Relativistic classical theory II. Holonomy-flux representation of gravitational degrees of freedom

TL;DR

This paper develops a regularized, nonperturbative holonomy-flux formalism for classical gravity based on loop quantum gravity ideas, ensuring gauge invariance and precise relations between geometric quantities.

Contribution

It introduces a new holonomy-flux representation of gravitational degrees of freedom with improved relations and quantization procedures, extending the CLQG framework.

Findings

Demonstrates a regularization of gravity on a lattice using holonomies and fluxes.

Establishes precise relations between holonomies, connections, and densitized dreibeins.

Provides a more accurate representation of the densitized dreibein determinant than previous models.

Abstract

This article describes the regularization of the generally relativistic gauge field representation of gravity on a piecewise linear lattice. It is a part of the program concerning the classical relativistic theory of fundamental interactions, represented by minimally coupled gauge vector field densities and half-densities. The correspondence between the local Darboux coordinates on phase space and the local structure of the links of the lattice, embedded in the spatial manifold, is demonstrated. Thus, the canonical coordinates are replaceable by links-related quantities. This idea and the significant part of formalism are directly based on the model of canonical loop quantum gravity (CLQG). The first stage of this program is formulated regarding the gauge field, which dynamics is independent of other fundamental fields, but contributes to their dynamics. This gauge field, which determines systems equivalence in the actions defining all fundamental interactions, represents Einsteinian gravity. The related links-defined quantities depend on holonomies of gravitational connections and fluxes of densitized dreibeins. This article demonstrates how to determine these quantities, which lead to a nonpertubative formalism that preserves the general postulate of relativity. From this perspective, the formalism presented in this article is analogous to the Ashtekar-Barbero-Holst formulation on which CLQG is based. However, in this project, it is additionally required that the fields' coordinates are quantizable in the standard canonical procedure for a gauge theory and that any approximation in the construction of the model is at least as precisely demonstrated as the gauge invariance. These requirements lead to new relations between holonomies and connections, and the representation of the densitized deibein determinant that is more precise than the volume representation in CLQG.