Loop Quantum Gravity a la Aharonov-Bohm

TL;DR

This paper explores the connection between Loop Quantum Gravity and topological field theories with defect lines, demonstrating how the quantum states relate to the Aharonov-Bohm effect and spin-network structures.

Contribution

It shows that the LQG state space can be derived from quantizing a topological field theory with defect lines, linking holonomies and spin-networks.

Findings

Scalar product reduces to a finite-dimensional integral over moduli space.

Derived a measure involving the Faddeev-Popov determinant.

Explicitly connected single defect-line case to LQG spin-network states.

Abstract

The state space of Loop Quantum Gravity admits a decomposition into orthogonal subspaces associated to diffeomorphism equivalence classes of spin-network graphs. In this paper I investigate the possibility of obtaining this state space from the quantization of a topological field theory with many degrees of freedom. The starting point is a 3-manifold with a network of defect-lines. A locally-flat connection on this manifold can have non-trivial holonomy around non-contractible loops. This is in fact the mathematical origin of the Aharonov-Bohm effect. I quantize this theory using standard field theoretical methods. The functional integral defining the scalar product is shown to reduce to a finite dimensional integral over moduli space. A non-trivial measure given by the Faddeev-Popov determinant is derived. I argue that the scalar product obtained coincides with the one used in Loop Quantum Gravity. I provide an explicit derivation in the case of a single defect-line, corresponding to a single loop in Loop Quantum Gravity. Moreover, I discuss the relation with spin-networks as used in the context of spin foam models.