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

TL;DR

This paper introduces a new framework called spinfoam stack to analyze Lorentzian spinfoam amplitudes, demonstrating triangulation independence and reduction to topological SU(2) BF theory in a specific limit, advancing understanding of quantum gravity.

Contribution

It proposes the spinfoam stack framework and proves triangulation independence of amplitudes in the infinite limit, connecting spinfoam models to topological quantum field theories.

Findings

Amplitude localizes onto SU(2) flat connections

Amplitude factorizes into boundary-dependent and normalization factors

Triangulation independence achieved in the infinite limit

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.