Path algebras, wave-particle duality, and quantization of phase space

TL;DR

This paper develops a mathematical model linking path algebra deformations to wave-particle duality, resulting in a quantized, singular phase space that explains phenomena like spin and variable wavelength.

Contribution

It introduces a novel deformation of semigroup algebras that models wave-particle duality and quantization of phase space, connecting algebraic structures to physical phenomena.

Findings

Wave length matches de Broglie's hypothesis for free particles

Phase space is quantized and singular at the Planck scale

Model explains spin and variable wavelength phenomena

Abstract

Semigroup algebras admit certain `coherent' deformations which, in the special case of a path algebra, may associate a periodic function to an evolving path; for a particle moving freely on a straight line after an initial impulse, the wave length is that hypothesized by de Broglie's wave-particle duality. This theory leads to a model of "physical" phase space of which mathematical phase space, the cotangent bundle of configuration space, is a projection. This space is singular, quantized at the Planck level, its structure implies the existence of spin, and the spread of a packet can be described as a random walk. The wavelength associated to a particle moving in this space need not be constant and its phase can change discontinuously.