Phase space descriptions for simplicial 4d geometries

TL;DR

Starting from the canonical phase space of discrete $SO(4)$ BF-theory, the authors implement a canonical version of the simplicity constraints to construct a phase space of geometric, simplicial 4d geometries and study its relation to loop quantum gravity and Regge calculus variants. They show that, on a fixed triangulation, the gauge-invariant sector of LQG corresponds to area–angle Regge calculus prior to gluing constraints, rather than to length Regge calculus, reflecting incomplete implementation of simplicity constraints in the canonical LQG framework. Through a two-step reduction—first enforcing left/right sector equality and then imposing gluing constraints—the paper derives a reduced phase space described by areas and 4d dihedral angles, which can be re-expressed in terms of edge lengths and conjugate momenta; for simple triangulations, first-class constraints can generate flat 4d dynamics. General triangulations introduce additional area-only constraints, suggesting a path to a length-based Regge-like phase space with potential dynamics via either Hamiltonian constraints or Pachner moves; the work clarifies the connections and tensions between LQG, spin foams, and Regge calculus in a canonical, geometrical setting.

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.