On the spin content of the classical massless Rarita--Schwinger system

TL;DR

This paper investigates the spin structure of the massless Rarita--Schwinger system using both Lagrangian and Hamiltonian frameworks, revealing how gauge fixing and the Dirac conjecture influence the propagation of spin-1/2 and spin-3/2 components.

Contribution

It provides a detailed analysis of the Rarita--Schwinger theory's spin content and clarifies the impact of the Dirac conjecture on the Hamiltonian formulation.

Findings

Gauge fixing leaves spin-1/2 and spin-3/2 propagating modes.

Assuming the Dirac conjecture removes the spin-1/2 sector.

Without the conjecture, spin-1/2 propagates and matches Euler-Lagrange equations.

Abstract

We analyze the Rarita--Schwinger (RS) massless theory in the Lagrangian and Hamiltonian approaches. At the Lagrangian level, the standard gamma-trace gauge fixing constraint leaves a spin-1/2 and a spin-3/2 propagating Poincar\'e group helicities. At the Hamiltonian level, the result depends on whether the Dirac conjecture--that all first class constraints generate gauge symmetries--is assumed or not. In the affirmative case, a secondary first class constraint must be added to the total Hamiltonian and a corresponding gauge fixing condition must be imposed, completely removing the spin-1/2 sector. In the opposite case, the spin-1/2 field propagates and the Hamilton field equations match the Euler-Lagrange equations.