Coarse graining free theories with gauge symmetries: the linearized case

TL;DR

This work develops a gauge-covariant coarse graining framework to construct discrete actions that preserve continuum gauge symmetries, starting from quadratic (free) theories and applying it to scalar fields, electromagnetism, and linearized gravity. By introducing coarse graining maps and projectors onto gauge (transverse) and physical (gauge-invariant) subspaces, the authors derive explicit formulas for the coarse grained actions and demonstrate invariance or form-invariance under coarse graining in representative cases, notably 2D electromagnetism and 3D linearized gravity. The approach leverages block-averaging and blocking-from-the-continuum to obtain perfect actions that mirror continuum dynamics at coarse scales, with potential implications for lattice gravity and diffeomorphism symmetry emergence. The results indicate that appropriate variable choices and projector-based coarse graining can control gauge degrees of freedom and yield nonlocal but physically faithful lattice theories, highlighting directions for extending to higher orders and non-topological settings.

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.

Figures (3)

• Figure 1: The variables used in discrete electromagnetism (here for $d=2$).
• Figure 2: The variables in electromagnetism are coarse grained along the lines of the lattice, since they are naturally one-forms. (here with $D=2$ and $L=3$)
• Figure 3: For gravity, the lattice (solid lines) needs to be enhanced by introducing (e.g. for $D=3$) face-diagonals (dashed lines) and body-diagonals (dotted lines) in order to capture all triangulation degrees of freedom of Regge calculus.