138 research outputs found
Analytic Results in 2D Causal Dynamical Triangulations: A Review
We describe the motivation behind the recent formulation of a nonperturbative
path integral for Lorentzian quantum gravity defined through Causal Dynamical
Triangulations (CDT). In the case of two dimensions the model is analytically
solvable, leading to a genuine continuum theory of quantum gravity whose ground
state describes a two-dimensional "universe" completely governed by quantum
fluctuations. One observes that two-dimensional Lorentzian and Euclidean
quantum gravity are distinct. In the second part of the review we address the
question of how to incorporate a sum over space-time topologies in the
gravitational path integral. It is shown that, provided suitable causality
restrictions are imposed on the path integral histories, there exists a
well-defined nonperturbative gravitational path integral including an explicit
sum over topologies in the setting of CDT. A complete analytical solution of
the quantum continuum dynamics is obtained uniquely by means of a double
scaling limit. We show that in the continuum limit there is a finite density of
infinitesimal wormholes. Remarkably, the presence of wormholes leads to a
decrease in the effective cosmological constant, reminiscent of the suppression
mechanism considered by Coleman and others in the context of a Euclidean path
integral formulation of four-dimensional quantum gravity in the continuum. In
the last part of the review universality and certain generalizations of the
original model are discussed, providing additional evidence that CDT define a
genuine continuum theory of two-dimensional Lorentzian quantum gravity.Comment: 66 pages, 17 figures. Based on the author's thesis for the Master of
Science in Theoretical Physics, supervised by R. Loll and co-supervised by J.
Ambjorn, J. Jersak, July 200
Counting entropy in causal set quantum gravity
The finiteness of black hole entropy suggest that spacetime is fundamentally
discrete, and hints at an underlying relationship between geometry and
"information". The foundation of this relationship is yet to be uncovered, but
should manifest itself in a theory of quantum gravity. We review recent
attempts to define a microscopic measure for black hole entropy and for the
maximum entropy of spherically symmetric spacelike regions, within the causal
set approach to quantum gravity.Comment: 5 pages, 1 figure. Talk given by S. Zohren at the Eleventh Marcel
Grossmann Meeting on General Relativity at the Freie U. Berlin, July 23 - 29,
200
A note on weak convergence results for uniform infinite causal triangulations
We discuss uniform infinite causal triangulations and equivalence to the size
biased branching process measure - the critical Galton-Watson branching process
distribution conditioned on non-extinction. Using known results from the theory
of branching processes, this relation is used to prove weak convergence of the
joint length-area process of a uniform infinite causal triangulations to a
limiting diffusion. The diffusion equation enables us to determine the physical
Hamiltonian and Green's function from the Feynman-Kac procedure, providing us
with a mathematical rigorous proof of certain scaling limits of causal
dynamical triangulations.Comment: 23 pages, 2 figure
Nonperturbative sum over topologies in 2D Lorentzian quantum gravity
The recent progress in the Causal Dynamical Triangulations (CDT) approach to
quantum gravity indicates that gravitation is nonperturbatively renormalizable.
We review some of the latest results in 1+1 and 3+1 dimensions with special
emphasis on the 1+1 model. In particular we discuss a nonperturbative
implementation of the sum over topologies in the gravitational path integral in
1+1 dimensions. The dynamics of this model shows that the presence of
infinitesimal wormholes leads to a decrease in the effective cosmological
constant. Similar ideas have been considered in the past by Coleman and others
in the formal setting of 4D Euclidean path integrals. A remarkable property of
the model is that in the continuum limit we obtain a finite space-time density
of microscopic wormholes without assuming fundamental discreteness. This shows
that one can in principle make sense out of a gravitational path integral
including a sum over topologies, provided one imposes suitable kinematical
restrictions on the state-space that preserve large scale causality.Comment: 10 pages, 4 figures. Talk given by S. Zohren at the Albert Einstein
Century International Conference (Paris, July 18-22 2005
A tight Tsirelson inequality for infinitely many outcomes
We present a novel tight bound on the quantum violations of the CGLMP
inequality in the case of infinitely many outcomes. Like in the case of
Tsirelson's inequality the proof of our new inequality does not require any
assumptions on the dimension of the Hilbert space or kinds of operators
involved. However, it is seen that the maximal violation is obtained by the
conjectured best measurements and a pure, but not maximally entangled, state.
We give an approximate state which, in the limit where the number of outcomes
tends to infinity, goes to the optimal state for this setting. This state might
be potentially relevant for experimental verifications of Bell inequalities
through multi-dimenisonal entangled photon pairs.Comment: 5 pages, 2 figures; improved presentation, change in title, as
published
Proper time is stochastic time in 2d quantum gravity
We show that proper time, when defined in the quantum theory of 2d gravity,
becomes identical to the stochastic time associated with the stochastic
quantization of space. This observation was first made by Kawai and
collaborators in the context of 2d Euclidean quantum gravity, but the relation
is even simpler and more transparent in he context of 2d gravity formulated in
the framework of CDT (causal dynamical triangulations).Comment: 30 pages, Talk presented at the meeting "Foundations of Space and
Time", Cape Town, 10-14 August 2009. To appear in the proceedings, CU
A Causal Alternative for c=0 Strings
We review a recently discovered continuum limit for the one-matrix model
which describes "causal" two-dimensional quantum gravity. The behaviour of the
quantum geometry in this limit is different from the quantum geometry of
Euclidean two-dimensional quantum gravity defined by taking the "standard"
continuum limit of the one-matrix model. Geodesic distance and time scale with
canonical dimensions in this new limit, contrary to the situation in Euclidean
two-dimensional quantum gravity. Remarkably, whenever we compare, the known
results of (generalized) causal dynamical triangulations are reproduced exactly
by the one-matrix model. We complement previous results by giving a geometrical
interpretation of the new model in terms of a generalization of the loop
equation of Euclidean dynamical triangulations. In addition, we discuss the
time evolution of the quantum geometry.Comment: 10 pages, 4 figures, Presented at "The 48th Cracow School of
Theoretical Physics: Aspects of Duality", June 13-22, 2008, Zakopane, Polan
- …