Plane wave holonomies in quantum gravity. I. A model

This is the first of two papers which study the behavior of the SU(2) holonomies of canonical quantum gravity, when they are acted upon by a unidirectional, plane gravity wave. The present paper constructs a model based on holonomies and fluxes having support on a lattice (LHF denotes lattice-holonomy flux, rather than loop quantum gravity, since there are no loops). Initially, the flux-holonomy variables are treated as classical, commuting functions rather than quantized operators, in a limit where variation from vertex to vertex is small and fields are weak. We impose symmetries and fix gauges at the classical level. Despite the weakness of the fields, the field equations are not linear. Also, the theory can be quantized, and the expectation values of the quantum operators behave like their classical analogs. Exact LHF theories may be either local or nonlocal. The present paper argues that a wide class of nonlocal theories share nonlocal features which survive to the semiclassical limit, and these nonlocal features are included in the near-classical limit studied here. An appendix computes the surface term required when the propagation direction is the real line rather than S-1. Paper II introduces coherent states, constructs a damped sine wave solution to the model, and solves for the behavior of the holonomies in the presence of the wave.