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 2-1 projection. This space is singular, quantized at the Planck level, and its structure implies the existence of spin.