Dimensionally restricted causal set quantum gravity: examples in two and three dimensions

Abstract

We study dimensionally restricted non-perturbative causal set quantum dynamics in 2 and 3 spacetime dimensions with non-trivial global spatial topology. The causal set sample space is generated from causal embeddings into latticisations of flat background spacetimes with global spatial topology S 1 and T 2 in 2 and 3 dimensions, respectively. The quantum gravity partition function over these sample spaces is studied using Markov Chain Monte Carlo (MCMC) simulations via an analytic continuation of a parameter β analogous to an inverse temperature. In both 2 and 3 dimensions we find a phase transition that separates the dominance of the action from that of the entropy. The action dominated phase is characterised by "layered" posets with a high degree of connectivity, while the causal sets in the entropy dominated phase are manifold-like. These results are similar in character to those obtained for topologically trivial causal set dynamics over the sample space of 2-orders. The current simulations use a newly developed framework for causal set MCMC calculations, and provide the first implementation of a 3 dimensional causal set dynamics.

Critical Reflections on Asymptotically Safe Gravity

“…In particular, in spin foams, a search for interacting fixed points in numerical simulations has started recently in reduced configuration spaces, see, e.g., [267][268][269]. In causal sets, investigating the phase diagram and the order of potential phase transitions has only shifted into focus more recently, with indications for first-order phase transitions in restricted configuration spaces for lower-dimensional causal set quantum gravity [270][271][272][273].…”

“…This motivates the search for a universal continuum limit, linked to a second-order phase transition in the phase diagram for causal sets. Monte Carlo studies of the phase diagram for restricted configuration spaces in low dimensionalities can be found in [270][271][272][273].…”

Boundary contributions in the causal set action

“…One is to construct a quantal version of the classical sequential growth models for causal sets [1][2][3][4] in which causal sets grow in a process of accretion of new elements. In contrast to this process approach, the other strategy is to construct what might be called state sum models where a sum over causal sets -usually of a fixed cardinality -is defined using a weight for each causal set given by the exponential of (i times) an action for the causal set [5][6][7][8].…”

Ollivier curvature of random geometric graphs converges to Ricci curvature of their Riemannian manifolds

“…In order to compare the two different Wasserstein metrics in (7) we extend the graph distance d G to a distance d M defined on a sufficiently large part of M. In particular, we will consider the ball B M (x * ; Qδ n ), with Q > 3 from Definition 4.1. The extension is such that for any two nodes…”

“…Here one wants to find a discrete geometry that converges in the continuum limit to the geometry of physical spacetime. To this end, Ollivier-Ricci curvature and its variations have been extensively investigated recently [18,40,7].…”