to space inversion, and that for suchinvariant tensors it is the same as complex conjugation. Thus, the tensor partof P(j′j) is real, and the pseudotensor part pure imaginary.

III. The 2(2j+1)-Component Formalism

It is most convenient to start from the Dirac equation in the van der Waerdenrepresentation [14]. Then the 4×4 Dirac matrices are written in terms of the2×2 Pauli matrices, (II.2) and (II.3),

γµ =

(0 σµσµ 0

), (1)

8

with the Dirac equation for a free particle of mass m and spin 12 being given

by

iγµ∂µψ(x) ≡ iγ ·∂ ψ(x) = mψ(x) ,

∂µ ≡∂

∂xµ.

(2)

By using the formal definition (II.12) for Dj(a·σ) with an operator argument,a·σ, this can be written(

0 D12 (iσ ·∂)

D12 (iσ ·∂) 0

)ψ(x) = mψ(x) . (3)

The generalization for any half-integer spin and nonzero mass is immediate [7, 8]:

(0 Dj(iσ ·∂)

Dj(iσ ·∂) 0

)ψ(x) = m2jψ(x) , (4a)

−�ψ(x) ≡ −∂µ∂µψ(x) = m2ψ(x) . (4b)

We refer to this pair of equations as “the wave equation”.The Klein-Gordon equation, (4b), is needed as a subsidiary condition be-

cause (4a) is an equation of order 2j, and thus requires extra boundary con-ditions. The subsidiary condition guarantees the uniqueness of the mass, andthat ψ(x) has the usual Fourier decomposition in terms of positive and negativeenergy, plane-wave solutions.

Although (4b) is not automatically satisfied by the solutions of (4a) if j > 12 ,

it is easy to see that the solutions of (4a) identically satisfy

i4j �2jψ(x) = m4jψ(x) . (5)

This follows by applying the operator on the left side of (4a) to both sides ofthat equation and then making use of the identity

Dj(a·σ) Dj(a·σ) = Dj(a·σ a·σ) = Dj(a·a) = (a·a)2j , (6)

which holds whenever the components commute,

[ aµ, aν ] = 0 = [ aµ, σν ] . (7)

The analogy with the Dirac equation in the usual form (2) is emphasizedwhen we use the definitions (II.15) and (II.16) to write (4a) as

i2j

(0 σµ1···µ2j (jj)

σµ1···µ2j(jj) 0

)∂µ1 · · · ∂µ2jψ(x) = m2jψ(x) . (8)

9

and then define

γ(µ)(jj) =

(0 σµ1···µ2j

(jj)

σµ1···µ2j (jj) 0

). (9)

Because σ(jj) and σ(jj) are completely symmetric and traceless in their tensorindices, so is γ(jj). The label (jj) is written with two j’s because later weshall consider matrices that are not square, but rectangular. This label is oftensuppressed when the context is clear. Then (4a) becomes

i2j γ(µ)∂(µ)ψ(x) = m2jψ(x) , (10)

where we use the notation

∂(µ) ≡ ∂µ1 · · · ∂µ2j . (11)

The matrices γ(µ) are the same as those arrived at by Weinberg, except fora factor −i2j . The algebra generated by these matrices is studied in detail inSection IV.

In order to set the context for that discussion, I shall list the basic propertiesof the solutions of the spin-j wave equation. For the most part, this amounts tosubstituting j for 1

2 in the appropriate places in the Dirac theory. Some of thesepoints are developed in more detail in the papers of Joos [7] and Weinberg [8].

A. Lorentz Covariance

The wave equation is so constructed that the spinor indices of its solutionstransform according to (j, 0)⊕ (0, j).

Thus, if we define

Sj(A) =

(Dj(A) 0

0 Dj(A)−1†

), A ∈ SL(2, C) , (12)

then the transformation laws for σ(jj) and σ(jj) imply that

Sj(A) γ(µ) Sj(A)−1 = Λ(ν)(µ)(A) γ(ν) ≡ Λν1

µ1 · · ·Λν2j

µ2jγν1···ν2j . (13)

From this, it follows that if ψ(x) is a solution of (4a) and (4b), then so is

[U(a,A)ψ](x) ≡ Sj(A)ψ[Λ−1(x− a)] , (14)

where a is an arbitrary real four-vector, and Λ = Λ(A). It is clear that theoperators U(a,A) defined in this way form a representation of the connected

part of the Poincare group, P↑+, with the multiplication law

U(a,A)U(a′, A′) = U [a+ Λ(A)a′, AA′] .

We shall see that these operators form a unitary representation of P↑+ in aHilbert space, and we shall see how to extend the set to include the discrete

10

transformations: space inversion (P ), time inversion (T ), and total inversion(Y ).

The analog of the Dirac adjoint spinor is defined with the help of the matrix

B =

(0 II 0

)= γ0···0(jj) = γ(0) , (15)

which has the properties

B = B†, B−1γ(µ)B = γ(µ)†, detB = 1 , (16)

from (15), (9), and the fact that σ(jj) and σ(jj) are Hermitean. As in thej = 1

2 case, it is only in the van der Waerden representation of γ(jj) thatB = γ(0). The matrix γ(0) has the properties

γ(0)2 = I, γ(0)γ(µ)γ(0) = γ(µ), (17)

from the definitions.The adjoint spinor ψ(x) is formed in the usual way,

ψ(x) = ψ(x)†B ; (18)

and because of (16), it satisfies

(−i)2j ∂(µ)ψ(x) γ(µ) = m2j ψ(x) . (19)

From the definitions (12) and (15) we get

B−1Sj(A)†B = Sj(A)−1. (20)

Thus ψ(x) transforms under P↑+ according to

[U(a,A)ψ](x) = ψ[Λ−1(x−a)] Sj(A)−1. (21)

For the discrete transformations, we again take over the discussion from theDirac theory by introducing, in addition to B given above, a set of 2(2j+1)×2(2j+1) matrices, γ5, K, C, R, M , having the following effects on γ(µ):

γ5 γ(µ) γ5−1 = −γ(µ) , (22a)

K γ(µ)TK−1 = γ(µ) , (22b)

C γ(µ)T C−1 = (−1)2j γ(µ) , (22c)

Rγ(µ)R−1 = γ(µ), (22d)

M γ(µ)∗M−1 = (−1)2j γ(µ) . (22e)

In the Dirac theory, these matrices are essentially determined by (16) and(22), once a representation of γµ is chosen. The same is most likely true

11

here. Although we have so far considered only a special representation for γ(µ),we shall indicate later that the generalized γ matrices, for give j, satisfy ananticommutation relation that appears to determine an algebra, with only oneirreducible representation for each j, unique up to a similarity transformation.If this is true, then, for any γ(µ) related to our special choice by a similaritytransformation, the matrices in (16) and (22) can be chosen to have certainproperties. These are listed below, along with the specific expressions for thematrices in the generalized van der Waerden representation.4

B = B† =

(0 II 0

), (23a)

γ5 = γ5−1 =

(I 00 −I

), (23b)

K = (−1)2jKT =

([ j ] 00 [ j ]

), (23c)

R = R−1 =

(0 II 0

)= γ(0), (23d)

C = (−1)2jCT =

([ j ] 0

0 (−1)2j [ j ]

)=

{γ5K, fermions

K, bosons(23e)

M = B−1C∗−1 =

(0 [ j ]

(−1)2j [ j ] 0

), (23f)

detB = (−1)2j det γ5 = detK = detR = detC = detM = 1 .

In these equations, [ j ] is the metric symbol defined in (II.13).The proof that these properties can be satisfied with a general represen-

tation of γ(µ), assuming that up to an equivalence there is only one that is

irreducible, is not given here, because it is so similar to the proof for j = 12 . In

any event, they are easily verified for our specific representation.Using these matrices, we define the transformation of ψ(x) under P , T , Y ,

and charge conjugation C just as for spin 12 ,

ψP (x) = R−1ψ(x) , xµ ≡ xµ (24a)

ψT (x) = KTRT ψT(−x) , (24b)

ψY (x) = KT ψT(−x) , (24c)

ψC(x) = CT ψT(x) . (24d)

It is easy to verify that if ψ(x) is a solution of the wave equation then so areψP,T,Y,C(x), and that P , T , and Y leave the spaces of positive and negativeenergy solutions invariant, whle C interchanges the two spaces.

4Our conventions are slightly different from those of Weinberg in [8].

12

The adjoint solutions can be calculated from these, with the aid of suchidentities as

BTC†B = (−1)2jCT,

BTK†B = K−1,(25)

and it is also a simple matter to write down the transformation laws of bilinearcovariants, which look exactly the same as in the case of j = 1

2 .

B. Plane-Wave Solutions

The space of solutions of the wave equation can be constructed from superpo-sitions of plane-wave solutions, in just the same way as for j = 1

2 . Here, wewrite down the plane-wave solutions explicitly.

By virtue of the Klein-Gordon equation, the positive and negative energysolutions, ψ±, respectively, take the form

ψ±(x) = e∓ik·x u±(k) , k0 = +(m2 + k2)12 ,

ψ±(x) = e±ik·x u±(k) ,(26)

where u±(k) and u±(k) satisfy

k(µ)γ(µ)u±(k) =

(0 Dj(k ·σ)

Dj(k ·σ) 0

)u±(k) = (±1)2jm2j u±(k) ,

u±(k) γ(µ)k(µ) = (±1)2jm2j u±(k) .

(27)

The functions u±(k) are labeled by the physical energy-momentum, k.It is easy to write down the solutions, once a direction for the spin axis

in a particular rest frame of the particle is chosen. For each k, we specify arest frame, the “standard frame”, by the requirement that k be obtained fromit by a pure velocity transformation, or “boost” as Weinberg [8] calls it. Theelement of SL(2, C) that corresponds to this transformation (up to a sign) isthe Hermitean, positive-definite square root

(k ·σ/m)12 = (m+ k ·σ)/[2m(m+ k0)]

12 . (28)

To each spin direction in the standard frame, represented by a unit three-vectore(k), we assign an element U(k) ∈ SU(2, C) corresponding to a rotation fromthe 3-direction to e:

e(k)·σ = U(k)σ3U(k)†. (29)

This assignment is of course determined only up to rotations about the 3-direction.

13

The solutions of (27) are then

u±(k) =1√2

(Dj [A(k)]

(±1)2jDj [A(k)]−1†

),

u±(k) =1√2

((±1)2jDj [A(k)]−1, Dj [A(k)]†

),

A(k) ≡ (k ·σ/m)12 U(k) .

(30)

The right-hand, column index of the 2(2j+1)×(2j+1) matrix u±(k) we call the“spin index”, and the left-hand, row index we call the “bispinor index”. Thebispinor index transforms like the direct sum of a spin-j spinor with a lowerundotted index and a spin-j spinor with an upper dotted index, from (14). Thespin index transforms according to a spin-j representation of the little group ofthe vector (m, 0, 0, 0), in the language of the Wigner [15] representation theory

for P↑+.Finally, we note the orthogonality relations

u±(k)u±(k) = (±1)2jI ,

u±(k)u∓(k) =(12 + (±1)2j 12

)I ,

(31)

where I is the (2j+1)-dimensional unit matrix. Note that for fermions, thepositive and negative energy solutions are orthogonal in the bispinor space,whereas for bosons, u+(k) = u−(k).

C. Conserved Currents and the Scalar Products

All that we have to do in order to make a Hilbert space from the space ofsolutions is to introduce a suitable, positive-definite, scalar product, the otheraxioms for a Hilbert space being trivially satisfied. This can be done withthe help of the following “conserved current density”.5 Let φ and ψ denotesolutions of the wave equation, and define

Jµ(x :φ, ψ) = (i)2j−1{φ(x) γµµ1···µ2j−1 ∂

µ1 · · · ∂µ2j−1ψ(x)

−[∂µ1φ(x)

]γµµ1···µ2j−1 ∂

µ2 · · · ∂µ2j−1ψ(x) + · · ·

+ (−1)2j+1[∂µ1 · · · ∂µ2j−1φ(x)

]γµµ1···µ2j−1 ψ(x)

}.

(32)

Because of the wave equation,

∂µJµ(x :φ, ψ) = 0 . (33)

It is straightforward to see that Jµ transforms as a vector under P↑+:

Jµ[x :U(a,A)φ,U(a,A)ψ] = Λµν Jν [Λ−1(x− a) :φ, ψ] , (34)

5This expression has also been considered by S. Weinberg (private communication).

14

and for the discrete transformations:

Jµ(x :φP , ψP ) = Jµ(x :φ, ψ) = Jµ(x :φ, ψ) ,

Jµ(x :φT , ψT ) = Jµ(−x :ψ, φ) = Jµ(−x :ψ, φ) ,

Jµ(x :φY , ψY ) = Jµ(−x :ψ, φ) ,

Jµ(x :φC , ψC) = (−1)2j+1Jµ(x :ψ, φ) .

(35)

Note that when j > 12 , J0(x :ψ,ψ) is not positive definite, so that it cannot

be used as a probability density as in the Dirac theory. But as usual that posesno problem when one goes over to a field theory, where it may represent acharge density.

Consider the bilinear form

〈φ, ψ〉 =

∫d3xJ0(x :φ, ψ) = 〈ψ, φ〉∗ . (36)

This form is constant in time, from (33). Moreover, from (34), U(a,A) isunitary within the bilinear form, as well as P , while T and Y are antiunitary.Charge conjugation is antiunitary for fermions; and up to a sign, which willbe eliminated shortly, it is antiunitary for bosons. (In the second quantizedtheory, charge conjugation becomes unitary, as usual [8].)

The above bilinear form is not positive definite, but it is trivially convertedinto one that is. For this purpose, we use the plane-wave decomposition of thesolutions ψ and φ. Thus, for example,

ψ(x) = ψ+(x) + ψ−(x) ,

ψ±(x) =

∫d3k

k0u±(k) f±(k) e∓ik·x,

k0 = (m2 + k2)12 ,

(37)

where (±) indicate, respectively, positive and negative energy parts, and wherewe use the matrix notation described before for the momentum space solu-tions u±, so that f± are (2j+ 1)-dimensional column vectors. Appropriateintegrability properties are taken for granted.

Then it is straightforward to show that the bilinear form has the followingproperties:

(φ+, ψ−) = (φ−, ψ+) ,

(±1)2j+1 (ψ±, ψ±) ≥ 0 ,(38)

with equality in the latter expression holding if and only if ψ± vanishes. Thus,we can define a positive-definite, scalar product:

〈φ, ψ〉 = (φ+, ψ+) + (±1)2j+1(φ−, ψ−) . (39)

15

Within this scalar product, U(a,A) and P are unitary; and T , Y , and C areantiunitary.

Finally, note that, as one expects [2], the choice of a conserved currentdensity is by no means unique. In our case, we could just as well have chosen

J ′µ(x :φ, ψ) = i[φ∂µψ − (∂µφ)ψ

]. (40)

Then Jµ and J ′µ have the same transformation laws, both are divergenceless,and both lead to the same bilinear form. We leave the verification to theinterested reader.

IV. The Generalized Dirac Algebra

Here we begin to study the algebras generated by the matrices γ(µ). Our generalprocedure is first to classify and construct a complete set of 2(2j′+1)×2(2j+1)matrices according to spins, and then to study the various algebraic relationsamong the basis elements.

The basic point of the discussion, which we use again and again, is thatthe complete set of generalized Dirac matrices is nothing but a collection ofCG coefficients, expressed partly in Cartesian form, for the reduction of adirect product of representations irreducible under L↑ into a direct sum ofsuch representations. This fact makes it possible to compute all matrices inthe complete set, and all traces of products of such matrices, in terms of theordinary CG coefficients for finite-dimensional representations of L↑+.

The objects to be studied are 2(2j′+1)×2(2j+1) matrices H, transformingunder L↑ according to

Sj′(A)H Sj(A)−1 = HA , j′+j = integer . (1)

We restrict j′+j to be an integer because otherwise H cannot be representedas a sum of tensors. If we write out the bispinor indices of H as Hab, then the

index a transforms according to Sj′(A), and b according to Sj(A)−1T. From

the definitions of B and C, (III.15) and (III.23e), and from (II.10), we havethat

CB Sj(A)−1TB−1C−1 = Sj(A) , (2)

so that Sj(A)−1T is unitary-equivalent to Sj(A). Thus, H transforms accordingto the representation [(j′, 0) ⊕ (0, j′)] ⊗ [(j, 0) ⊕ (0, j)]. By a Clebsch-Gordananalysis, we get from this representation the following invariant subspaces,irreducible under L↑+:

[(j′+j, 0)⊕ (0, j′+j)] ⊕ [(j′+j−1, 0)⊕ (0, j′+j−1)]

⊕ · · · ⊕ [(|j′−j|, 0)⊕ (0, |j′−j|)] ⊕ [(j′, j)⊕ (j, j′)] .

Of course, when j′ = j, the subspaces (0, 0) ⊕ (0, 0) and (j, j) ⊕ (j, j) areno longer irreducible under L↑, but rather each subspace (0, 0) and (j, j) isseparately irreducible.

16

Our first task is to construct matrices spanning each of these irreduciblesubspaces. Before doing that we recall how the classification works in thefamiliar case j′ = j = 1

2 . Then we have the following correspondence:

(i) scalar: I ↔ (0, 0);

(ii) vector: γµ ↔ ( 12 ,

12 );

(iii) antisymmetric tensor (or pseudotensor):σµν(or γ5σµν)↔ (1, 0)⊕ (0, 1);

(iv) pseudovector: γ5γµ ↔ ( 12 ,

12 );

(v) pseudoscalar: γ5 ↔ (0, 0);

In the following we shall see how this classification generalizes for any j′ andj, with j′+j = integer.

A. Construction of a Complete Set

Because of the spinor character of the diagonal and off-diagonal blocks of H,indicated schematically below, it is easy to guess how the invariant subspacesare to be characterized.

H =

(Xα

β YαβY ′αβ X ′αβ

). (3)

The first natural division is between the space of H’s with only diagonal blocksnonvanishing, HD, and the space of H’s with only off-diagonal blocks nonvan-ishing, HO. Then HD and HO are clearly invariant subspaces under L↑, fromthe form (III.12) of Sj(A) and the form (III.23d) of γ(0), which represents spaceinversion.

To construct a complete set, we shall use the generalized Pauli matricesgiven in Section II, which we normalize for convenience as follows:

ρ(j′j) ≡ 2−M σ(j′j) ,

ρ(j′j) ≡ 2−M σ(j′j) ,

M = max[j′, j] .

(4)

Consider first the space HO. From the spinor character of the off-diagonalblocks in (3), HO corresponds to the representation (j′, j) ⊕ (j, j′). To get acomplete set of matrices in HO, note that the matrices ρ(µν)(λ)(j′j), definedby (II.24) and (4) above, span the upper right-hand block, because of theorthogonality relation (II.27), and that the matrices ρ(µν)(λ)(j′j) span the lowerleft-hand block, for the same reason. Thus, the following are a complete set in

17

HO:

κ(µν)(λ)(j′j) ≡

(0 ρ(µν)(λ)(j′j)

ρ(µν)(λ)(j′j) 0

),

γ5(j′)κ(µν)(λ)(j′j) ≡

(0 ρ(µν)(λ)(j′j)

−ρ(µν)(λ)(j′j) 0

)

= −κ(µν)(λ)(j′j) γ5(j) ,

(5)

where γ5(j′) and γ5(j) are the square matrices of appropriate dimension definedin (III.23b).

From the transformation laws of ρ and ρ under L↑+, and the space inversionrelation (II.25) between ρ and ρ, which gives

γ(0)(j′)κ(µν)(λ)(j

′j) γ(0)(j) = κ(µν)(λ)(j′j) ,

γ(0)(j′) γ5(j′)κ(µν)(λ)(j

′j) γ(0)(j) = −κ(µν)(λ)(j′j) ,(6)

it follows that each of the sets of matrices κ(µν)(λ) and γ5κ(µν)(λ) spans an

invariant subspace with respect to L↑, the first forming a tensor and the seconda pseudotensor with respect to space inversion, P . Now HO corresponds to(j′, j) ⊕ (j, j′), so that if j′ 6= j the γ5κ

(µν)(λ) are not linearly independentfrom the κ(µν)(λ), because there is only one invariant subspace under L↑. Butif j′ = j, there are two invariant subspaces; and the two sets of matrices aretherefore independent.

Indeed this follows from our construction. When j′ 6= j, ρ(µν)(λ)(j′j) isselfdual (j′ > j) or antiselfdual (j′ < j) with respect to (µν), while the reverseis true of ρ(j′j). Thus,

γ5(j′)κ(µν)(λ)(j′j) = ± i2εµ1ν1

στ κστµ2ν2...µ∆ν∆(λ)(j′j) ,

(±), resp: j′ > j, j′ < j ; ∆ ≡ |j′−j| .(6a)

From the properties of ρ and ρ, κ(µν)(λ) and γ5κ(µν)(λ) are traceless in their

tensor indices, with the symmetries indicated by the notation introduced inSection II. Of course κ(jj) = 2−jγ(jj).

To complete our set of matrices, we must still span the spaceHD of matriceswith diagonal blocks, X and X ′, having spinor indices Xα

β and X ′αβ . Keepingin mind the earlier decomposition of H into irreducible parts, we reduce X andX ′, making use of the completeness of the CG coefficients:

Xαβ =

∑J

(−1)2j′[ Jj′j ]γα

β [ Jj′j ]γα′β′ Xα′

β′ ,

≡∑J

X(J)γ [ Jj′j ]γαβ ,

X ′αβ =∑J

X ′(J)γ [ Jj′j ]γαβ .

(7)

18

Now the spinors ρ(µν)(J) = 2−Jσ(µν)(J) and ρ(µν)(J) constructed in (II.21)span the (2J+1)-dimensional spaces corresponding to X(J) and X ′(J), becauseof the orthogonality relations (II.22). Thus, if we define (2j′+1)×(2j+1) matrices

η(µν)(J : j′j)αβ ≡ ρ(µν)(J)γ [ Jj′j ]γα

β ,

η(µν)(J : j′j)αβ ≡ ρ(µν)(J)γ [ Jj′j ]γ

αβ

= η(µν)(J : j′j)αβ ,

(8)

we get a complete set in HD by choosing

τ (µν)(J : j′j) =

(η(µν)(J : j′j) 0

0 η(µν)(J : j′j)

), (9a)

and

γ5(j′) τ (µν)(J : j′j) = τ (µν)(J : j′j) γ5(j)

=

(η(µν)(J : j′j) 0

0 −η(µν)(J : j′j)

),

(9b)

with J = |j′−j|, |j′−j|+1, . . . , j′+j.As before, τ(J : j′j) and γ5τ(J : j′j) are, respectively, tensors and pseu-

dotensors under L↑. Both sets of matrices, for a given J 6= 0, span invariantsubspaces under L↑, which are at least 2(2j+1)-dimensional, from the originaldecomposition of H; and hence the sets τ (µν)(J : j′j) and γ5(j′)τ (µν)(J : j′j)cannot be linearly independent. In fact, from Section II, the upper blocks areselfdual with respect to (µν) and the lower blocks antiselfdual, so we have therelation

γ5(j′) τ (µν)(J : j′j) =i

2εµ1ν1

στ τστµ2ν2···µJνJ (J : j′j) ,

J 6= 0 .

(10)

From the properties of η and η, defined in (8), the τ (µν)(J : j′j) are tracelesstensors with the indicated symmetry.

Thus, our final complete sets can be chosen as follows, with the symmetriesindicated by the notation of Section II, and all tensors having vanishing trace:

Case I: j′ = j

(i) scalar: I ↔ (0, 0);

(ii) 2j-th rank tensor: κ(µ)(jj)↔ (j, j);

(iii) 2J-th rank tensor (or pseudotensor):τ (µν)(J : jj) [or γ5(j) τ (µν)(J : jj)] ↔ (J, 0)⊕ (0, J);J = 1, . . . , 2j;

19

(iv) 2j-th rank pseudotensor: γ5(j)κ(µ)(jj)↔ (j, j);

(v) pseudoscalar: γ5(j)↔ (0, 0).

Case II: j′ 6= j

(i) 2M -th rank tensor (or pseudotensor), M = max[j′, j]:κ(µν)(λ)(j′j) [or γ5(j)κ(µν)(λ)(j′j)] ↔ (j′, j)⊕ (j, j′);

(ii) 2J-th rank tensor (or pseudotensor):τ (µν)(J : j′j) [or γ5(j′) τ (µν)(J : j′j)] ↔ (J, 0)⊕ (0, J);J = |j′−j|, . . . , j′+j.

B. Infinitesimal Generators

It is hardly surprising that the components of the second rank, antisymmetrictensor τ (µν)(1 : jj) are the infinitesimal generators of the representation (j, 0)⊕(0, j) of L↑, except for a normalization.

Take the standard conventions for the Hermitian matrices J(j) that arethe infinitesimal generators of the (2j+1)-dimensional representation of therotation group [11]. The six infinitesimal generators for the representations

(j, 0) and (0, j) of L↑+ are J(j) and K(j) = ∓iJ(j), respectively. These formthe usual antisymmetric tensor Mµν , with

Mij = εijk Jk , i, j, k = 1, 2, 3 , ε123 = +1 ,

M i0 = Ki .(11)

A slightly tedious, but straightforward calculation gives

Mµν(j, 0) = ζ(j) ηµν(1 : jj) ,

Mµν(0, j) = ζ(j) ηµν(1 : jj) ,

ζ(j) ≡ 2i[(2j+1)j(j+1)/3]12 .

(12)

The generators Mµν(j) for (j, 0)⊕ (0, j) in the van der Waerden represen-tation are then

Mµν(j) =

(Mµν(j, 0) 0

0 Mµν(0, j)

). (13)

C. Representations of 2(2j′+1)×2(2j+1) Matrices

Traces are particularly easy to study in the van der Waerden representation.The most general traces that we need in this section are those of products oftwo matrices.

Traces of (square) matrices in HO vanish trivially. This means that thetrace of any product of matrices with an odd number of factors from HO, that

20

is an odd number of factors of the form κ(j′j) or γ5κ(j′j), must vanish. Thusin particular

Trκ(µ)(jj) = Tr[γ5(j)κ(µ)(jj)

]= 0 ,

Tr[κ(µν)(λ)(j′j) τ (στ)(J : jj′)

]= 0 ,

Tr[γ5(j′)κ(µν)(λ)(j′j) τ (στ)(J : jj′)

]= 0 .

(14)

Furthermore, because [ Jjj ]αββ = 0, and the expression (III.23b) for γ5,

Tr τ (µν)(J : jj) = Tr[γ5(j) τ (µν)(J : jj)

]= 0 , J 6= 0 ,

Tr γ5(j) = 0 .

(15)

Finally, we consider traces of products of matrices in the same class:

Tr[τ (µν)(J : j′j) τ (µ

′ν′)(J ′ : jj′)]

= Tr[γ5(j′) τ (µν)(J : j′j) γ5(j) τ (µ

′ν′)(J ′ : jj′)]

= (−1)J+j′−j δJJ ′

[P(J)(µν):(µ

′ν′) + P(J)(µν):(µ′ν′)]

≡ (−1)J+j′−j δJJ ′ T (J)(µν):(µ

′ν′), J 6= 0 ,

(16)

where P(J) and P(J) are the projection operators for the selfdual and antiself-dual tensors corresponding to (J, 0) and (0, J), defined in (II.23). This result isimmediate from the definitions. (The factor (−1)J+j

′−j comes partly from thesymmetry of the CG coefficients under the interchange of j′ and j, partly fromtheir orthogonality relations.) Note that T (J) is an invariant tensor under L↑,and that it is a projection operator, corresponding to the space of tracelesstensors with symmetry (µν).

The last case is calculated in the same way:

Tr[κ(µν)(λ)(j′j)κ(µ

′ν′)(λ′)(jj′)]

= −Tr[γ5(j′)κ(µν)(λ)(j′j) γ5(j)κ(µ

′ν′)(λ′)(jj′)]

= P(j′j)(µν)(λ):(µ′ν′)(λ′) + P(j′j)(µν)(λ):(µ

′ν′)(λ′)

≡ T (j′j)(µν)(λ):(µ′ν′)(λ′).

(17)

Again, T (j′j) is an invariant tensor, and for j′ 6= j a projection operator fortraceless tensors with the indicated symmetry, while for j′ = j, T (jj)/2 is the

projection operator, because P(jj) = P(jj), from (II.28).

21

Note that

T (j′j) = T (jj′) . (18)

We have already seen that the generalized Dirac matrices form a completeset. With the traces above, we can give an explicit representation for any2(2j′+1)×2(2j+1) matrix, just as for the spin- 12 case, as a sum of terms withdefinite symmetries:

H(j′j) =δj′j

2(2j+1){Tr[H(j′j)] I + Tr [H(j′j) γ5(j)] γ5(j)}

+∑J 6=0

(−1)J+j′+j Tr

[H(j′j) τ(µν)(J : jj′)

]τ (µν)(J : j′j)

+ ζ(jj′) Tr[H(j′j)κ(µν)(λ)(jj

′)]κ(µν)(λ)(j′j)

− ζ(jj′) Tr[H(j′j) γ5(j)κ(µν)(λ)(jj

′)]γ5(j′)κ(µν)(λ)(j′j) ,

ζ(jj′) ≡

{1 j 6= j′,12 j = j′.

(19)

To prove this relation, multiply both sides by one of the basis elements andtake the trace, using the results described above.

D. γ-Algebra: Threefold Traces

Now that we know how to represent any matrix in terms of the basis above,we can give formulas that represent any of the basis elements as linear combi-nations of products of γ matrices. To do this we need certain threefold traces.For completeness, we compute all such traces, in terms of CG coefficients.

Consider the following quantities:

C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ)

≡ ρ(στ)(J)α [ Jj′j ]αβγ[ρ(µν)(λ)(jj′′) ρ(µ

′ν′)(λ′)(j′′j)]γ

β ;(20)

and define C(Jj′j[j′′]) by the usual rule of respectively raising and lowering alllower and upper tensor indices. Of course j′′+j′ and j′′+j are integers.

These quantities are just CG coefficients [ Jj′j ] in various Cartesian repre-sentations, labeled by j′′, with orthogonality relations

C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ) C(J ′j′j[j′′])(σ′τ ′):(µ′ν′)(λ′):(µν)(λ)

= δJJ ′ (2j′′+1)P(J)(στ):(σ′τ ′) .

(21)

Similar relations hold for C; and in addition, C and C are orthogonal to eachother:

C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ) C(J ′j′j[j′′])(σ′τ ′):(µ′ν′)(λ′):(µν)(λ)

= 0 .(22)

22

These equations follow from the defintions, from orthogonality relations suchas (II.27), and from the properties of the CG coefficients.

When J , j′, and j are all integers, the special case j′′ = 0 in the aboveleads to Cartesian representations of [ Jj′j ] in terms of selfdual and antiselfdualtensors:

C(Jj′j)(στ)(µ′ν′):(µν) ≡ ρ(στ)(J)α [ Jj′j ]αβγ ρ(µ

′ν′)(j′)β ρ(µν)(j)γ ,

J, j′, j = integers ,(23)

with an analogous expression for C(Jj′j). These quantities satisfy (21) and(22) with j′′ = 0 in (21).

The threefold traces of basis matrices become:

Tr[τ(στ)(J : j′j)κ(µν)(λ)(jj′′)κ(µ

′ν′)(λ′)(j′′j)]

= C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ) + C(Jj′j[j′′])...

... ,

(24a)

Tr [γ5(j′) τ(J : j′j)κ(jj′′)κ(j′′j′)]

= C(Jj′j[j′′])− C(Jj′j[j′′]) ,(24b)

Tr[τ (µν)(J : j′j) τ (µ

′ν′)(J ′ : jj′′) τ (µ′′ν′′)(J ′′ : j′′j′)

]= C(JJ ′J ′′)(µν):(µ

′ν′):(µ′′ν′′) + C(JJ ′J ′′)... ,

(24c)

Tr [γ5(j′) τ(J : j′j) τ(J ′ : jj′′) τ(J ′′ : j′′j′)]

= C(JJ ′J ′′)− C(JJ ′J ′′) ,(24d)

where we have occasionally suppressed tensor indices. Those traces withouta factor γ5 are of course (real) tensors, the tensor parts of C(Jj′j[j′′]) andC(JJ ′J ′′), while those having a factor γ5 are the corresponding (pure imagi-nary) pseudotensor parts. Traces with more factors γ5 can be written as oneof the above. The remaining threefold traces have an odd number of factors κ,and hence vanish.

In effect, we get an expression for τ(J : j′j) in terms of κ(j′j′′)κ(j′′j) byinverting (24a). Note that the matrices η defined in (8) satisfy

η(στ)(J : j′j) =(2j′′+1)−1 C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ)

× ρ(µ′ν′)(λ′)(j′j′′) ρ(µν)(λ)(j′′j) ,

(25)

from the definition of C and the orthogonality properties of ρ and ρ. By similararguments,

C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ)

× ρ(µ′ν′)(λ′)(j′j′′) ρ(µν)(λ)(j′′j) = 0 , J 6= 0 .

(26)

23

Analogous equations hold for η. Thus, from the definition of τ in terms of ηand η, Eq. (9a), we get for J 6= 0:

τ (στ)(J : j′j) =1

2j′′+1

[C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ)

+ C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ)

]× κ(µ

′ν′)(λ′)(j′j′′)κ(µν)(λ)(j′′j) ,

(27a)

γ5(j′) τ (στ)(J : j′j) =1

2j′′+1

[C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ)

− C(Jj′j[j′′])(στ)(µ′ν′)(λ′):(µν)(λ)

]× κ(µ

′ν′)(λ′)(j′j′′)κ(µν)(λ)(j′′j) .

(27b)

These equations say that τ(J) and γ5τ(J), for J 6= 0, are generated byκ(j′j′′)κ(j′′j) for any j′′ with j′′+j′ and j′′+j being integers. The only remainingbasis matrices are I and γ5. From the orthogonality relations (II.27) we get forthe 2(2j+1)-dimensional unit matrix

I(j) = (2j′+1)−1 κ(µν)(λ)(jj′)κ(µν)(λ)(j′j) , (28)

and from similar arguments, using properties of the CG coefficients, we get

γ5(j) = (−1)1−j′+j 2j+1

3

i

2εµνλρ τ

µν(1 : jj′) τλρ(1 : j′j) . (29)

Thus all basis elements are generated by products of κ(j′j) matrices, whichcan be chosen with any row and column dimensions that yield the row andcolumn dimensions desired for the basis elements.

E. Abstract Characterization of the Algebras

We restrict our attention to the case j′ = j, so that the κ(µ)(jj) are squarematrices that generate the algebra of 2(2j+1)×2(2j+1) matrices. The CartesianCG coefficients defined in (20) can be used to define projection operators, forJ 6= 0,

T (Jjj)(λ)(ρ):(λ′)(ρ′) ≡ C(Jjj[j])(µν):(λ):(ρ) C(Jjj[j])(µν)

(λ′):(ρ′)

+ C(Jjj[j])(µν):(λ):(ρ) C(Jjj[j])(µν)(λ′):(ρ′).

(30)

These operators are pure tensors under parity, from the definition of C; andhence they can be expanded as (real) combinations of the metric symbol gµν .

24

A straightforward calculation gives the identity:

2j∑J=1

T (Jjj)(µ)(ν)(µ′)(ν′) κ(µ′)(jj)κ(ν

′)(jj) + g(j)(µ)(ν) I

= (2j+1)κ(µ)(jj)κ(ν)(jj) ,

(31)

where g(j) is the metric symbol for the space of symmetric and traceless tensorsof rank 2j, i.e.,

g(j)(µ)(ν) = Tr ρ(µ)(jj) ρ(ν)(jj) . (32)

This quantity is proportional to C(0jj[j]).The familiar symmetry of the CG coefficients,

[ Jjj ]αβγ = (−1)J−2j [ Jjj ]α

γβ ,

and the definition (20), imply that

C(Jjj[j])(µν)(λ):(ρ) = (−1)J C(Jjj[j])(µν)(ρ):(λ) , (33)

with a similar expression for C. In particular

g(j)(µ)(ν) = g(j)(ν)(µ). (34)

These symmetries imply that (31) is equivalent to the expression

(2j+1)[κ(µ)(jj)κ(ν)(jj) + κ(ν)(jj)κ(µ)(jj)

]= 2g(j)(µ)(ν) I + 2

2J≤2j∑J=1

T (2J, jj)(µ)(ν)(µ′)(ν′) κ(µ′)(jj)κ(ν

′)(jj) .(35)

For j = 12 , this reduces to the usual Dirac anticommutation relation.

We have arrived at this equation by means of our special representationfor κ(jj). Instead, we could have postulated this anticommutation relation asfundamental and asked for the irreducible representations of κ(jj). We do notas yet have a complete proof (for j > 1

2 ) that all irreducible representations areequivalent to the 2(2j+1)-dimensional representation constructed in previoussections, but it seems plausible. It does not seem appropriate, however, toemphasize this possibility very much, because of our view of the wave equationas merely a convenient summary of covariance properties rather than as afundamental equation of motion. We present it only as a perhaps interestingside remark.

V. Plane-Wave Matrix Elements

To complete our study, we present a few remarks on the representation ofplane-wave matrix elements, which illustrate the fact that the usual concepts

25

from the Dirac theory go over without change. We continue to use the van derWaerden representation.

Because we want to use wave functions with different masses and spins,we introduce the notation ujη(k), where η = (±)2j , for the spin-j, plane-wave,± energy solutions with physical four-momentum k defined earlier in (III.30).The label η gives a consistent notation because, for bosons, u+ = u−.

Consider the 2(2j+1)-dimensional square matrices

Λjη(k) ≡ η ujη(k)ujη(k) . (1)

From the orthogonality properties (III.31), it follows that

Λjη(k) Λjη(k) = Λjη(k) , (2)

and that for fermions,

Λj±(k) Λj∓(k) = 0 . (fermions) (3)

From the completeness of the u’s, it follows that for any positive or negativeenergy, momentum space solution of the wave equation vjη(k), we get

Λjη(k) vjη(k) = vjη(k) ,

vjη(k) Λjη(k) = vjη(k) ,(4)

and for fermions

Λjη(k) vj−η(k) = vj−η(k) Λjη(k) = 0 . (fermions) (5)

In other words, for fermions Λj±(k) are respectively positive and negative energyprojection operators for plane-wave solutions. For bosons, because u+ = u−,we have only one plane-wave projection operator.

It is clear from what we have just said that Λjη(k) does not depend on any

special choice of the spin axis in the definition of ujη(k). In fact, from theexplicit forms for the solutions (III.30) we have

Λjη(k) =1

2

I ηDj

(k ·σm

)ηDj

(k ·σm

)I

. (6)

For fermions, this implies that

Λj+(k) + Λj−(k) = I , (fermions) (7)

whereas for bosons

Λj+(k) + γ5 Λj−(k) γ5 = I . (bosons) (8)

26

In the latter case, γ5Λj+(k)γ5 is the projection operator for plane-wave solutionsof momentum k corresponding to the equation

(i)2j γ(µ)∂(µ) ψ(x) = −m2j ψ(x) , (9)

an equation that we have no occasion to consider at present.Except for a normalization, Λjη(k) are the same matrices that occur in

Weinberg’s expressions for spin-j propagators [8].These projection operators can be used to simplify the representation of

(2j′+1)×(2j+1) plane-wave matrix elements of the form

R = uj′

η′(k′)H ujη(k)

= uj′

η′(k′) Λj

′

η′(k′)H Λjη(k)ujη(k) ,

(10)

where H is a 2(2j′+1)×2(2j+1) matrix, which we assume to transform covari-antly according to (IV.1), and j′+j is an integer.

If the blocks of H are written as in (IV.3), the following relations are easyto verify:

H Λjη(k) =

X + η Y Dj

(k ·σm

)0

0 η Y ′Dj

(k ·σm

)+X ′

, (11)

=

0 η X Dj

(k ·σm

)+ Y

Y ′ + η X ′Dj

(k ·σm

)0

Λjη(k) , (12)

Λj′

η′(k′)H = Λj

′

η′(k′)

X + η′Dj′(k′ ·σm′

)Y ′ 0

0 η′Dj′(k′ ·σm′

)Y +X ′

, (13)

= Λj′

η′(k′)

0 Y + η′Dj′(k′ ·σm′

)X ′

η′Dj′(k′ ·σm′

)X + Y ′ 0

. (14)

These relations can also be derived by taking note of the spinor character ofthe various blocks and observing that Dj(k·σ/m) and Dj(k·σ/m) are “metric”symbols for converting dotted into undotted or undotted into dotted indices [7,16].

Since H is a 2(2j′+ 1)× 2(2j+ 1) matrix, it can be expanded in termsof the basis set constructed in previous sections. The effect of the relationsabove is that plane-wave matrix elements of H can be represented in terms of

27

basis matrices with only off-diagonal blocks nonzero, or alternatively, with onlydiagonal blocks nonzero, that is, in terms of κ(j′j) (and γ5κ(jj), if j′ = j), oralternatively in terms of the set τ(J : j′j) (plus I and γ5 if j′ = j).

The generalization of this procedure to plane-wave matrix elements witharbitrary numbers of incoming and outgoing particles is of course straightfor-ward.

Finally, we indicate how to represent any S-matrix element for the scat-tering of particles in plane-wave states as an expression similar to (10). Forsimplicity, we consider the case where one incoming particle and one outgoingparticle have nonzero spin, but the generalization will be clear.

To do this, we use the spinor representation [7] for scattering amplitudes,which is the same as the M -function representation of Stapp [16]. In this repre-

sentation, the spin index of the particle, which transforms under P↑+ accordingto Wigner rotations [15] (representations of the little group of the momentumvector in the rest frame), is converted into a covariant spinor index. For plane-wave solutions of the wave equation, this amounts to defining new solutions,

wjη(k) ≡ ujη(k) Dj [A(k)]† =1√2

Dj

(k ·σm

)η I

wjη(k) = Dj [A(k)]ujη(k) =

1√2

(η I, Dj

(k ·σm

)),

(15)

where A(k) is defined by (III.30).

Now let M j′j

αβbe any spinor function, where α is a spinor index of spin j′,

and β an index of spin j. Then is it easy to see that

M j′j = wj′

η′(k′)

η′M j′j Dj

(k ·σm

)0

0 ηDj′(k′ ·σm′

)M j′j

wjη(k) ,

= wj′

η′(k′)

0 ηη′M j′j

Dj′(k′ ·σm′

)M j′j Dj

(k ·σm

)0

wjη(k) .

(16)

From these expressions it is straightforward to work out the effects of thediscrete symmetries on the scattering amplitudes in the spinor representation,by making use of (III.24a)–(III.24d). The interested reader should consult thepapers of Joos [7] and Stapp [16] for the results.

Acknowledgments

I wish to thank Professor C. Zemach for first encouraging me to pursue thismatter, and Professor S. Weinberg for encouraging me to complete it. It is a

28

pleasure to thank Professor M. Fierz for instructive discussions on higher-spinwave equations.

References

[1] P. A. M. Dirac, Proc. Roy. Soc. A 155, 447 (1936).

[2] M. Fierz, Helv. Phys. Acta 12, 3 (1939).

[3] M. Fierz and W. Pauli, Proc. Roy. Soc. A 173, 211 (1939).

[4] V. Bargmann and E. Wigner, Proc. Natl. Acad. Sci. U.S. 34, 211 (1949).

[5] W. Rarita and J. Schwinger, Phys. Rev. 60, 61 (1941).

[6] H. Umezawa, Quantum Field Theory, (Interscience Publishers, Inc., NewYork, 1956).

[7] H. Joos, Fortschr. d. Phys. 10, 65 (1962).

[8] S. Weinberg, Phys. Rev. B 133, 1318 (1964); 134, 882 (1964).

[9] W. L. Bade and H. Jehle, Rev. Mod. Phys. 25, 714 (1953).

[10] E. M. Corson, Introduction to Tensors, Spinors, and Relativistic WaveEquations, (Blackie and Son, Ltd., London, 1953).

[11] M. E. Rose, Elementary Theory of Angular Momentum, (John Wiley andSons, Inc., New York, 1957).

[12] E. P. Wigner, Group Theory and Its Application to the Quantum Mechan-ics of Atomic Spectra, (Academic Press, Inc., New York, 1959), pp. 292–296.

[13] A. O. Barut, I. J. Muzinich, and D. N. Williams, Phys. Rev. 130, 442(1963).

[14] B. L. van der Waerden, Die gruppentheoretische Methode in der Quan-tenmechanik, (Berlin, 1932).

[15] E. P. Wigner, Ann. Math. 40, 149 (1939).

[16] H. P. Stapp, Phys. Rev. 125, 2139 (1962).

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