Towards a Discrete Spacetime

TL;DR

This paper introduces a formalism for creating a discrete spacetime with an inherent length scale, utilizing symmetry breaking and higher derivative gravity to break translation symmetry.

Contribution

It proposes a novel approach combining Landau theory and higher derivative Einstein actions to generate a discrete spacetime structure.

Findings

A formalism for discrete spacetime with a length scale is developed.

Spontaneous symmetry breaking at non-zero momentum is used to break translation symmetry.

Higher derivative terms are essential for the proposed formalism.

Abstract

A formalism is proposed to generate (the first step of) a discrete spacetime: spacetime with an inbuilt length scale. We follow the celebrated Landau theory of liquid - solid phase transition induced by Spontaneous Symmetry Breaking by a condensate whose Fourier transform has support at a {\it{non-zero}} momentum. The latter requirement is essential for breaking the translation symmetry. This, in turn, compels us to generalize Einstein action to higher derivative terms.