Phase space descriptions for simplicial 4d geometries

TL;DR

This paper develops phase space descriptions for 4d simplicial geometries from BF-theory, revealing differences between loop quantum gravity and Regge calculus, and constructing constraints for flat spacetime solutions.

Contribution

It introduces a canonical phase space framework for simplicial geometries, connecting loop quantum gravity with Regge calculus and analyzing the implementation of simplicity constraints.

Findings

Loop quantum gravity phase space is larger than length and area Regge calculus.

The phase space corresponds to area-angle Regge calculus before gluing constraints.

First class Hamiltonian and Diffeomorphism constraints can produce flat 4d spacetimes.

Abstract

Starting from the canonical phase space for discretised (4d) BF-theory, we implement a canonical version of the simplicity constraints and construct phase spaces for simplicial geometries. Our construction allows us to study the connection between different versions of Regge calculus and approaches using connection variables, such as loop quantum gravity. We find that on a fixed triangulation the (gauge invariant) phase space associated to loop quantum gravity is genuinely larger than the one for length and even area Regge calculus. Rather, it corresponds to the phase space of area-angle Regge calculus, as defined by Dittrich and Speziale in [arXiv:0802.0864] (prior to the imposition of gluing constraints, that ensure the metricity of the triangulation). We argue that this is due to the fact that the simplicity constraints are not fully implemented in canonical loop quantum gravity. Finally, we show that for a subclass of triangulations one can construct first class Hamiltonian and Diffeomorphism constraints leading to flat 4d space-times.