Quantum Mechanical versus Stochastic Processes in Path Integration

TL;DR

This paper compares quantum and stochastic processes in path integration, highlighting how quantum probabilities involve non-positive-definite functionals, illustrating the fundamental differences between quantum and classical stochastic models.

Contribution

It formally rewrites quantum evolution as a stochastic differential equation with a non-positive-definite probability functional, revealing the nature of quantum probabilities.

Findings

Quantum probability densities can be expressed via stochastic differential equations.

The stochastic force in quantum path integrals is associated with a non-positive-definite functional.

This approach illustrates the fundamental differences between quantum and classical stochastic processes.

Abstract

By using path integrals, the stochastic process associated to the time evolution of the quantum probability density is formally rewritten in terms of a stochastic differential equation, given by Newton's equation of motion with an additional multiplicative stochastic force. However, the term playing the role of the stochastic force is defined by a non-positive-definite probability functional, providing a clear example of the negative (or "extended") probabilities characteristic of quantum mechanics.