Combinatorial space from loop quantum gravity

TL;DR

This paper revisits the canonical quantization of diffeomorphism invariant theories using loop variables, resulting in a combinatorial quantum model that extends to gravity and gauge fields.

Contribution

It introduces a new representation of the observable algebra in a separable Hilbert space, emphasizing a combinatorial approach to space in loop quantum gravity.

Findings

Operators respect diffeomorphism invariance or covariance.

Quantum theory is equivalent to a combinatorial gauge theory model.

Representation of observables is in a separable Hilbert space.

Abstract

The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectably invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.