The Random Discrete Action for 2-Dimensional Spacetime

TL;DR

This paper introduces a Discrete Action for 2D Lorentzian spacetime, analyzing its expectation value and topological properties, and demonstrating its behavior in different spacetime regions including flat and singular geometries.

Contribution

It defines a new Discrete Action parameterized by a scale, calculates its expectation in various regions, and explores its topological properties and limitations.

Findings

Mean Discrete Action matches null tiling topology in Minkowski regions

Predicted zero mean for flat Lorentzian cylinder verified

Topological character breaks down near singularities and non-causal regions

Abstract

A one-parameter family of random variables, called the Discrete Action, is defined for a 2-dimensional Lorentzian spacetime of finite volume. The single parameter is a discreteness scale. The expectation value of this Discrete Action is calculated for various regions of 2D Minkowski spacetime. When a causally convex region of 2D Minkowski spacetime is divided into subregions using null lines the mean of the Discrete Action is equal to the alternating sum of the numbers of vertices, edges and faces of the null tiling, up to corrections that tend to zero as the discreteness scale is taken to zero. This result is used to predict that the mean of the Discrete Action of the flat Lorentzian cylinder is zero up to corrections, which is verified. The ``topological'' character of the Discrete Action breaks down for causally convex regions of the flat trousers spacetime that contain the singularity and for non-causally convex rectangles.