Simplicity and closure constraints in spin foam models of gravity

TL;DR

This work argues that 4D spin foam gravity requires explicit treatment of second-class (secondary) constraints and a holonomy-based augmentation of simplicity constraints, implemented via a covariant discretized measure depending on tetrahedral normals $x_t\in G/H$. The authors derive simplex boundary states as projected spin networks and show that the closure constraint must be relaxed to maintain covariance, aligning spin foam boundary data with covariant loop quantum gravity. A concrete SU(2) BF theory example demonstrates how the vertex amplitude naturally reduces to a projected spin network structure and reproduces the CLQG boundary Hilbert space, validating the approach. Overall, the paper provides a more consistent bridge between spin foam dynamics and covariant canonical quantization by incorporating second-class constraints into the path integral framework and clarifying boundary state spaces.

Abstract

We revise imposition of various constraints in spin foam models of 4-dimensional general relativity. We argue that the usual simplicity constraint must be supplemented by a constraint on holonomies and together they must be inserted explicitly into the discretized path integral. At the same time, the closure constraint must be relaxed so that the new constraint expresses covariance of intertwiners assigned to tetrahedra by spin foam quantization. As a result, the spin foam boundary states are shown to be realized in terms of projected spin networks of the covariant loop approach to quantum gravity.