Non-dispersive retarded interactions as manifestations of unique symmetry of our space associated with its 3-dimensionality

TL;DR

The paper demonstrates that the unique non-dispersive retarded interactions, linked to the symmetry of our 3-dimensional space, emerge naturally from the mathematical properties of wave equations and causality principles.

Contribution

It establishes a special integro-differential formula that explains why only three-dimensional space supports non-dispersive retarded interactions.

Findings

SIDF depends on space dimension n and relates to wave solutions only when n=3.

Retarded solutions of wave equations are uniquely valid in three-dimensional space.

The 3D space symmetry underpins the physical reality of non-dispersive retarded interactions.

Abstract

A special integro-differential formula (SIDF) of the class of Green's theorems is established for a pair of functional forms (FFs) defined in a real affine Euclidean space of any given dimension n>=1 (nDRAES); one of the FFs is the Green FF for a Poisson equation in the unbounded nDRAES, and the other one is an arbitrary twice-continuously-differentiable time biased (retarded or advanced) FF. The SIDF is an identity, which depends on n, but which is not associated with any specific differential equation for the arbitrary FF. If, however, the latter is assumed to satisfy an inhomogeneous wave equation (IWE) in nDRAES then the SIDF expresses the forced retarded or advanced solution of the IWE if and only if n=3. Based on this fact and also on the pertinent properly defined causality principle, it is concluded that an nDRAES can be regarded as a receptacle of Nature, i.e. as a receptacle of matter along with the principle of special relativity and also along with all metamorphoses, which occur to matter in time in the irreversible direction from past through present to future - metamorphoses that are called physical, chemical, biological, etc. processes - if and only if n=3.