Deformation stability of BRST-quantization

TL;DR

This paper establishes the deformation stability of BRST-quantization in gauge theories, providing a local construction of observables that remains consistent under perturbations, with applications to QED in finite volumes.

Contribution

It introduces a local, perturbation-theoretic construction of BRST observables that preserves the Hilbert space structure under deformations, avoiding the adiabatic limit.

Findings

Hilbert space structure is stable under perturbations.

BRST-quantization is incompatible with periodic boundary conditions for massless gauge fields.

The construction applies to QED in finite spatial volume.

Abstract

To avoid the problems which are connected with the long distance behavior of perturbative gauge theories we present a local construction of the observables which does not involve the adiabatic limit. First we construct the interacting fields as formal power series by means of causal perturbation theory. The observables are defined by BRST invariance where the BRST-transformation $\tilde s$ acts as a graded derivation on the algebra of interacting fields. Positivity, i.e. the existence of Hilbert space representations of the local algebras of observables is shown with the help of a local Kugo-Ojima operator $Q_{\rm int}$ which implements $\tilde s$ on a local algebra and differs from the corresponding operator $Q$ of the free theory. We prove that the Hilbert space structure present in the free case is stable under perturbations. All assumptions are shown to be satisfied in QED in a finite spatial volume with suitable boundary conditions. As a by-product we find that the BRST-quantization is not compatible with periodic boundary conditions for massless free gauge fields.