Proposed proper Engle-Pereira-Rovelli-Livine vertex amplitude

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 Engle-Pereira-Rovelli-Livine 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 ($\mathrm{II}+$) with positive orientation or sector ($\mathrm{II}-$) 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 semiclassical limit, leading in general to a non-Regge action and hence non-Regge and nongravitational 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 $e^{i{S}_{\mathrm{Regge}}}$. 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 Engle-Pereira-Rovelli-Livine 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 $e^{i{S}_{\mathrm{Regge}}}$, and all degenerate configurations are exponentially suppressed. A natural generalization to the Lorentzian signature is also presented.