Unimodular Proca Theory: Breaking the U(1) gauge symmetry of unimodular gravity via a mass term

TL;DR

This paper explores the Hamiltonian structure of unimodular-like theories, demonstrating that breaking the $U(1)$ gauge symmetry introduces a massless degree of freedom and alters the dynamics of the zero-mode, with implications for cosmological constants.

Contribution

It provides a detailed Hamiltonian analysis of unimodular gravity, showing how breaking the $U(1)$ gauge symmetry leads to new physical degrees of freedom and modified zero-mode dynamics.

Findings

Gauge fixing restricts the spatial components of the four-volume time vector.

Breaking $U(1)$ gauge invariance introduces a massless propagating degree of freedom.

Modified dynamics for the gauge-invariant zero-mode depend on the choice of the constant.

Abstract

We study the Hamiltonian structure of unimodular-like theories, where the cosmological constant (or other supposed constants of nature) are demoted from fixed parameters to classical constants of motion. No new local degrees of freedom are present as a result of a $U(1)$ gauge invariance of the theory. Hamiltonian analysis of the action reveals that the only possible gauge fixing that can be enforced is setting the spatial components of the four-volume time vector ${\cal T}^{i}\approx0$. As a consequence of this, the gauge-fixed unimodular path integral is equivalent to the minisuperspace unimodular path integral. However, should we break the $U(1)$ gauge invariance, two things happen: a massless propagating degree of freedom appears, and the (gauge-invariant) zero-mode receives modified dynamics. The implications are investigated, with the phenomenology depending crucially on the target ``constant''.