Recurrence relations for spin foam vertices

TL;DR

This paper derives recurrence relations for spin foam vertex symbols, linking topological invariance, classical constraints, and geometric properties, with applications to boundary states and correlation functions in quantum gravity models.

Contribution

It introduces new recurrence relations for Wigner 3nj-symbols, connecting topological invariance, classical constraints, and geometric interpretations in spin foam models.

Findings

Recurrence relations derived from Pachner move invariance.

Relations applicable to any SU(2) invariant symbol.

Extension to boundary states resembling Ward identities.

Abstract

We study recurrence relations for various Wigner 3nj-symbols and the non-topological 10j-symbol. For the 6j-symbol and the 15j-symbols which correspond to basic amplitudes of 3d and 4d topological spin foam models, recurrence relations are obtained from the invariance under Pachner moves and can be interpreted as quantizations of the constraints of the underlying classical field theories. We also derive recurrences from the action of holonomy operators on spin network functionals, making a more precise link between the topological Pachner moves and the classical constraints. Interestingly, our recurrence relations apply to any SU(2) invariant symbol, depending on the cycles of the corresponding spin network graph. Another method is used for non-topological objects such as the 10j-symbol and pseudo-isoceles 6j-symbols. The recurrence relations are also interpreted in terms of elementary geometric properties. Finally, we discuss the extension of the recurrences to take into account boundary states which leads to equations similar to Ward identities for correlation functions in the Barrett-Crane model.