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

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.

51 pages, 18 figures

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Cited by in corpus (11)

• From the discrete to the continuous - towards a cylindrically consistent dynamics
• Holonomy Spin Foam Models: Definition and Coarse Graining
• Hypercuboidal renormalization in spin foam quantum gravity
• Holonomy spin foam models: Asymptotic geometry of the partition function
• Quantum group spin nets: refinement limit and relation to spin foams
• Renormalization of symmetry restricted spin foam models with curvature in the asymptotic regime
• Linking covariant and canonical LQG II: Spin foam projector
• On the role of the Barbero-Immirzi parameter in discrete quantum gravity
• Bubble divergences and gauge symmetries in spin foams
• Topological lattice field theories from intertwiner dynamics
• Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models