Constraints on the Dimensionality of Space

TL;DR

This paper derives constraints on the number of spatial dimensions necessary for bound states to exist, showing that higher dimensions can support binding under certain conditions, challenging the common assumption of three-dimensional space.

Contribution

It provides a general derivation of dimensional constraints for bound states and explores conditions under which higher-dimensional spaces can support binding.

Findings

Bound states are generally restricted to three dimensions.

Higher dimensions can support bound states under specific force conditions.

The paper offers a framework to analyze the dimensionality constraints for various potentials.

Abstract

Complex structures can only form in a universe that allows for bound states. While this is clearly observed in three-dimensions, added degrees of freedom in a higher-dimensional space preclude the immediate assumption that binding potentials can in fact exist. In this paper, we derive a constraint on the dimensionality of a universe in the presence of an arbitrary set of forces. We then apply this constraint to systems with several example potentials. In doing so, we find that bound states in higher than 3 dimensions are in fact possible under specific circumstances which we characterize.