Face amplitude of spinfoam quantum gravity

TL;DR

The paper addresses the longstanding ambiguity of the face amplitude in spinfoam quantum gravity and shows that, given a boundary Hilbert space, a composition law for gluing two-complexes, and a locality constraint on face holonomies, the spinfoam partition function must take a local form $Z_\sigma=\int dU_f^v\ \prod_v A_v(U_f^v)\ \prod_f \delta(U_f^{v_1}\dots U_f^{v_k})$. By inserting the vertex expansion, the face amplitude is fixed to $d_j=\dim(j)$, and for gravity with boundary space ${\cal H}_\Gamma=L_2[SU(2)^L,dU_l]$ this becomes $d_j=2j+1$, eliminating the $SO(4)$-type choice and reducing divergences. The BF theory case is shown to fit within this general framework, illustrating the broad applicability of the result. Overall, the work provides a principled, measure-consistent determination of the face amplitude that respects locality and composition in spinfoam models.

Abstract

The structure of the boundary Hilbert-space and the condition that amplitudes behave appropriately under compositions determine the face amplitude of a spinfoam theory. In quantum gravity the face amplitude turns out to be simpler than originally thought.

Figures (2)

• Figure 1: Schematic definition of the group elements $h_{ve}$, $U^v_f$ and $U_e$ associated to a portion of a face $f$ of the two-complex.
• Figure 2: Cutting of a face of the two-complex. The holonomy $U_l$ is attached to a link of the boundary spin-network and satisfies equation \ref{['locality']}.