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On the summation and triangulation independence of Lorentzian spinfoam amplitudes for all LQG

Muxin Han

TL;DR

This work tackles triangulation dependence in Lorentzian spinfoam quantum gravity by introducing the spinfoam stack, a framework that sums over an infinite class of 2-complexes formed by stacking faces on a root complex. In the limit of large internal area cutoffs $A_f$, the path integral localizes onto the space of SU(2) flat connections, effectively yielding a topological BF-like theory and, for trivial topology, a renormalized amplitude independent of bulk triangulation. The key technical steps involve representing spin-network stacks in the LQG Hilbert space, constructing the spinfoam stack with SL(2, C) integrals and a Laplace transform to perform a stationary phase analysis, and proving that the resulting moduli space coincides with the SU(2) flat connection moduli space. For trivial topology, the amplitude factorizes into a bulk normalization and a boundary-only finite part, leading to explicit triangulation independence after renormalization. The results point to a potential UV fixed point in the spinfoam framework and clarify how continuum, topological features emerge from a regulated sum over discrete complexes, while preserving consistency with known IR semiclassical limits and their relation to Regge calculus.

Abstract

This paper investigates the fundamental issue of triangulation dependence in spinfoam quantum gravity. It introduces a novel framework, named spinfoam stack, to systematically sum spinfoam amplitudes over an infinite class of 2-complexes. These complexes are generated by stacking an arbitrary number of faces upon a simpler root complex. The central result is obtained by analyzing the amplitude of spinfoam stack in the limit where an upper cut-off on the area of internal faces is taken to infinity. In this limit, the amplitude as an integral localizes via a stationary phase mechanism onto a critical manifold. This manifold is shown to be the space of SU(2) flat connections on the underlying complex. This localization effectively reduces the bulk dynamics from a theory of quantum geometry to a topological theory akin to SU(2) BF theory. For spinfoams on topologically trivial manifolds, this result has a powerful consequence: the spinfoam stack amplitude factorizes into a triangulation-dependent normalization factor and a finite part that depends only on the boundary data. Renormalizing the amplitude yields a finite result that is manifestly independent of the bulk structure of the 2-complex. This provides a concrete realization of triangulation independence in a well-defined limit, suggesting the possibility of existing a non-trivial fixed point of quantum gravity within the spinfoam formalism.

On the summation and triangulation independence of Lorentzian spinfoam amplitudes for all LQG

TL;DR

This work tackles triangulation dependence in Lorentzian spinfoam quantum gravity by introducing the spinfoam stack, a framework that sums over an infinite class of 2-complexes formed by stacking faces on a root complex. In the limit of large internal area cutoffs , the path integral localizes onto the space of SU(2) flat connections, effectively yielding a topological BF-like theory and, for trivial topology, a renormalized amplitude independent of bulk triangulation. The key technical steps involve representing spin-network stacks in the LQG Hilbert space, constructing the spinfoam stack with SL(2, C) integrals and a Laplace transform to perform a stationary phase analysis, and proving that the resulting moduli space coincides with the SU(2) flat connection moduli space. For trivial topology, the amplitude factorizes into a bulk normalization and a boundary-only finite part, leading to explicit triangulation independence after renormalization. The results point to a potential UV fixed point in the spinfoam framework and clarify how continuum, topological features emerge from a regulated sum over discrete complexes, while preserving consistency with known IR semiclassical limits and their relation to Regge calculus.

Abstract

This paper investigates the fundamental issue of triangulation dependence in spinfoam quantum gravity. It introduces a novel framework, named spinfoam stack, to systematically sum spinfoam amplitudes over an infinite class of 2-complexes. These complexes are generated by stacking an arbitrary number of faces upon a simpler root complex. The central result is obtained by analyzing the amplitude of spinfoam stack in the limit where an upper cut-off on the area of internal faces is taken to infinity. In this limit, the amplitude as an integral localizes via a stationary phase mechanism onto a critical manifold. This manifold is shown to be the space of SU(2) flat connections on the underlying complex. This localization effectively reduces the bulk dynamics from a theory of quantum geometry to a topological theory akin to SU(2) BF theory. For spinfoams on topologically trivial manifolds, this result has a powerful consequence: the spinfoam stack amplitude factorizes into a triangulation-dependent normalization factor and a finite part that depends only on the boundary data. Renormalizing the amplitude yields a finite result that is manifestly independent of the bulk structure of the 2-complex. This provides a concrete realization of triangulation independence in a well-defined limit, suggesting the possibility of existing a non-trivial fixed point of quantum gravity within the spinfoam formalism.

Paper Structure

This paper contains 13 sections, 6 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spin-network stack
  3. Spinfoam stack
  4. Stack vertex amplitude
  5. Stack amplitudes on arbitrary complex
  6. Localization to the space of flat connections
  7. Trivial topology and triangulation independence
  8. Discussion
  9. Explicit computations for trivial topology
  10. Parametrizations
  11. Nondegenerate Hessian matrix
  12. Analyticity of $s_h(g)$
  13. Complete the proof of Theorem \ref{['SU2flatconn']}

Key Result

Lemma 4.1

$|\zeta_k^{(h)}|\leq \lambda_h d_k^2$, the equality holds if and only if $g_h$ satisfies

Figures (4)

  • Figure 1: The spin-network stack.
  • Figure 2: The spin-network stack evolves to the spinfoam stack: The spin-network link $\mathfrak{l}$ evolves to the spinfoam face $f$. the spin-network nodes $\mathfrak{n}_1,\mathfrak{n}_2$ evolve to the spinfoam edges $e_1,e_2$. The faces evolves from the dashed links are not shown on this figure. The leftmost complex is the root complex. The power of coupling constant $\lambda_f$ counts the number of stacked faces.
  • Figure 3: The intuitive picture of the vertex amplitude $\mathscr{A}_v$ for a valent-5 vertex $v$: $B_+\cup B_-=\partial B_4$ (drawn in purple lines) is the 3d boundary of a neighborhood of the vertex. $B_+$ is in the causal future of $B_-$. $B_+\cap B_-$ (the purple dashed circle) is the corner of the causal diamond (the red dashed diamond). The edges and faces of the spinfoam leaves the nodes and links (in blue) in $\partial B_4$. Here only the root graph is drawn, so each blue link corresponds to arbitrarily number of links for $\mathscr{A}_v$. Some of the links are across the conner. Each nodes corresponds to a closed subregion such that $B_\pm$ are the unions of subregions.
  • Figure 4: (a)$\to$(b): Gluing a pair of faces in two different vertex amplitudes. (c) Generalization to gluing the faces in six vertex amplitudes to form an internal face.

Theorems & Definitions (6)

  • Lemma 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 7.1
  • Lemma 7.2
  • Lemma A.1