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2d manifold-independent spinfoam theory

Etera R. Livine, Alejandro Perez, Carlo Rovelli

TL;DR

This paper investigates how several background-independent quantization methods apply to a simple 2d topological BF theory, demonstrating their mutual consistency by deriving manifold- and topology-independent transition amplitudes. It shows that canonical loop/spin-network quantization, spin-foam path integrals on triangulations, and an auxiliary group-field theory yield the same physical content, with the latter providing a natural sum over spacetime topologies. The transition amplitudes serve as physical observables and permit reconstruction of the canonical structure via a GNS construction, akin to Wightman functions in standard QFT. The study offers a controlled setting that illuminates how similar diffeomorphism-invariant structures could underpin nonperturbative 4d quantum gravity, linking loop quantum gravity boundary states to a topology-summing spinfoam framework.

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.

2d manifold-independent spinfoam theory

TL;DR

This paper investigates how several background-independent quantization methods apply to a simple 2d topological BF theory, demonstrating their mutual consistency by deriving manifold- and topology-independent transition amplitudes. It shows that canonical loop/spin-network quantization, spin-foam path integrals on triangulations, and an auxiliary group-field theory yield the same physical content, with the latter providing a natural sum over spacetime topologies. The transition amplitudes serve as physical observables and permit reconstruction of the canonical structure via a GNS construction, akin to Wightman functions in standard QFT. The study offers a controlled setting that illuminates how similar diffeomorphism-invariant structures could underpin nonperturbative 4d quantum gravity, linking loop quantum gravity boundary states to a topology-summing spinfoam framework.

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.

Paper Structure

This paper contains 10 sections, 90 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Classical theory
  3. Hamiltonian analysis
  4. Canonical quantization
  5. Discretization
  6. Boundaries
  7. Auxiliary field theory
  8. Transition amplitudes
  9. Canonical and algebraic structure
  10. Conclusions and perspectives

Figures (5)

  • Figure 1: A spacetime manifold of genus 1 with two boundary components.
  • Figure 2: Structure of kinetic and potential term in the action.
  • Figure 3: Propagator and vertex.
  • Figure 4: A 2d two-complex (part of it).
  • Figure 5: Two-complexes corresponding in 2d to the sphere $S$, the torus $T$, and the projective plane $RP^2$