Quantum trajectories

TL;DR

This paper introduces a phase space approach to quantum trajectories using Moyal's formulation, deforming classical flow actions to describe quantum dynamics on a classical phase space.

Contribution

It develops a novel formalism for quantum trajectories based on phase space deformation, extending classical Hamiltonian flow concepts to quantum systems.

Findings

Formalism applied to specific quantum systems

Deformation of classical flow to quantum flow

Examples illustrating the approach

Abstract

This paper presents a new approach to phase space trajectories in quantum mechanics. A Moyal description of quantum theory is used, where observables and states are treated as classical functions on a classical phase space. A quantum trajectory being an appropriate solution to quantum Hamiltonian equations is also a function defined on a classical phase space. The difference, when compare with classical theory, is in a deformation of a classical action of a flow on observables and states to an appropriate quantum action. This leads also to deformation of a multiplication rule for any quantum trajectory treated as a one-parameter group of diffeomorphisms. Moreover, several examples are given, presenting the developed formalism for particular quantum systems.