Duality of momentum-energy and space-time on an almost complex manifold

Abstract

We proved that under quantum mechanics a momentum-energy and a space-time are dual vector spaces on an almost complex manifold in position representation, and the minimal uncertainty relations are equivalent to the inner-product relations of their bases. In a microscopic sense, there exist locally a momentum-energy conservation and a space-time conservation. The minimal uncertainty relations refer to a local equilibrium state for a stable system, and the relations will be invariable in the special relativity. A supposition about something having dark property is proposed, which relates to a breakdown of time symmetry.