Gupta-Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times

TL;DR

This paper develops a comprehensive framework for Gupta-Bleuler quantization of the electromagnetic field in globally hyperbolic space-times, ensuring states satisfy the microlocal spectrum condition without spectral gap assumptions.

Contribution

It introduces a new method for constructing Gupta-Bleuler representations that does not rely on spectral gaps or absence of zero modes, applicable to a broad class of space-times.

Findings

Constructed ground states on ultrastatic space-times without spectral gap.

Proved stability of zero-resonance absence under topology and metric perturbations.

Extended the quantization framework to a wide class of globally hyperbolic space-times.

Abstract

We give a complete framework for the Gupta-Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta-Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero modes. The only requirement, the absence of zero-resonance states, is shown to be stable under compact perturbations of topology and metric. Usual deformation arguments based on the time-slice axiom then lead to a construction of Gupta-Bleuler representations on a large class of globally hyperbolic space-times. As usual, the field algebra is represented on an indefinite inner product space, in which the physical states form a positive semi-definite subspace. Gauge transformations are incorporated in such a way that the field can be coupled perturbatively to a Dirac field. Our approach does not require any topological restrictions on the underlying space-time.