Maxwell field with gauge fixing term in the radiation- and matter-dominant stages: exact solution and stress tensor

TL;DR

This paper derives exact solutions and analyzes the stress tensor of the Maxwell field with gauge fixing in radiation- and matter-dominant universe stages, showing the vacuum stress tensor vanishes after regularization, independent of gauge fixing.

Contribution

It provides the first exact solutions and covariant quantization of Maxwell fields with gauge fixing in RD and MD stages, extending previous de Sitter space results.

Findings

Vacuum stress tensor becomes zero after regularization in both RD and MD stages.

Transverse stress tensor is similar across flat Robertson-Walker spacetimes.

Longitudinal-temporal stress tensor cancels out, resulting in zero particle contribution.

Abstract

We study the Maxwell field with a general gauge fixing (GF) term in the radiation-dominant (RD) and matter-dominant (MD) stages of expanding Universe, as a continuation to the previous work in de Sitter space. We derive the exact solutions, perform the covariant canonical quantization and obtain the stress tensor in the Gupta-Bleuler (GB) physical states, which is independent of the GF constant and is also invariant under the quantum residual gauge transformation. The transverse stress tensor is similar in all flat Robertson-Walker spacetimes, and its vacuum part is $\propto k^4$ and becomes zero after the 0th-order adiabatic regularization. The longitudinal-temporal stress tensor, in both RD and MD stages, is zero due to a cancelation between the longitudinal and temporal parts in the GB states, and so is the particle part of the GF stress tensor. The vacuum GF stress tensor, in the RD stage, contains $k^4,k^2$ divergences and becomes zero by the 2nd-order regularization, however, in the MD stage, contains $k^4, k^2, k^0$ divergences and becomes zero by the 4th-order regularization. So, the order of adequate regularization depends not only upon the type of fields, but also upon the background spacetimes. In summary, in both the RD and MD stages, like in de Sitter space, the total regularized vacuum stress tensor is zero, only the transverse photon part remains, and this holds independent of the GF term.