Spin foams with timelike surfaces

TL;DR

This paper extends 4d gravity spin foam models to include timelike surfaces, expressing the partition function via vertex amplitudes and coherent states, and explores methods to impose quantum simplicity constraints.

Contribution

It introduces a new type of coherent state for timelike surfaces and derives the associated completeness relation, advancing the spin foam formalism for 4d gravity.

Findings

Coherent states characterized by normals in 2-sphere or hyperboloids.

Derived the completeness relation for timelike surface states.

Demonstrated three methods to impose quantum simplicity constraints.

Abstract

Spin foams of 4d gravity were recently extended from complexes with purely spacelike surfaces to complexes that also contain timelike surfaces. In this article, we express the associated partition function in terms of vertex amplitudes and integrals over coherent states. The coherent states are characterized by unit 3--vectors which represent normals to surfaces and lie either in the 2--sphere or the 2d hyperboloids. In the case of timelike surfaces, a new type of coherent state is used and the associated completeness relation is derived. It is also shown that the quantum simplicity constraints can be deduced by three different methods: by weak imposition of the constraints, by restriction of coherent state bases and by the master constraint.