$2+1$ Covariant Lattice Theory and t'Hooft's Formulation

TL;DR

The paper investigates how 't Hooft's gauge-fixed polygon representation of $(2+1)$-dimensional gravity relates to covariant lattice gravity. By mapping covariant lattice variables to scalar variables and analyzing first-class constraints, it shows that the Hamiltonian is a linear sum of deficit angles and that time discretization is a gauge-choice artifact dependent on the quantization order. It demonstrates that the spectra for space and time depend on whether one quantizes before or after gauge fixing, with covariant quantization predicting discrete link spectra in Euclidean gravity and other routes predicting continuous spectra. The work also outlines a 3+1 extension in the lattice/topological gravity context, suggesting broader implications for quantization of discrete gravity models.

Abstract

We show that 't Hooft's representation of (2+1)-dimensional gravity in terms of flat polygonal tiles is closely related to a gauge-fixed version of the covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of which is defined modulo $2 π$. A cyclic Hamiltonian implies that ``time'' is quantized. However, it turns out that this Hamiltonian is {\it constrained}. If one chooses an internal time and solves this constraint for the ``physical Hamiltonian'', the result is not a cyclic function. Even if one quantizes {\it a la Dirac}, the ``internal time'' observable does not acquire a discrete spectrum. We also show that in Euclidean 3-d lattice gravity, ``space'' can be either discrete or continuous depending on the choice of quantization. Finally, we propose a generalization of 't Hooft's gauge for Hamiltonian lattice formulations of topological gravity dimension 4.