# Algebra and Geometry Seminar

Event Type:
Seminar
Speaker:
Roman Sverdlov
Event Date:
Wednesday, September 27, 2017 -
3:00pm to 4:00pm
Location:
SMLC 124
Audience:
General Public
A.Buium

### 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

Journal-ref: Class.Quant.Grav.26:075011,2009 ) we dealt with this issue by selecting a pair of causally-related points, and restricting myself to the region that is defined by the intersection of light cones of those two points. This, however, sets up a preferred frame: in particular, the preferred t-axis is the one passing through those two points. That is why the Lagrangian that we came up with was non-relativistic, unless we did some tricks to make it relativistic -- which is less than one could expect from manifestly relativistic theory. A couple of years later I proposed a different approach (see arXiv:0910.2498): instead of embedding partially ordered set into the spacetime, I decided to embed it into what I call a "phase space time": in particular, an element of phase space time is a point together with timelike velocity vector or, equivalently, phase space time is  a timelike subset of a tangent bundle to Lorentzian manifold. I also re-defined causal relations: two vectors (x,u) and (y,v) are causally related if I can start at x, with velocity u, and reach y, with velocity v, without an accelleration exceeding a certain constant a_{max} and also without the length of the path exceeding a constant \tau_{max}. Thus, each point will have naturally-attached preferred frame to itself (and like I said I need a preferred frame in order to avoid lightcone non-locality). Yet, one might argue that this type of preferred frame doesn't violate relativity: after all, the location of a point, x, doesn't violate translational invariance, so why should its preferred velocity, v, be any "worse" if we view velocity as nothing but just another set of coordinates? In this talk I will discuss both spacetime approach and phase-spacetime approach that I been proposing.

### Event Contact

Contact Name: A. Buium

Contact Email: buium@math.unm.edu