Quantum Mechanical versus Stochastic Processes in Path Integration

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.