A local (perturbative) construction of observables in gauge theories: the example of QED

TL;DR

This paper develops a local, perturbative method for constructing gauge theory observables, exemplified by QED, avoiding infrared issues and ensuring stability of the Hilbert space structure.

Contribution

It introduces a local construction of gauge observables using causal perturbation theory and the Kugo-Ojima formalism, without relying on the adiabatic limit.

Findings

Constructs interacting fields as formal power series.

Defines BRST transformations locally via the Kugo-Ojima formalism.

Proves stability of the Hilbert space structure under perturbations in QED.

Abstract

Interacting fields can be constructed as formal power series in the framework of causal perturbation theory. The local field algebra $\tilde {\cal F}({\cal O})$ is obtained without performing the adiabatic limit; the (usually bad) infrared behavior plays no role. To construct the observables in gauge theories we use the Kugo-Ojima formalism; we define the BRST-transformation $\tilde s$ as a graded derivation on the algebra of interacting fields and use the implementation of $\tilde s$ by the Kugo-Ojima operator $Q_{\rm int}$. Since our treatment is local, the operator $Q_{\rm int}$ 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.