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The Structure of the Continuum Limit of Spin Foams

Matteo Bruno, Eugenia Colafranceschi, Fabio M. Mele, Carlo Rovelli

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.

The Structure of the Continuum Limit of Spin Foams

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.
Paper Structure (26 sections, 27 theorems, 154 equations, 6 figures)

This paper contains 26 sections, 27 theorems, 154 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A Primer of Topological Quantum Field Theories
  3. Axioms for Spin Foams
  4. The Spin Foam Formulation of Loop Quantum Gravity
  5. Axioms in the Discrete
  6. A First Step in the Continuum
  7. Weak Order
  8. Properties of the Limit and TQFT
  9. Example: The Ponzano-Regge Model
  10. The Model
  11. State Sum Invariants
  12. Functional Nature of the Invariants
  13. Refinement and Continuum Limit
  14. Subdivision and Inductive Limit
  15. A No-Go Theorem
  16. ...and 11 more sections

Key Result

Lemma 4.1

Let $M\simeq N_1\cup_{\Sigma}N_2$ be a $d$-manifold with $\partial N_1=\Sigma_1^*\sqcup\Sigma$, $\partial N_2=\Sigma^*\sqcup\Sigma_2$ and $\partial M=\Sigma_1^*\sqcup \Sigma_2$. The subset of $\mathcal{T}_{M}^{\delta_1^* \sqcup \delta_2}$ consisting of triangulations that can be written as the glui The map $\gamma$ is order-preserving and cofinal. As a consequence, $Z_{\gamma(\bullet)}$ is a subn

Figures (6)

  • Figure 1: In a TQFT, the gluing of two $d$-manifolds $M_1$ and $M_2$ along their boundary $\Sigma$ is associated to a numerical invariant $Z(M)$, which only depends on the topology of the closed manifold $M=M_1\cup_{\Sigma}M_2$ and is given by the transition amplitude between the quantum states $Z(M_1)$ and $Z(M_2)$ in $\mathcal{H}_{\Sigma}$ respectively associated to $M_1$ and $M_2$.
  • Figure 2: Graphical representation of a spin foam. The boundary spin network states associated to the graphs $\Gamma_0$ and $\Gamma_1$ (in blue) encode the quantum geometry of the initial and final hyspersurfaces. The 2-complex interpolating between them represents a quantum spacetime history (in black). Vertices $v$ can generate new links and nodes as highlighted in grey.
  • Figure 3: Gluing of triangulated manifolds in a spin foam model.
  • Figure 4: Graphic representation of the weak order. Notice that the increment of simplices could be localised.
  • Figure 5: Illustration of the property \ref{['eq:ZPZgluingconv']} where, for simplicity, we take $\Sigma_1=\varnothing=\Sigma_2$. The first equality illustrates a key technical step discussed in Appendix \ref{['AppSec:gluing']} consisting of the restriction to the subnet defined over the directed set $S_\mathscr{C}$ of triangulations of $M\cup_\Sigma N$ that split in a suitable manner (see Lemma \ref{['lem:gen_cutsubnet']} in Appendix \ref{['AppSec:gluing']} for details). The second equality illustrates the propagator-like convolution gluing \ref{['eq:ZPZgluingconv']} to be contrasted with the inner product gluing in TQFT (cfr. Fig. \ref{['fig:tqftgluing']}). The second equality follows from the first provided that the sums and the limit can be exchanged (cfr. Appendix \ref{['AppSec:gluing']}).
  • ...and 1 more figures

Theorems & Definitions (72)

  • Definition 2.1: Topological Quantum Field Theory
  • Definition 3.1: Triangulation
  • Definition 3.2: Spin foam model
  • Lemma 4.1
  • Proposition 4.2
  • Corollary 4.3
  • Proposition 4.4
  • Corollary 4.5
  • Definition 5.1: Inductive limit
  • Proposition 5.1
  • ...and 62 more