Canonical Field Anticommutators in the Extended Gauged Rarita-Schwinger Theory

TL;DR

This paper investigates the canonical quantization of the gauged Rarita-Schwinger theory with an auxiliary field, revealing singularities and pathologies that challenge conventional positive semi-definite quantization methods.

Contribution

It introduces an extended gauged Rarita-Schwinger theory with an auxiliary field and analyzes its quantization, highlighting issues with positivity and singularities in the Dirac brackets.

Findings

Dirac brackets are singular in the original theory

In radiation gauge, brackets are nonsingular but not positive semi-definite

The theory exhibits pathologies preventing standard quantization

Abstract

We reexamine canonical quantization of the gauged Rarita-Schwinger theory using the extended theory, incorporating a dimension $\frac{1}{2}$ auxiliary spin-$\frac{1}{2}$ field $\Lambda$, in which there is an exact off-shell gauge invariance. In $\Lambda=0$ gauge, which reduces to the original unextended theory, our results agree with those found by Johnson and Sudarshan, and later verified by Velo and Zwanziger, which give a canonical Rarita-Schwinger field Dirac bracket that is singular for small gauge fields. In gauge covariant radiation gauge, the Dirac bracket of the Rarita-Schwinger fields is nonsingular, but does not correspond to a positive semi-definite anticommutator, and the Dirac bracket of the auxiliary fields has a singularity of the same form as found in the unextended theory. These results indicate that gauged Rarita-Schwinger theory is somewhat pathological, and cannot be canonically quantized within a conventional positive semi-definite metric Hilbert space. We leave open the questions of whether consistent quantizations can be achieved by using an indefinite metric Hilbert space, by path integral methods, or by appropriate couplings to conventional dimension $\frac{3}{2}$ spin-$\frac{1}{2}$ fields.