Three dimensional loop quantum gravity: physical scalar product and spin foam models

TL;DR

This paper rigorously defines the physical scalar product in three-dimensional loop quantum gravity, connecting canonical quantization with spin-foam models without divergences or cut-offs, advancing the understanding of quantum gravity dynamics.

Contribution

It provides a rigorous construction of the physical scalar product in 3D loop quantum gravity and links it explicitly to spin-foam models without divergences.

Findings

Physical scalar product expressed as finite spin-foam sum

No cut-off or bubble divergences in the spin-foam representation

Establishes a direct connection between canonical and spin-foam quantizations

Abstract

In this paper, we address the problem of the dynamics in three dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space--corresponding to the quantization of the infinite dimensional kinematical configuration space of the theory--to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canonical quantization of three dimensional gravity and spin-foam models. We emphasize two main properties of the result: first that no cut-off in the kinematical degrees of freedom of the theory is introduced (in contrast to standard `lattice' methods), and second that no ill-defined sum over spins (`bubble' divergences) are present in the spin foam representation.