Quantization of the Proca field in curved spacetimes - A study of mass dependence and the zero mass limit

TL;DR

This paper rigorously studies the classical and quantum Proca field in curved spacetimes, focusing on the zero mass limit and the conditions needed for the field to converge to Maxwell's theory, revealing gauge invariance and conservation properties.

Contribution

It provides a rigorous construction of the classical and quantum Proca fields in curved spacetimes, analyzing the zero mass limit and the role of gauge invariance in this process.

Findings

Zero mass limit exists under co-closed test form restriction

Quantum Proca field theory is local and covariant

Fields do not satisfy Maxwell's equations in the limit without further modifications

Abstract

In this thesis we investigate the Proca field in arbitrary globally hyperbolic curved spacetimes. We rigorously construct solutions to the classical Proca equation, including external sources and without restrictive assumptions on the topology of the spacetime, and investigate the classical zero mass limit. We find that the limit exists if we restrict the class of test one-forms to those that are co-closed, effectively implementing a gauge invariance by exact distributional one-forms of the vector potential. In order to obtain also the Maxwell dynamics in the limit, one has to restrict the initial data such that the Lorenz constraint is well behaved. With this, we naturally find conservation of current. For the quantum problem we first construct the generally covariant quantum Proca field theory in curved spacetimes in the framework of Brunetti, Fredenhagen and Verch and show that the theory is local. We define a precise notion of continuity of the quantum Proca field with respect to the mass. With this notion we investigate the zero mass limit in the quantum case and find that, like in the classical case, the limit exists if and only if the class of test one-forms is restricted to co-closed ones, again implementing a gauge equivalence relation by exact distributional one-forms. It turns out that in the limit the fields do not solve Maxwell's equation in a distributional sense. We will discuss the reason from different perspectives and suggest possible solutions to find the correct Maxwell dynamics in the zero mass limit.