Path algebras and de Broglie waves

TL;DR

This paper explores how de Broglie waves can be modeled as a deformation in the path algebra of phase space, linking wave phenomena to algebraic structures on manifolds, with implications for quantum mechanics.

Contribution

It introduces a novel algebraic framework connecting de Broglie waves to path algebra deformations on Riemannian manifolds with a 2-form.

Findings

Path algebra deformation relates to de Broglie wavelength in flat space

De Broglie waves are approximations in curved space

Path product deformation encodes wave-like behavior

Abstract

De Broglie waves may be a reflection of a deformation inherent in the path algebra of phase space. On a Riemannian manifold equipped with a suitable 2-form, the product of paths, which is ordinarily their concatenation, can be deformed by multiplication by a scalar weight giving rise to a function on paths. In flat phase space the associated function is periodic with period the de Broglie wave length. The de Broglie description may only be approximate in curved space.