Non-Commutative Geometry, Multiscalars, and the Symbol Map

TL;DR

This paper develops a framework for differential forms on quantum phase-space using non-commutative geometry, establishing a parallel to classical tensor fields in Hamiltonian dynamics.

Contribution

It introduces a construction of differential forms on quantum phase-space based on the universal exterior algebra in non-commutative geometry, extending classical concepts.

Findings

Differential forms on quantum phase-space mirror classical tensor fields.

Establishes a natural relationship between quantum differential forms and quantum dynamics.

Provides a geometric framework for quantum systems using non-commutative geometry.

Abstract

Starting from the concept of the universal exterior algebra in non-commutative differential geometry we construct differential forms on the quantum phase-space of an arbitrary system. They bear the same natural relationship to quantum dynamics which ordinary tensor fields have with respect to classical hamiltonian dynamics.