A Proposal for a Differential Calculus in Quantum Mechanics

TL;DR

This paper introduces a quantum-deformed exterior calculus on phase-space using the Weyl-Wigner-Moyal formalism, incorporating super-manifolds and a Moyal super-star product to extend classical tensor calculus into quantum mechanics.

Contribution

It develops a novel quantum-deformed differential calculus on phase-space, integrating super-manifolds and Moyal products to generalize classical exterior calculus for quantum systems.

Findings

Defines a super-manifold structure related to phase-space

Constructs a Moyal super-star product for differential forms

Derives quantum-deformed rules for exterior calculus operations

Abstract

In this paper, using the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a {\it quantum-deformed} exterior calculus on the phase-space of an arbitrary hamiltonian system. Introducing additional bosonic and fermionic coordinates we construct a super-manifold which is closely related to the tangent and cotangent bundle over phase-space. Scalar functions on the super-manifold become equivalent to differential forms on the standard phase-space. The algebra of these functions is equipped with a Moyal super-star product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra in order to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus.