Quantum Symmetry Reduction for Diffeomorphism Invariant Theories of Connections

TL;DR

This paper develops a quantum symmetry reduction method for diffeomorphism invariant connection theories, using symmetric states and spin networks, with applications to lower-dimensional gravity and spherically symmetric models.

Contribution

It introduces a new quantum symmetry reduction procedure for diffeomorphism invariant theories based on symmetric states and spin networks.

Findings

Constructed symmetric states with a scalar product from the Ashtekar-Lewandowski measure.

Defined a quantum symmetry reduction procedure within the spin network framework.

Analyzed the structure of reduced quantum theories in specific models.

Abstract

Given a symmetry group acting on a principal fibre bundle, symmetric states of the quantum theory of a diffeomorphism invariant theory of connections on this fibre bundle are defined. These symmetric states, equipped with a scalar product derived from the Ashtekar-Lewandowski measure for loop quantum gravity, form a Hilbert space of their own. Restriction to this Hilbert space yields a quantum symmetry reduction procedure in the framework of spin network states the structure of which is analyzed in detail. Three illustrating examples are discussed: Reduction of 3+1 to 2+1 dimensional quantum gravity, spherically symmetric electromagnetism and spherically symmetric gravity.