On the Non-Existence of Unbounded Discrete Space-Time

TL;DR

This paper investigates scalar fields on n-scales, concluding that unbounded discrete space-time cannot support massive scalar fields, implying such space-times must be bounded in each dimension.

Contribution

It introduces a modified field equation for scalar fields on n-scales and demonstrates the non-existence of massive scalar fields in unbounded discrete space-time.

Findings

Massive scalar fields cannot exist in unbounded discrete space-time.

Discrete space-time, if it exists, must be bounded in each dimension.

Provides mode solutions for scalar fields on n-scales.

Abstract

$n$-scales are a generalization of time-scales that has been put forward to unify continuous and discrete analyses in higher dimensions. In this paper we investigate massive scalar field theory on a regular $n$-Scale. We have given the modified field equation that is appropriate to this space-time structure and gave the mode solutions. If the space-time is discrete we have found that no massive scalar field can exist. Hence, we concluded that unbounded discrete space-time cannot exist. If discrete space-time exists, it has to be bounded in each dimension.