Massless Rarita–Schwinger field from a divergenceless antisymmetric tensor-spinor of pure spin-3/2

Abstract

We construct the Rarita-Schwinger basis vectors, U µ , spanning the direct product space, U µ := A µ ⊗ u M , of a massless four-vector, A µ , with massless Majorana spinors, u M , together with the associated field-strength tensor, T µν := p µ U ν − p ν U µ . The T µν space is reducible and contains one massless subspace of a pure spin-3/2 ∈ (3/2, 0) ⊕ (0, 3/2). We show how to single out the latter in a unique way by acting on T µν with an earlier derived momentum independent projector, P (3/2,0) , properly constructed from one of the Casimir operators of the algebra so(1, 3) of the homogeneous Lorentz group. In this way it becomes possible to describe the irreducible massless (3/2, 0) ⊕ (0, 3/2) carrier space by means of the anti-symmetric-tensor of second rank with Majorana spinor components, defined as w (3/2,0) µν := P (3/2,0) µν γδ T γδ . The conclusion is that the (3/2, 0) ⊕ (0, 3/2) bi-vector spinor field can play the same role with respect to a U µ gauge field as the bi-vector, (1, 0) ⊕ (0, 1), associated with the electromagnetic field-strength tensor, F µν , plays for the Maxwell gauge field, A µ . Correspondingly, we find the free electromagnetic field equation, p µ F µν = 0, is paralleled by the free massless Rarita-Schwinger field equation, p µ w (3/2,0) µν = 0, supplemented by the additional condition, γ µ γ ν w (3/2,0) µν = 0, a constraint that invokes the Majorana sector.

Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

“…The projector that finds this sector in B [µν] ⊗ ψ, earlier obtained in [22], was reported in Equation (62). As a recent application, the presentation of (3/2, 0) ⊕ (0, 3/2) as a totally-antisymmetric Lorentz tensor of second rank with Dirac spinor components was employed in [40] as the field-strength tensor of the gravitino, described by a massless Rarita-Schwinger field and in complete parallel to the description in the quantum electrodynamics of a massless photon by means of a field strength tensor transforming as (1, 0) ⊕ (0, 1). Moreover, calculations of Compton scattering off such a spin-3/2 revealed differences to spin-3/2 embedded by the four-vector-spinor ψ µ , and the conclusion could be drawn that particles of the same spin transforming in distinct carrier spaces possess distinct physical characteristics, among them the electromagnetic multipole moments.…”

On the choice of relativistic wave equation for the particle having spin s = 3/2

“…The main problem is an interaction with external fields. These difficulties of the Rarita-Schwinger equation were mentioned by many researchers, see, for example, [24][25][26][27][28][29][30][31][32][33][34][35].…”

“…This equation was suggested in [9] for the particle having spin 3/2. It is the Dirac-like equation with 24component The system of equation ( 12) was put into consideration in order to overcome the difficulties mentioned in [ [24][25][26][27][28][29][30][31][32][33][34][35]. Note only partial and rather conditional success.…”

“…It should be noted especially that the theoretical description of the elementary particles with spin 3/2 in the frameworks of field theory still deals with a number of fundamental problems (see, e.g., [24][25][26][27][28][29][30][31][32][33][34][35], as well as [13][14][15][16]). This is typical for all fields with higher spins that are larger than s = 1.…”

“…It is clear that the authors of publications [ [13][14][15][16][24][25][26][27][28][29][30][31][32][33][34][35][36][37] tried to improve the theory. We have the same goal.…”