Loop Quantum Gravity a la Aharonov-Bohm

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.

Figures (1)

• Figure 1: A portion of the manifold $\Sigma'=\Sigma-l$. The defect-network consists of the single line $l$. As the connection is locally-flat, the magnetic field vanishes everywhere except along the line $l$. The line can be thought as a thin solenoid which confines the magnetic field. The flux of the magnetic field through a surface $S$ (in figure) determines the holonomy of the connection along a curve $\gamma$ bounding the surface.