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

Sum over topologies and double-scaling limit in 2D Lorentzian quantum gravity

We construct a combined non-perturbative path integral over geometries and topologies for two-dimensional Lorentzian quantum gravity. The Lorentzian structure is used in an essential way to exclude geometries with unacceptably large causality violations. The remaining sum can be performed analytically and possesses a unique and well-defined double-scaling limit, a property which has eluded similar models of Euclidean quantum gravity in the past.Comment: 9 pages, 3 Postscript figures; added comments on strip versus bulk partition functio

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

Topology change in causal quantum gravity

The role of topology change in a fundamental theory of quantum gravity is still a matter of debate. However, when regarding string theory as two-dimensional quantum gravity, topological fluctuations are essential. Here we present a third quantization of two-dimensional surfaces based on the method of causal dynamical triangulation (CDT). Formally, our construction is similar to the c = 0 non-critical string field theory developed by Ishibashi, Kawai and others, but physically it is quite distinct. Unlike in non-critical string theory the topology change of spatial slices is well controlled and regulated by Newton's constant.Comment: 4 pages, proceedings of the workshop JGRG 17 (Nagoya, Japan, December 2007

New aspects of two-dimensional quantum gravity

Causal dynamical triangulations (CDT) can be used as a regularization of quantum gravity. In two dimensions the theory can be solved anlytically, even before the cut-off is removed and one can study in detail how to take the continuum limit. We show how the CDT theory is related to Euclidean 2d quantum gravity (Liouville quantum gravity), how it can be generalized and how this generalized CDT model has a string field theory representation as well as a matrix model representationof a new kind, and finally how it examplifies the possibility that time in quantum gravity might be the stochastic time related to the branching of space into baby universes.Comment: Lectures presented at the 49th Cracow School of Theoretical Physics, "Non-Perturbative Gravity and Quantum Chromodynamics", Zakopane May 31-June 10, 2009. To appear in Acta Physica Polonica B 40 (2009) 1001-103

Shocks in economic growth=shocking effects for food security?

The recent economic and financial turmoil raises the question on how global economic growth affects agricultural commodity markets and, hence, food security. To address this question, this paper assesses the potential impacts of faster economic growth in developed and emerging economies on the one hand and a replication of the recent economic downturn on the other hand. The empirical analysis uses AGLINK-COSIMO, a recursive-dynamic, partial equilibrium, supply–demand model. Simulation results demonstrate that higher economic growth influences demand more than supply, resulting in higher world market prices for agricultural commodities. Emerging economies tend to import more and to stock less in order to cover their demand needs, while the rest of the world increases its exports. The modelled faster economic growth also helps developing countries to improve their trade balance, but does not necessarily give them the incentive to address domestic food security concerns by boosting domestic consumption. A replication of an economic downturn leads to lower world prices, and while the magnitude of the effects decreases over time, markets do not regain their baseline levels within a 5-year period. Due to the lower world market prices, developing countries import more and increase their per capita food calorie intake. However, as developing countries become more import dependent, this also implies that they become more vulnerable to disruptions in agricultural world markets

Taming the cosmological constant in 2D causal quantum gravity with topology change

As shown in previous work, there is a well-defined nonperturbative gravitational path integral including an explicit sum over topologies in the setting of Causal Dynamical Triangulations in two dimensions. In this paper we derive a complete analytical solution of the quantum continuum dynamics of this model, obtained uniquely by means of a double-scaling limit. We show that the presence of infinitesimal wormholes leads to a decrease in the effective cosmological constant, reminiscent of the suppression mechanism considered by Coleman and others in the four-dimensional Euclidean path integral. Remarkably, in the continuum limit we obtain a finite spacetime density of microscopic wormholes without assuming fundamental discreteness. This shows that one can in principle make sense of a gravitational path integral which includes a sum over topologies, provided suitable causality restrictions are imposed on the path integral histories.Comment: 19 pages, 4 figures. Comments on general covariance added. To be published in Nucl. Phys.

Causal random geometry from stochastic quantization

In this short note we review a recently found formulation of two-dimensional causal quantum gravity defined through Causal Dynamical Triangulations and stochastic quantization. This procedure enables one to extract the nonperturbative quantum Hamiltonian of the random surface model including the sum over topologies. Interestingly, the generally fictitious stochastic time corresponds to proper time on the geometries.Comment: 5 pages, 2 figures, presented at XI Latin American Workshop on Nonlinear Phenomena, Buzios, 2009, accepted for publication in Journal of Physics: Conference Proceeding

A Matrix Model for 2D Quantum Gravity defined by Causal Dynamical Triangulations

A novel continuum theory of two-dimensional quantum gravity, based on a version of Causal Dynamical Triangulations which incorporates topology change, has recently been formulated as a genuine string field theory in zero-dimensional target space (arXiv:0802.0719). Here we show that the Dyson-Schwinger equations of this string field theory are reproduced by a cubic matrix model. This matrix model also appears in the so-called Dijkgraaf-Vafa correspondence if the superpotential there is required to be renormalizable. In the spirit of this model, as well as the original large-N expansion by 't Hooft, we need no special double-scaling limit involving a fine tuning of coupling constants to obtain the continuum quantum-gravitational theory. Our result also implies a matrix model representation of the original, strictly causal quantum gravity model.Comment: 13 pages, 1 figur