  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.
