Holonomy Spin Foam Models: Boundary Hilbert spaces and Time Evolution Operators

TL;DR

This paper introduces a holonomy-based formulation of Spin Foam models for lattice gauge gravity, defining a universal boundary Hilbert space and analyzing the structure of time evolution operators influenced by simplicity constraints.

Contribution

It develops a novel holonomy formulation, constructs a universal boundary Hilbert space, and derives general transfer operators for Spin Foam models, clarifying the role of simplicity constraints.

Findings

Defined a natural basis for simplicity constraints

Mapped boundary Hilbert spaces to spin network spaces

Derived general form of transfer operators for time evolution

Abstract

In this and the companion paper a novel holonomy formulation of so called Spin Foam models of lattice gauge gravity are explored. After giving a natural basis for the space of simplicity constraints we define a universal boundary Hilbert space, on which the imposition of different forms of the simplicity constraints can be studied. We detail under which conditions this Hilbert space can be mapped to a Hilbert space of projected spin networks or an ordinary spin network space. These considerations allow to derive the general form of the transfer operators which generates discrete time evolution. We will describe the transfer operators for some current models on the different boundary Hilbert spaces and highlight the role of the simplicity constraints determining the concrete form of the time evolution operators.