Holonomy spin foam models: boundary Hilbert spaces and time evolution operators

Abstract

In this and the companion paper, a novel holonomy formulation of the so-called spin foam models of lattice gauge gravity is 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 us 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.