The Random Discrete Action for 2-Dimensional Spacetime

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.

20 pages, 10 figures, Typos corrected

References in corpus (3)

• Quantum Gravity Phenomenology, Lorentz Invariance and Discreteness
• The Scalar Curvature of a Causal Set
• Complex actions in two-dimensional topology change

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• A closed form expression for the causal set d'Alembertian
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• Finite Size Scaling in 2d Causal Set Quantum Gravity
• Causal Set Generator and Action Computer
• Inference of Boundaries in Causal Sets
• Singularity at the demise of a black hole
• Manifold Properties from Causal Sets using Chains
• Timelike boundary and corner terms in the causal set action
• Numerical Evaluation of the Causal Set Propagator in 2D Anti-de Sitter Spacetime