Non-Commutative Geometry on Quantum Phase-Space

TL;DR

This paper develops a non-commutative analogue of classical differential forms on quantum phase-space, linking quantum dynamics with geometric structures and providing a unified framework for classical and quantum forms.

Contribution

It introduces a universal non-commutative differential form framework on quantum phase-space using Weyl--Wigner symbols, connecting quantum and classical geometries.

Findings

Constructed non-commutative differential forms on quantum phase-space.

Established a relation between quantum forms and classical tensor fields.

Provided a physical interpretation and potential applications of quantum differential forms.

Abstract

A non--commutative analogue of the classical differential forms is constructed on the phase--space of an arbitrary quantum system. The non--commutative forms are universal and are related to the quantum mechanical dynamics in the same way as the classical forms are related to classical dynamics. They are constructed by applying the Weyl--Wigner symbol map to the differential envelope of the linear operators on the quantum mechanical Hilbert space. This leads to a representation of the non--commutative forms considered by A.~Connes in terms of multiscalar functions on the classical phase--space. In an appropriate coincidence limit they define a quantum deformation of the classical tensor fields and both commutative and non--commutative forms can be studied in a unified framework. We interprete the quantum differential forms in physical terms and comment on possible applications.