Lattice diffeomorphism invariance

TL;DR

This paper introduces a lattice formulation of diffeomorphism invariance for quantum gravity, ensuring the continuum limit respects general coordinate transformations without fundamental metrics.

Contribution

It develops a lattice action framework that maintains diffeomorphism invariance independent of lattice point positioning, without fundamental geometric objects.

Findings

Lattice diffeomorphism invariance leads to continuum general coordinate invariance.

The approach formulates gravity without fundamental metrics, deriving them as collective field expectations.

Examples include invariant actions for sigma-models and spinor gravity.

Abstract

We propose a lattice counterpart of diffeomorphism symmetry in the continuum. A functional integral for quantum gravity is regularized on a discrete set of space-time points, with fermionic or bosonic lattice fields. When the space-time points are positioned as discrete points of a continuous manifold, the lattice action can be reformulated in terms of average fields within local cells and lattice derivatives. Lattice diffeomorphism invariance is realized if the action is independent of the positioning of the space-time points. Regular as well as rather irregular lattices are then described by the same action. Lattice diffeomorphism invariance implies that the continuum limit and the quantum effective action are invariant under general coordinate transformations - the basic ingredient for general relativity. In our approach the lattice diffeomorphism invariant actions are formulated without introducing a metric or other geometrical objects as fundamental degrees of freedom. The metric rather arises as the expectation value of a suitable collective field. As examples, we present lattice diffeomorphism invariant actions for a bosonic non-linear sigma-model and lattice spinor gravity.