Unitarity of Maxwell theory on curved spacetimes in the covariant formalism

TL;DR

This paper demonstrates the unitarity of the covariant Maxwell theory in curved spacetimes using a Euclidean path integral approach, resolving previous conflicts with canonical methods and highlighting the importance of zero mode contributions.

Contribution

It establishes the unitarity of the covariant Maxwell theory in curved spacetime by carefully analyzing gauge fixing and zero modes, including an overlooked geometric factor.

Findings

The covariant approach agrees with canonical results on ultrastatic manifolds.

An extra geometric factor from zero modes affects entropy and stress-energy calculations.

The covariant formalism is unitarily consistent if a canonical formulation exists.

Abstract

Quantum field theory in curved spacetime may be defined either through a manifestly unitary canonical approach or via the manifestly covariant path integral formalism. For gauge theories, these two approaches have produced conflicting results, leading to the question of whether the canonical approach is covariant, and whether the path integral approach is unitary. We show the unitarity of the covariant U(1) Maxwell theory, defined via the Wick rotation of a Euclidean path integral. We begin by gauge-fixing the path integral, taking care with zero modes, large gauge transformations, and nontrivial bundles. We find an extra geometric factor in the partition function that has been overlooked in previous work, coming from the zero mode of the gauge symmetry, which affects the entropy and stress-energy tensor. With this extra factor, the covariant calculation agrees with the canonical result for ultrastatic manifolds, and in D = 2. Finally, we argue that if there exists a unitary (but not necessarily covariant) canonical formulation, then the covariant formulation must also be unitary, even if the two approaches disagree.