Finiteness of spinfoam vertex amplitude with timelike polyhedra, and the full amplitude

TL;DR

This paper proves the finiteness of the vertex amplitude in a 4D extended spinfoam model with timelike polyhedra, introducing a quantum cut-off scheme to regularize divergences from summing over intertwiners, ensuring a finite full amplitude.

Contribution

It establishes the absolute convergence of the vertex amplitude with timelike polyhedra and constructs a regularized, finite spinfoam amplitude on arbitrary complexes.

Findings

Vertex amplitude is absolutely convergent with specific boundary states.

A quantum cut-off scheme effectively regularizes divergences.

The full spinfoam amplitude with timelike polyhedra is finite under imposed cut-offs.

Abstract

This work focuses on Conrady-Hnybida's 4-dimensional extended spinfoam model with timelike polyhedra, while we restrict all faces to be spacelike. Firstly, we prove the absolute convergence of the vertex amplitude with timelike polyhedra, when SU(1,1) boundary states are coherent states or the canonical basis, or their finite linear combinations. Secondly, based on the finite vertex amplitude and a proper prescription of the SU(1,1) intertwiner space, we construct the extended spinfoam amplitude on arbitrary cellular complex, taking into account the sum over SU(1,1) intertwiners of internal timelike polyhedra. We observe that the sum over SU(1,1) intertwiners is infinite for the internal timelike polyhedron that has at least 2 future-pointing and 2 past-pointing face-normals. In order to regularize the possible divergence from summing over SU(1,1) intertwiners, we develop a quantum cut-off scheme based on the eigenvalue of the ``shadow operator''. The spinfoam amplitude with timelike internal polyhedra (and spacelike faces) is finite, when 2 types of cut-offs are imposed: one is imposed on $j$ the eigenvalue of area operator, the other is imposed on the eigenvalue of shadow operator for every internal timelike polyhedron that has at least 2 future-pointing and 2 past-pointing face-normals.