Classical stochastic representation of quantum mechanics

TL;DR

This paper demonstrates that quantum dynamics can be represented by a classical Hamiltonian system through a specific canonical transformation, with the wave function treated as a stochastic variable to account for quantum probabilistic behavior.

Contribution

It introduces a novel classical representation of quantum mechanics by transforming phase space into a Hilbert space via a canonical transformation, incorporating stochastic wave functions.

Findings

Quantum dynamics represented by classical Hamiltonian systems.

Wave function treated as a stochastic variable.

Norm-preserving dynamics of the underlying classical system.

Abstract

We show that the dynamics of a quantum system can be represented by the dynamics of an underlying classical systems obeying the Hamilton equations of motion. This is achieved by transforming the phase space of dimension $2n$ into a Hilbert space of dimension $n$ which is obtained by a peculiar canonical transformation that changes a pair of real canonical variables into a pair of complex canonical variables which are complex conjugate of each other. The probabilistic character of quantum mechanics is devised by treating the wave function as a stochastic variable. The dynamics of the underlying system is chosen so as to preserve the norm of the state vector.