Restoring Poincar\'e Symmetry to the Lattice

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.