Coarse graining free theories with gauge symmetries: the linearized case

TL;DR

This paper explores a perturbative coarse graining method for linearized gauge theories, demonstrating that a specific discretization of linearized gravity maintains gauge symmetry invariance under coarse graining.

Contribution

It develops a formalism for coarse graining gauge systems and provides a discretization of linearized gravity that preserves gauge symmetry at the linearized level.

Findings

3D linearized gravity action is invariant under coarse graining.

The developed formalism handles gauge symmetries in discretized theories.

Perturbative approach enables analytical insights into gauge symmetry preservation.

Abstract

Discretizations of continuum theories often do not preserve the gauge symmetry content. This occurs in particular for diffeomorphism symmetry in general relativity, which leads to severe difficulties both in canonical and covariant quantization approaches. We discuss here the method of perfect actions, which attempts to restore gauge symmetries by mirroring exactly continuum physics on a lattice via a coarse graining process. Analytical results can only be obtained via a perturbative approach, for which we consider the first steps, namely the coarse graining of the linearized theory. The linearized gauge symmetries are exact also in the discretized theory, hence we develop a formalism to deal with gauge systems. Finally we provide a discretization of linearized gravity as well as a coarse graining map and show that with this choice the 3D linearized gravity action is invariant under coarse graining.