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

TL;DR

This paper explores the relationship between 't Hooft's polygonal tile representation of (2+1)-dimensional gravity and a covariant lattice theory, revealing insights into Hamiltonian constraints, quantization of time, and potential extensions to higher dimensions.

Contribution

It establishes a connection between 't Hooft's gauge and covariant lattice theory, analyzes Hamiltonian constraints, and proposes a generalization for 4D topological gravity.

Findings

Hamiltonian is a sum of vertex Hamiltonians modulo 2π

Time is quantized in 't Hooft's formulation but not after solving constraints

Space can be discrete or continuous depending on quantization in 3D Euclidean gravity

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 $2 \pi$. 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.