On diffeomorphism invariance for lattice theories

TL;DR

This paper emphasizes the importance of explicitly imposing the diffeomorphism constraint in lattice formulations of diffeomorphism invariant theories to ensure physically meaningful quantum theories, demonstrated through 2+1 gravity and gauge theories.

Contribution

It shows that automatic handling of the diffeomorphism constraint in abstract lattices is insufficient, advocating for explicit imposition to achieve valid quantum models.

Findings

Explicit diffeomorphism constraint imposition is necessary for physical validity.

Lattice formulations without explicit constraints can lead to unphysical results.

Comparison between continuum and lattice theories confirms the importance of the constraint.

Abstract

We consider the role of the diffeomorphism constraint in the quantization of lattice formulations of diffeomorphism invariant theories of connections. It has been argued that in working with abstract lattices, one automatically takes care of the diffeomorphism constraint in the quantum theory. We use two systems in order to show that imposing the diffeomorphism constraint is imperative to obtain a physically acceptable quantum theory. First, we consider $2+1$ gravity where an exact lattice formulation is available. Next, general theories of connections for compact gauge groups are treated, where the quantum theories are known --for both the continuum and the lattice-- and can be compared.