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

Han M, Kaminski W, Liu H (2022)

Publication Type: Journal article

Publication year: 2022

Journal

Physical Review D American Physical Society

Book Volume: 105

Journal Issue: 8

DOI: 10.1103/PhysRevD.105.084034

Abstract

This work focuses on Conrady-Hnybida's four-dimensional extended spinfoam model with timelike polyhedra, while we restrict all faces to be spacelike. First, 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. Second, based on the finite vertex amplitude and a proper prescription of the SU(1, 1) intertwiner space, we construct the extended spinfoam amplitude on an 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 two future-pointing and two past-pointing face normals. In order to regularize the possible divergence from summing over SU(1, 1) intertwiners, we develop a quantum cutoff scheme based on the eigenvalue of the "shadow operator". The spinfoam amplitude with timelike internal polyhedra (and spacelike faces) is finite, when two types of cutoffs are imposed. One is imposed on j the eigenvalue of area operator, and the other is imposed on the eigenvalue of the shadow operator for every internal timelike polyhedron that has at least two future-pointing and two past-pointing face normals.

Authors with CRIS profile

Hongguang Liu Chair for Theoretical Physics III (Quantum Gravity)

Involved external institutions

Florida Atlantic University US United States (USA) (US) University of Warsaw / Uniwersytet Warszawski PL Poland (PL)

How to cite

APA:

Han, M., Kaminski, W., & Liu, H. (2022). Finiteness of spinfoam vertex amplitude with timelike polyhedra and the regularization of full amplitude. Physical Review D, 105(8). https://dx.doi.org/10.1103/PhysRevD.105.084034

MLA:

Han, Muxin, Wojciech Kaminski, and Hongguang Liu. "Finiteness of spinfoam vertex amplitude with timelike polyhedra and the regularization of full amplitude." Physical Review D 105.8 (2022).

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