Restoring Poincar\'e Symmetry to the Lattice

TL;DR

This paper develops a framework to incorporate Poincaré symmetry into lattice theories, enabling discrete models of spacetime that preserve fundamental symmetries, and applies it to lattice gravity.

Contribution

It introduces the discrete principal Poincaré bundle and lattice 5-vector theory, enabling Poincaré-invariant lattice models including gravity.

Findings

Demonstrates Poincaré symmetry in discrete lattice dynamics

Develops lattice gauge theory of gravity with Poincaré invariance

Recasts spacetime data as matter field data in lattice models

Abstract

The following work demonstrates the viability of Poincar\'e symmetry in a discrete universe. We develop the technology of the discrete principal Poincar\'e bundle to describe the pairing of (1) a hypercubic lattice `base manifold' labeled by integer vertices-denoted $\{\mathbf{n}\}=\{(n_t,n_x,n_y,n_z)\}$-with (2) a Poincar\'e structure group. We develop lattice 5-vector theory, which describes a non-unitary representation of the Poincar\'e group whose dynamics and gauge transformations on the lattice closely resemble those of a scalar field in spacetime. We demonstrate that such a theory generates discrete dynamics with the complete infinitesimal symmetry-and associated invariants-of the Poincar\'e group. Following our companion paper, we `lift' the Poincar\'e gauge symmetries to act only on vertical matter and solder fields, and recast `spacetime data'--stored in the $\partial_\mu\phi(x)$ kinetic terms of a free scalar field theory--as `matter field data'-stored in the $\phi^\mu[\mathbf{n}]$ components of the 5-vector field itself. We gauge 5-vector theory to describe a lattice gauge theory of gravity, and discuss the physical implications of a discrete, Poincar\'e-invariant theory.