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

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.