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IC/68/13

INTERNAL REPORT

(Limited distribution)

A LAGRANGIAN FORMULATION OF THE

JOOS-WEINBERG WAVE EQUATIONS FOR SPIN-S PARTICLES

D. Shay

ABSTRACT

The Lorentz covariant, spin-s, Joos-Weinberg equations are derived

from a Lagrangian. This Lagrangian is written entirely in terms of an

auxiliary field <f> (x) and the Joos-Weinberg wave function i^x) is defined

in terms of <ft(x) . The function ip{x) describes a free,massive, spin-s

particle, while $(x) describes a free, massive, spin-s particle "decorated"

with massless, spin-s neutrinos. The question of whether 0(x) or ^(x)

is the wave function corresponding to reality is discussed. The inter-

pretation of 0(x) as the actual wave function is favoured here.

TRIESTE

February 1968

International Centre for Theoretical Physics, Trieste.

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A LAGRANGIAN FORMULATION OF THE JOOS-WEINBERG WAVE

EQUATIONS FOR SPIN-S PARTICLES

I. INTRODUCTION

Equivalent, Lorentz covariant, spinor formulations of the theory

of a free mass-m, spin-s particle have been given by DIRAC , FIERZ and

PAULI2, RARITA and SCHWINGER3, BARGMANN and WIGNER 4, JOOS 5,

WEINBERG6 and WEAVER, HAMMER and GOOD7 . In fact it has been

Q

shown by PURSEY that an infinite number of covariant formulations

exist. These covariant formulations have the property that, for

s > 1 , the wave function must satisfy a wave equation plus one or more

auxiliary conditions which are required to guarantee a physical mass and/or

the correct spin.

Since the spin-s wave function must satisfy a set of auxiliary con-

ditions in addition to the wave equation, a simple Lagrangian approach to

these formulations does not work. Lagrangian approaches to the Dirac-

Fierz-Pauli equations, Rarita-Schwinger and tensor equations for hall-

integral and integral spin, respectively, and the Bargmann-Wigner equa-

tions have been given by FIERZ and PAULI2, CHANG11 and GURALNIK

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and KIBBLE , respectively. These approaches all make use of one or

more auxiliary fields which appear in the Lagrangian, along with the wave

function, in order to get the wave equations and auxiliary conditions. The

auxiliary fields vanish in the free-particle case but do not in the presence

of an electromagnetic field.

Here a Lagrangian approach to the Joos-Weinberg equations is

developed. The Lagrangian is written entirely in terms of an auxiliary

field <£(x), and the wave function \p(x) is defined in terms of #x) . While

t//(x) is taken to be the actual wave function here, this question is open to

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*) While the Hamiltonian form of the Weaver-Hammer-Good formulation involves a unique Hamlltonian

without auxiliary conditions, the manifestly covariant form does involve a wave equation plus an

auxiliary equation, in the form of the Klein-Gordon equation, to guarantee a physical mass.

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debate. HAMMER, McDONALD and PURSEY ' derive essentially the

same equations for ^(x) by a different method and interpret $(x) as the

wave function. The difference is this: For s > 1 , ^/(x) describes a

free, raass-m, spin-s particle; on the other hand, $(x) describes a free,

mass-m, spin-s particle with spin-s, massless particles mixed in; that

is, the massive particle is "decorated" with neutrinos. For s = 1/2, 1,

^(x) and 0(x) are identical and describe a simple massive particle with-

out neutrinos. The ^(x) used here is slightly different from the Hammer-

McDonald-Purs ey $(x) , but this does not affect the discussion for the

arbitrary spin case.

In the following, two a priori restrictions are made. The equa-

/—r- i

2 — ia/at/H

1

tions are not to involve operators of the form (9 9 f or idfdt/ -^ + m^ ,

the energy sign operator. The former restriction necessitates slightly

different treatments for integral and half-integral spin. The latter pre-

vents the application of this method to the Weaver-Hammer-Good covariant

formalism for integral spin. The Joos-Weinberg and Weaver-Hammer-

Good equations are identical for half-integral spin and differ only by the

presence of the energy sign operator in the Weaver-Hammer-Good integral

spin equations. These restrictions are made because, while both operators

are well defined in the free-particle case, they may not be so when electro-

magnetic interactions are considered.

II. HALF-INTEGRAL SPIN

The Joos-Weinberg equations for a particle of mass m and half-

integral spin s may be obtained from the following Lagrangian:

L = i 3 ./. 8 ?(x)7 8 ... 8 *(x)-

I s+2 1 2s s+j- 2s

a ..a

1

itae ...8 * w

S-2

,

S-"2 1

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where x = (x, it) and d = - — . The 2(2s + 1) x 2(2s + 1) covariant

matrices 7 are symmetric with respect to the interchange of any

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two tensor indices and are the generalizations of the Dirac y~matrices

discussed by WEINBERG 6 and BARUT, MUZINICH and WILLIAMS14.

The matrix ^(x) is a 2 (2s + 1) column matrix and

* ( x ) = ^ ( x )T ,

where " t " denotes hermitian conjugation.

The function $(x) is the massive, neutrino "decorated", spin-s

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particle wave function considered by HAMMER, McDONALD and PURSEY

Its transformation properties are complicated because of the mixture of

massive and massless particles. It transforms according to a generalized

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Lorentz transformation A given by HAMMER, McDONALD and PURSEY

and

Ay A = a . . . a 7 ,

M1--^2S V l w2a1/28 VV"V2s

where the a are the ordinary four-vector transformation coefficients.

Essentially A is an ordinary Lorentz transformation operator in which the

1/2

mass has been replaced by a mass operator, for example (-p p ) ' .

As a result then,the Lagrangian is a scalar and the derived wave equations

are covariant.

The Lagrangian implies the following equation:

[ SM • • • % V...M, +»P,V*1*WO .

I AS ± 2,S

(2)

The equation given by Hammer, McDonald and Pursey is their eq. (66),

This equation leads to spin-1/2 particles "decorated" with neutrinos while

eq. (2) reduces to the ordinary Dirac equation.

made in terms of eq. (2).

The discussion will be