2019/01/26 by Glasser, Alexander S., Qin, Hong
#FOS: Physical sciences #General Physics (physics.gen-ph)
paper · doi:10.48550/arxiv.1902.04396
The following work demonstrates the viability of Poincaré symmetry in a discrete universe. We develop the technology of the discrete principal Poincaré bundle to describe the pairing of (1) a hypercubic lattice `base manifold' labeled by integer vertices-denoted \n\=\(nt,nx,ny,nz)\-with (2) a Poincaré structure group. We develop lattice 5-vector theory, which describes a non-unitary representation of the Poincaré 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 Poincaré group. Following our companion paper, we `lift' the Poincaré gauge symmetries to act only on vertical matter and solder fields, and recast `spacetime data'--stored in the ∂μϕ(x) kinetic terms of a free scalar field theory--as `matter field data'-stored in the ϕμ[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, Poincaré-invariant theory.