Spinor gravity and diffeomorphism invariance on the lattice

TL;DR

This paper develops a lattice formulation of quantum gravity using fermions, maintaining diffeomorphism invariance and local Lorentz symmetry, and explores how spacetime structure emerges dynamically.

Contribution

It introduces a fermion-based lattice functional integral for quantum gravity that preserves diffeomorphism invariance and extends to both Euclidean and Minkowski signatures.

Findings

Diffeomorphism invariance is realized if the action is independent of lattice point positioning.

The formulation maintains local Lorentz symmetry on the lattice.

The space-time signature difference emerges dynamically from the metric field's expectation value.

Abstract

The key ingredient for lattice regularized quantum gravity is diffeomorphism symmetry. We formulate a lattice functional integral for quantum gravity in terms of fermions. This allows for a diffeomorphism invariant functional measure and avoids problems of boundedness of the action. We discuss the concept of lattice diffeomorphism invariance. This is realized if the action does not depend on the positioning of abstract lattice points on a continuous manifold. Our formulation of lattice spinor gravity also realizes local Lorentz symmetry. Furthermore, the Lorentz transformations are generalized such that the functional integral describes simultaneously euclidean and Minkowski signature. The difference between space and time arises as a dynamical effect due to the expectation value of a collective metric field. The quantum effective action for the metric is diffeomorphism invariant. Realistic gravity can be obtained if this effective action admits a derivative expansion for long wavelengths.