Improved and perfect actions in discrete gravity

Abstract

We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they -although being defined on a triangulation -reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of General Relativity. We construct the perfect action in three dimensions with cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge Calculus with curved simplices, which arises naturally in this context.

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…With a cosmological constant one would, however, expect a vacuum describing a homogeneously curved geometry. There are a number of approaches to incorporate homogeneously curved geometry into the kinematical set-up of (loop) quantum gravity [18][19][20][21]. An important aspect, making such a construction very attractive [22], is that one expects the Hilbert space associated to a fixed triangulation 1 to be finite dimensional.…”

“…On the classical level this is realized in [19,20]. A different strategy for the quantum theory would be to start from the state space for a small cosmological constant (large k) and to impose curvature constraints with a large cosmological constant (corresponding to a smaller k).…”

“…To restore this symmetry one would have to consider a continuum limit [19,20,[104][105][106], which can be constructed via an auxiliary coarse graining flow [58]. For coarse graining schemes along the lines of [27][28][29][30][31][32][33] the finiteness of the state spaces constructed here facilitates a numerical implementation.…”

Fusion basis for lattice gauge theory and loop quantum gravity

“…In the context of (2 + 1) dimensional gravity models, a particularly intriguing question is how to flow via coarse-graining from models based on flat building blocks to models based curved building blocks, hence recovering in the quantum theory the classical result of [79,80]. More specifically, for spin-foam models one expects a transition from SU(2) to the quantum deformed SU(2) q .…”

“…Hence, we effectively obtain torsion, defined as a violation of the Gauß constraint, due to the presence of curvature. Such an effect, which was named curvature-induced torsion in [14], is strictly related to the need of deforming the Gauß constraint in phase spaces describing piecewise homogeneouslycurved (instead of piecewise flat) geometries [55,[79][80][81][82][83] (see also [84][85][86][87], for an analysis in four dimensions). In terms of defect excitations discussed in this paper, torsion excitations interpreted as spinning particles can arise from the fusion of two spinless defects, since two particles can have orbital angular momentum.…”