The Structure of the Continuum Limit of Spin Foams

TL;DR

This paper develops an axiomatic framework for understanding the continuum limit of spin foam models in quantum gravity, revealing structural insights and limitations of convergence approaches, and proposing a distributional method to define the physical Hilbert space.

Contribution

It introduces a model-independent axiomatic approach to the continuum limit of spin foams, analyzing convergence properties and proposing a distributional framework for the gravitational path integral.

Findings

Strong convergence assumptions lead to topological theories.

A weakened, distributional convergence defines a physical Hilbert space.

The continuum amplitudes act as distributions on the physical state space.

Abstract

The Spin Foam approach to quantum gravity aims at providing a covariant path-integral formulation of canonical Loop Quantum Gravity. Since spin foam amplitudes are defined through discretisations of spacetime, understanding the continuum limit of the theory remains a central open problem. In this work, we investigate the structural aspects of this limit in a model-independent manner. We begin by introducing an axiomatic framework for spin foam amplitudes inspired by Atiyah's formulation of Topological Quantum Field Theories (TQFTs). In this setting, Hilbert spaces and amplitudes are assigned to combinatorial and topological data associated with triangulated manifolds. By equipping the set of triangulations with suitable orders, this framework provides a precise notion of continuum limit and allows us to analyse its properties independently of any specific model. We proceed then to systematically investigate how the specifics of the limit procedure allow to go beyond TQFT in the continuum. Under natural assumptions on the convergence of spin foam amplitudes, we establish a no-go result: sufficiently strong notions of convergence necessarily lead to a topological theory. Motivated by this obstruction, we weaken the notion of convergence and consider the continuum limit of spin foam amplitudes in a distributional sense, in the spirit of Refined Algebraic Quantisation. Under this assumption, the amplitude associated with the cylinder defines a rigging map, yielding a canonical construction of the physical Hilbert space. The resulting continuum amplitudes act as well-defined distributions on this space of physical states, characterising this formulation of the gravitational path integral as physical in a precise sense.