Quantum Trajectories: Dirac, Moyal and Bohm

TL;DR

This paper explores the development of quantum trajectories through non-commutative algebra, connecting Dirac's ideas with weak values and advanced mathematical structures to deepen the understanding of quantum phenomena.

Contribution

It generalizes Dirac's early proposals, linking quantum trajectories to weak values and geometric algebra, advancing the mathematical framework of quantum mechanics.

Findings

Relation of quantum trajectories to weak values

Use of covering spaces to explain wave properties

Introduction of new mathematical techniques beyond Hilbert space

Abstract

We recall Dirac's early proposals to develop a description of quantum phenomena in terms of a non-commutative algebra in which he suggested a way to construct what he called `quantum trajectories'. Generalising these ideas, we show how they are related to weak values and explore their use in the experimental construction of quantum trajectories. We discuss covering spaces which play an essential role in accounting for the `wave' properties of quantum particles. We briefly point out how new mathematical techniques take us beyond Hilbert space and into a deeper structure which connects with the algebras originally introduced by Born, Heisenberg and Jordan. This enables us to bring out the geometric aspects of quantum phenomena.