A proposed proper EPRL vertex amplitude

TL;DR

The paper introduces a modified EPRL vertex amplitude that isolates the gravitational sector, ensuring correct semiclassical behavior and suppressing non-gravitational configurations in spin-foam models.

Contribution

It proposes the proper EPRL vertex amplitude with asymptotics containing only the Regge action term, improving the semiclassical limit of spin-foam models.

Findings

Asymptotics include only the $e^{iS_{Regge}}$ term.

Degenerate configurations are exponentially suppressed.

The vertex maintains SU(2) gauge invariance and boundary data dependence.

Abstract

As established in a prior work of the author, the linear simplicity constraints used in the construction of the so-called `new' spin-foam models mix three of the five sectors of Plebanski theory as well as two dynamical orientations, and this is the reason for multiple terms in the asymptotics of the EPRL vertex amplitude as calculated by Barrett et al. Specifically, the term equal to the usual exponential of $i$ times the Regge action corresponds to configurations either in sector (II+) with positive orientation or sector (II-) with negative orientation. The presence of the other terms beyond this cause problems in the semiclassical limit of the spin-foam model when considering multiple 4-simplices due to the fact that the different terms for different 4-simplices mix in the semi-classical limit, leading in general to a non-Regge action and hence non-Regge and non-gravitational configurations persisting in the semiclassical limit. To correct this problem, we propose to modify the vertex so its asymptotics include only the one term of the form $e^{iS_{\Regge}}$. To do this, an explicit classical discrete condition is derived that isolates the desired gravitational sector corresponding to this one term. This condition is quantized and used to modify the vertex amplitude, yielding what we call the `proper EPRL vertex amplitude.' This vertex still depends only on standard SU(2) spin-network data on the boundary, is SU(2) gauge-invariant, and is linear in the boundary state, as required. In addition, the asymptotics now consist in the single desired term of the form $e^{iS_{\Regge}}$, and all degenerate configurations are exponentially suppressed. A natural generalization to the Lorentzian signature is also presented.