# Algebra and Geometry Seminar

### Event Description:

Title: Causal set as a discretization of regular space-time and phase-space-time

Abstract: Causal set theory, originally introduced by Rafael Sorkin, is a discretization of spacetime. It is assumed that our spacetime is a partially ordered set: that is, an imaginary classical particle can travel from point a to point b without going faster than the speed of light if and only if a<b. At the same time, on very small scales, the geometry might break down due to quantum fluctuations of gravity: thus, the notion of "traveling with a certain speed" is no longer well defined; nevertheless, partial ordering continues to be well defined even then. Based on this motivation, Sorkin proposed a project of trying to rewrite key ingredients of physics in such a way that they are well defined for general partially ordered set and, at the same time, will return the physics as we know of in the special case of generating partial ordering through embedding the causal set into a manifold (which, let me stress, is only a special case -- but an important special case). Now, the embedding of causal set is simply some sort of choice of discretization of spacetime. While a number of discretization theories are based on some sort of lattice, that is not what Sorkin prefers since lattice has preferred directions determined by its edges, while one of the aims of causal set theory is manifest Lorentz invariance since its only key ingredient -- causal relation -- is manifestly invariant. For that reason, Sorkin proposed to discretize spacetime through Poisson distribution instad of lattice. But this leads to a problem: if we are dealing with discrete space, then the direct neighbors of any given point will roughly concentrate around a distance-neighborhood of some finite radius. If the neighborhood is based on Eucledian distance then its volume is finite, and, therefore, the number of direct neighbors is finite as well. But if the neighborhood is based on Lorentzian distance, which is the case in Sorkin's theory, then its volume is infinite and therefore each point has infinitely many neighbors which are arbitrarily far from a given point coordinate-wise. In my 2008 papers (see for example my first paper which I wrote in collaboration with Luca Bombelli -- arXiv:0801.0240