Combinatorial quantum gravity: geometry from random bits
HJE
Combinatorial quantum gravity: geometry from random bits
Carlo A. Trugenberger 0
0 chemin Diodati 10 , CH-1223 Cologny , Switzerland
I propose a quantum gravity model in which geometric space emerges from random bits in a quantum phase transition driven by the combinatorial Ollivier-Ricci curvature and corresponding to the condensation of short cycles in random graphs. This quantum critical point de nes quantum gravity non-perturbatively. In the ordered geometric phase at large distances the action reduces to the standard Einstein-Hilbert term.
Models of Quantum Gravity; Random Systems
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The Einstein equations of general relativity are the Euler-Lagrange equations of the
Einstein-Hilbert action. In a quantum treatment, the Einstein-Hilbert action is
perturbatively non-renormalizable. Within traditional quantum
eld theory the main approaches
to resolve this conundrum are to postulate new physics at short scales and try to
embed general relativity in a larger model, the string theory approach [1], or to look for a
non-Gaussian ultraviolet (UV) xed point that de nes the theory non-perturbatively, the
asymptotic safety approach [2]. This is perhaps best exempli ed by the causal dynamical
triangulations (CDT) program [
3
], the gravity equivalent of lattice gauge theories, in which
space-time is discretized in terms of (causal) simplicial complexes and the Einstein-Hilbert
action formulated by Regge calculus [4, 5].
In this paper I propose a di erent approach to quantum gravity and formulate a
proofof-concept toy model to show how this approach works. The idea is not to postulate
spacetime ab initio, but rather to consider it as an emergent property of purely combinatorial
fundamental degrees of freedom. In this spirit, quantum gravity would be a close cousin of
the Ising model: at short scales, physics is de ned by an UV
xed point for fundamental
constituents that are just random bits, the links of random graphs [6]; at large distances, the
interaction is weaker and long-range order emerges in form of random geometric graphs [7],
which are random graphs equipped with a metric and de ne a discretization of a manifold.
In particular, space and geometry are expected to emerge in the infrared (IR) limit due
to the condensation of short graph cycles, the number of triangles being, e.g. one of the
distinguishing features of random graphs vs. random geometric graphs [8, 9]. As a driver of
the quantum phase transition I will consider the combinatorial Ollivier-Ricci curvature [10{
15], which becomes the standard Ricci curvature scalar in the ordered phase. Note that
this program is totally di erent from previous approaches to quantum gravity based on
graph structures [16, 17]: there is no need of auxiliary group variables and the action is a
purely combinatorial version of the Einstein-Hilbert action.
To show how this idea works concretely I will consider here a simpli ed toy model in
which the con guration space CS is restricted to diluted random regular bipartite graphs.
Bipartite graphs have no odd cycles, the smallest, \elementary" loops being thus 4-cycles,
squares. Two di erent squares on a graph can share zero, one or two edges (if they share
three edges they must share also the fourth. i.e. they would be identical). By \diluted"
graphs I mean graphs in which two di erent elementary squares can share maximally one
edge. This is a loop-equivalent of a hard core requirement in a classical gas: the elementary
constituent can touch but not overlap.
Note that this restriction of the con guration
space has nothing to do with any fundamental requirement, it is just a mathematical
simpli cation that makes the toy model easily tractable.
The partition function of the model is then de ned by
Z =
X exp ( SEH=~) ;
CS
1
2g
SEH =
Tr w4 ;
(1)
where w denotes the adjacency matrices of the graphs in CS, ~ is the Planck constant and
g is the gravity coupling constant with dimension 1/action. The dimensionless quantity ~g
{ 1 {
will play the role of e ective \temperature" in this statistical eld theory model. From now
on I will focus on even connectivities of the regular graphs and denote these by k = 2d.
Random regular bipartite graphs are \small worlds", i.e. their diameter and
average distances on the graphs scale logarithmically with the number N of vertices (the
volume) [18, 19]. They have locally a tree structure with very sparse short cycles governed by
a Poisson distribution [18, 19] with mean (2d
1)l=l for cycles of length l. This behaviour
is clearly not what is expected of a geometric space. As I now show, however,
geometry emerges when the dimensionless coupling ~g is small, the crucial point being that the
interaction is nothing else than the discrete curvature scalar for graphs.
Random graphs are very di erent form simplicial complexes, which are regular con
gHJEP09(217)45
urations that can be always associated to a geometric realization. Because of their random
character, the Regge formulation o (...truncated)