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arXiv:1505.05859 · Letters in Mathematical Physics

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

Murray Gerstenhaber

Published Nov 19, 2016• 3 citations
arXivHTMLPDF
DOI
10.1007/s11005-016-0909-8

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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.
Light has historically resisted a single physical interpretation, exhibiting wave-likeinterference, particle-like energy exchange, and invariant high-speed propagation throughspace. These behaviors led to the concept of wave–particle duality, an interpretivecompromise rather than a unified physical explanation. In the SP3 framework, space itselfis not empty or passive but possesses phase, memory, and the capacity for conditioning.This paper reframes light as a localized, self-maintaining reconfiguration of space-phaserather than a particle moving through emptiness or a field oscillating in a void. Interference,quantization, constant propagation speed, and gravitational bending arise naturally fromthe properties of space-phase itself. Wave–particle duality is therefore not resolved bychoosing one description over another but is eliminated by replacing an incompleteconception of space with a physically active medium.
Abstract Duality games are a way of looking at wave–particle duality. In these games. Alice and Bob together are playing against the house. The house specifies, at random, which of two sub-games Alice and Bob will play. One game, Ways, requires that they obtain path information about a particle going through an N -path interferometer and the other, Phases, requires that they obtain phase information. In general, because of wave–particle duality, Alice and Bob cannot always win the overall game. However, if the required amount of path and phase information is not too great, for example specifying a set of paths or phases, one of which is the right one, then they can always win. Here we study examples of duality games that can always be won, and develop a wave–particle duality relation expressed only in terms of mutual information to help analyze these games.