Some Observations on Non-covariant Gauges and the epsilon-term

TL;DR

This paper analyzes the role of the epsilon-term in Minkowski space path-integrals for non-covariant gauges, showing its importance for correct BRST identities and the limitations of Wick rotation.

Contribution

It demonstrates the necessity of including an epsilon-term in path-integral formulations for certain gauges and explores its impact on BRST identities and Green's functions.

Findings

Epsilon-term inclusion is crucial for correct gauge treatment.

Additional constraints on Green's functions arise from epsilon-term considerations.

Some gauges cannot be obtained via Wick rotation from Euclidean space.

Abstract

We consider the Lagrangian path-integrals in Minkowski space for gauges with a residual gauge-invariance. From rather elementary considerations, we demonstrate the necessity of inclusion of an epsilon-term (even) in the formal treatments, without which one may reach incorrect conclusions. We show, further, that the epsilon-term can contribute to the BRST WT-identities in a nontrivial way (even as epsilon-->0). We also show that the (expectation value of the) correct epsilon-term satisfies an algebraic condition. We show by considering (a commonly used) example of a simple local quadratic epsilon -term, that they lead to additional constraints on Green's function that are not normally taken into account in the BRST formalism that ignores the epsilon-term, and that they are characteristic of the way the singularities in propagators are handled. We argue that for a subclass of these gauges, the Minkowski path-integral could not be obtained by a Wick rotation from a Euclidean path-integral.