Simplicial quantum gravity with higher derivative terms: Formalism and numerical results in four dimensions

Abstract

Higher derivative terms for Regge's formulation of latti~ gravity are discussed. The analytic weak-field expansion for the regular tessellation a s of the four-sphere is presented. Preliminary numerical results for some computations in four dimensions are also discussed.

The perturbative Regge-calculus regime of loop quantum gravity

“…By dimensional analysis we have that from a 0 can come only terms independent of G N , from a 1 only terms which scale as 8πG N and similarly for higher orders. This suggests that the contribution to the perturbative action coming from the measure µ l0 (δl mn ) can be written in terms of an higher-curvature Regge action [99,100], with specific coefficients for the higher-curvature terms fixed by the non-perturbative theory. While suggestive, this conjecture needs a detailed analysis 20 of the expansion in k mn of the a n (ū(k mn )) to be confirmed or rejected.…”

Introducing quantum Ricci curvature

“…Its integrated version appears in a simplicial analogue of the Einstein-Hilbert action, which is also used in quantum Regge calculus and Dynamical Triangulations [10]. In the context of Regge calculus, related simplicial representations have been constructed for more complicated curvature tensors (see, for example, [11,12,13]).…”

“…Recall that geodesics on S 2 are arcs of great circles and that the geodesic distance between two points (θ i , ϕ i ), i = 1, 2, is given by d((θ 1 , ϕ 1 ), (θ 2 , ϕ 2 )) = ρ arccos(cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos(ϕ 2 − ϕ 1 )). (11) The sphere distance of two -circles whose centres are a distance δ apart is given (12). Right: average sphere distance (14).…”

Conformal variations and quantum fluctuations in discrete gravity

“…Regge discretizations can be used directly to construct a lattice regularized path integral approach known as simplicial quantum gravity and introduced in the 1980s [26,[10][11][12][18][19][20]. Here we adopt the formalism and the notations as established in Hamber's book [13].…”

“…We refer to the references listed above for a comprehensive account of solved and still open questions regarding this approach (e.g. based on the addition of higher-curvature extra terms, see [10][11][12]). To our knowledge the approach of conformal variations in the PL context reported in Sec.…”