The degrees of freedom of area Regge calculus: dynamics, non-metricity, and broken diffeomorphisms

Abstract

Discretization of general relativity is a promising route towards quantum gravity. Discrete geometries have a finite number of degrees of freedom and can mimic aspects of quantum geometry. However, selection of the correct discrete freedoms and description of their dynamics has remained a challenging problem. We explore classical area Regge calculus, an alternative to standard Regge calculus where instead of lengths, the areas of a simplicial discretization are fundamental. There are a number of surprises: though the equations of motion impose flatness we show that diffeomorphism symmetry is broken for a large class of area Regge geometries. This is due to degrees of freedom not available in the length calculus. In particular, an area discretization only imposes that the areas of glued simplicial faces agrees; their shapes need not be the same. We enumerate and characterize these non-metric, or 'twisted', degrees of freedom and provide tools for understanding their dynamics. The non-metric degrees of freedom also lead to fewer invariances of the area Regge action-in comparison to the length action-under local changes of the triangulation (Pachner moves). This means that invariance properties can be used to classify the dynamics of spin foam models. Our results lay a promising foundation for understanding the dynamics of the non-metric degrees of freedom in loop quantum gravity and spin foams. * Electronic address: sasante@perimeterinstitute.ca † Electronic address: bdittrich@perimeterinstitute.ca ‡ Electronic address: haggard@bard.edu arXiv:1802.09551v1 [gr-qc]

Renormalization of symmetry restricted spin foam models with curvature in the asymptotic regime

“…Previous investigations [17,18] only allowed for quasilocal fluctuation of the metric which are, in the semiclassical limit, expected to turn to gauge degrees of freedom. It can be expected that these appear in the theory as spurious degrees of freedom, since it is well-known that the gauge symmetry of GR is broken in the EPRL model [45,73,77].…”

“…In particular, in [45], it was observed that the EPRL model breaks vertex displacement symmetry, which is the manifestation of diffeomorphisms on the lattice [45,[73][74][75][76][77]. While this breaking of symmetry is well-known in classical Regge calculus, where it appears whenever curvature is involved, the quantum theory breaks it even in the case of flat metrics.…”

Holographic description of boundary gravitons in (3+1) dimensions

“…To this end, one uses versions of Regge calculus based on other sets of variables than the edge lengths, e.g. areas and angles [41,45,69,81]. Another choice, possibly more suited for Lorentzian signature, would be variables related to spinors or twistors [82,83].…”

“…Turning on curvature, for instance by changing the boundary data, these eigenvalues will no longer vanish and will scale with the amount of curvature, as determined by the deficit angles [37,38]. A similar effect arises in the presence of torsion [45].…”

Cosmological Constant from Condensation of Defect Excitations

“…The states are peaked on homogeneous geometries. These geometries are however generalized in the same way as the standard loop quantum gravity geometries are, if understood in terms of piecewise flat simplicial building blocks [76][77][78][79][80].…”