Path integral representation of spin foam models of 4d gravity

TL;DR

This paper unifies recent 4D gravity spin foam models by expressing them as path integrals, clarifying their relations, and analyzing boundary states for different Immirzi parameters.

Contribution

It provides a unified path integral formulation for ELPR and FK spin foam models and explores their boundary state structures across various Immirzi parameters.

Findings

FK models are equivalent to path integrals of a discrete theory for all Immirzi values

Explicit formulas for the actions of FK models are derived

Boundary states are characterized for different Immirzi parameters

Abstract

We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira and Rovelli (ELPR) and by Freidel and Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter, equivalent to path integrals of a discrete theory and we provide an explicit formula for the associated actions. We discuss the relation between the FK and ELPR models and also study the corresponding boundary states. For general Immirzi parameter, these are given by Alexandrov's and Livine's SO(4) projected states. For 0 <= gamma < 1, the states can be restricted to SU(2) spin networks.