Non-Trivial Ghosts and Second Class Constraints

TL;DR

This paper investigates the quantization of a vector-tensor gauge model, revealing that ghost fields from second-class constraints affect physical degrees of freedom, and explores converting these constraints to first-class using Stueckelberg fields and BFT methods.

Contribution

It identifies the role of ghost fields from second-class constraints in a vector-tensor model and compares Stueckelberg and BFT approaches for constraint conversion.

Findings

Ghost fields from second-class constraints influence physical degrees of freedom.

Stueckelberg fields are not equivalent to BFT in the vector-tensor model.

A new gauge invariance is discovered via BFT in the pure tensor model.

Abstract

In a model in which a vector gauge field $W_\mu^a$ is coupled to an antisymmetric tensor field $\phi_{\mu\nu}^a$ possessing a pseudoscalar mass, it has been shown that all physical degrees of freedom reside in the vector field. Upon quantizing this model using the Faddeev-Popov procedure, explicit calculation of the two-point functions $<\phi\phi >$ and $<W \phi>$ at one-loop order seems to have yielded the puzzling result that the effective action generated by radiative effects has more physical degrees of freedom than the original classical action. In this paper we point out that this is not in fact a real effect, but rather appears to be a consequence of having ignored a "ghost" field arising from the contribution to the measure in the path integral arising from the presence of non-trivial second-class constraints. These ghost fields couple to the fields $W_\mu^a$ and $\phi_{\mu\nu}^a$, which makes them distinct from other models involving ghosts arising from second-class constraints (such as massive Yang-Mills (YM) models) that have been considered, as in these other models such ghosts decouple. As an alternative to dealing with second class constraints, we consider introducing a "Stueckelberg field" to eliminate second-class constraints in favour of first-class constraints and examine if it is possible to then use the Faddeev-Popov quantization procedure. In the Proca model, introduction of the Stueckelberg vector is equivalent to the Batalin-Fradkin-Tyutin (BFT) approach to converting second-class constraints to being first class through the introduction of new variables. However, introduction of a Stueckelberg vector is not equivalent to the BFT approach for the vector-tensor model. In an appendix, the BFT procedure is applied to the pure tensor model and a novel gauge invariance is found.