2d manifold-independent spinfoam theory

TL;DR

This paper explores various background independent quantization techniques in a simple 2D topological theory, introducing a novel auxiliary field approach that enables manifold-independent transition amplitude calculations.

Contribution

It introduces a new auxiliary field theory method that provides a rigorous, background-independent way to define and compute manifold-independent transition amplitudes in 2D quantum gravity.

Findings

The auxiliary field theory allows precise sum over topologies.

Transition amplitudes are identified as physical observables.

Canonical structure can be reconstructed from amplitudes.

Abstract

A number of background independent quantizations procedures have recently been employed in 4d nonperturbative quantum gravity. We investigate and illustrate these techniques and their relation in the context of a simple 2d topological theory. We discuss canonical quantization, loop or spin network states, path integral quantization over a discretization of the manifold, spin foam formulation, as well as the fully background independent definition of the theory using an auxiliary field theory on a group manifold. While several of these techniques have already been applied to this theory by Witten, the last one is novel: it allows us to give a precise meaning to the sum over topologies, and to compute background-independent and, in fact, "manifold-independent" transition amplitudes. These transition amplitudes play the role of Wightman functions of the theory. They are physical observable quantities, and the canonical structure of the theory can be reconstructed from them via a C* algebraic GNS construction. We expect an analogous structure to be relevant in 4d quantum gravity.