International Research Fellowship Program: Introduction of Field Theory into the Causal Set Context

Sverdlov, Roman M, Ann Arbor MI

Investigators

Abstract

0853079 Sverdlov This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twenty-four-month research fellowship by Dr. Roman M. Sverdlov to work with Dr. Sumati Surya at the Raman Research Institute in Bangalore, India. Causal Sets, originally proposed by Rafael Sorkin, is one model of discrete space time intended to develop quantum theory of gravity. According to this model, the space time we live in can be described (both macroscopically and microscopically) as a locally finite set of points and their causal relations. On a macroscopical level, a causal relation between the two points is an answer to the question as to whether or not it is possible to travel between them without going faster than the speed of light. It has been shown by Hawking that if the volumes of regions are specified which, in discrete setting, can be done by assuming each point takes up the same volume, one can deduce metric from the causal structure. Thus, causal structure can be viewed as a generalization of a metric for a microscopic setting, where manifold-like geometry breaks down due to quantum fluctuations. One unexplored area of the theory, however, is to show how the approximate manifold structure is restored on a larger scale. Scientists focus on addressing topological questions in a toy model of absolute vacuum, and have postponed introducing particles until topology is under control (although there are few exceptions, such as the work of Johnston and Jacobsen). On the first glance, this makes sense: geometry is needed in order for particles to propagate. This, however, is a point in which the PI?s research deviates from the norm substantially. Since geometry is identified with gravity, which is a field, it is not possible to have geometry literally without matter. This also means that the manifold-like topology on a large scale might well be a consequence of the dynamics matter is subjected to (in particular, gravitational field). Thus, he proposes to reverse the steps of the program: first introduce fields and Lagrangians in non-manifoldlike causal set, and later explore the large scale geometry once the non-manifoldlike dynamics is defined. However, in light of the fact that the prediction for a manifold-like scenario is the ultimate verification of a theory, he often considers ?special cases? where manifold structure is being put by hand; typically, these cases involve Poisson distribution of points on Lorentzian manifold. In his dissertation he proposed a way to define the known fields and their Lagrangians for a general causal set. However, the next step of going from Lagrangian to propagator is very problematic, due to the fact that the number of degrees of freedom of path integration is equal to the number of points (or even worse, pairs of points) of the entire universe. In regular quantum field theory this is done by imagining a regular cubic lattice, in which case it is possible to perform different integrals ?all at once?. However, due to the fact that the difference between the edges of the lattice and its diagonals violate relativity, such structure cannot be used, so in order to obtain some results one has to think of another method. This can be somewhat justified by analogy with Einstein?s equation, which does not have an exact solutions either. However, in order for this analogy to work, there has to be some simple cases that do have exact solution. The goal of his project is to find such cases. One direction of research is a toy model of a causal set consisting of two points A and B on a manifold singled out beforehand, together with random sprinkling of n other points. Then causal set based propagator is computed between points A and B, where all geometry is ignored, except for causal relations between relevant points. That result is compared to the propagator between A and B computed by ordinary methods of quantum field theory, where selection of n points is ignored. The above work can realistically be done by numeric simulations of toy models of causal sets consisting of very few points (10 points, 100 points, etc). However, he hopes to analytically show that for very large number of points the causal set based propagator satisfies some form of differential equation, which would allow the application of the theory to many point scenario.