Diffeomorphism invariant Quantum Field Theories of Connections in terms of webs

TL;DR

This paper extends the framework of diffeomorphism invariant quantum field theories of connections by including all piecewise smooth paths and loops, leading to new states and operators in quantum gravity.

Contribution

It introduces spin-web states, generalizes diffeomorphism averaging, and extends geometric and Hamiltonian operators within a more inclusive path framework.

Findings

Characterization of the Ashtekar-Isham configuration space spectrum

Introduction of spin-web states as a generalization of spin-network states

Extension of diffeomorphism invariance and operators in the theory

Abstract

In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise smooth finite paths and loops. In particular, we $(i)$ characterize the spectrum of the Ashtekar-Isham configuration space, $(ii)$ introduce spin-web states, a generalization of the spin-network states, $(iii)$ extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism invariant states and finally $(iv)$ extend the 3-geometry operators and the Hamiltonian operator.