Simplicity in simplicial phase space

TL;DR

This paper explores how simplicity constraints are implemented in a phase space framework for simplicial geometries in spin foam quantum gravity, clarifying their interpretation and relation to boundary Hilbert spaces.

Contribution

It provides a detailed analysis of gauge-variant and gauge-invariant simplicity constraints and their effects on gauge-invariant variables, linking them to known simplicity conditions.

Findings

Gauge-variant simplicity constraints resemble Ashtekar reality conditions.

Gauge-invariant constraints correspond to diagonal, cross, and edge simplicity.

Secondary constraints lead to gluing conditions in simplicial geometries.

Abstract

A key point in the spin foam approach to quantum gravity is the implementation of simplicity constraints in the partition functions of the models. Here, we discuss the imposition of these constraints in a phase space setting corresponding to simplicial geometries. On the one hand, this could serve as a starting point for a derivation of spin foam models by canonical quantisation. On the other, it elucidates the interpretation of the boundary Hilbert space that arises in spin foam models. More precisely, we discuss different versions of the simplicity constraints, namely gauge-variant and gauge-invariant versions. In the gauge-variant version, the primary and secondary simplicity constraints take a similar form to the reality conditions known already in the context of (complex) Ashtekar variables. Subsequently, we describe the effect of these primary and secondary simplicity constraints on gauge-invariant variables. This allows us to illustrate their equivalence to the so-called diagonal, cross and edge simplicity constraints, which are the gauge-invariant versions of the simplicity constraints. In particular, we clarify how the so-called gluing conditions arise from the secondary simplicity constraints. Finally, we discuss the significance of degenerate configurations, and the ramifications of our work in a broader setting.