Electromagnetic Lagrangian on a causal set that resides on edges rather than points

TL;DR

This paper proposes a novel approach to defining the electromagnetic Lagrangian on a causal set by assigning it to edges rather than points, aiming to address non-locality issues and preserve locality in the theory.

Contribution

It introduces a new method of assigning Lagrangian density to edges in a causal set, focusing on timelike edges to maintain locality and reduce non-locality inherent in point-based approaches.

Findings

Edges are only timelike, simplifying calculations.

Finitely many neighboring edges enable local Lagrangian definition.

The approach preserves Lorentz invariance in the hyperplane perpendicular to edges.

Abstract

The goal of this paper is to introduce one of the versions of the electromagnetic Lagrangian on a causal set in such a way that would address the non-locality issues inherent to causal set theory. The key idea is that Lagrangian density is assigned to the edges rather than points, and there is a way of defining the concept of "neighboring edges" of a given edge in such a way that each edge has only finitely many neighboring edges which would ultimately allow for the theory to be local. That is to be contrasted with points where every point has infinitely many direct neighbors which is a source of non-locality. The edges are needed in order to define electromagnetic Lagrangian anyway, regardless of the consideration of locality; the novelty of this paper is to assign Lagrangian density to the edges as well. Also, in the other papers edges were both spacelike and timelike, while in this paper they are only timelike. This makes calculations considerably more complicated, but it is crucial in preserving locality since the Lorentz group in a hyperplane perpendicular to the edge is compact only if the edge is timelike.