Extending Quantum Probability from Real Axis to Complex Plane

TL;DR

This paper extends quantum probability into the complex plane, deriving a stochastic differential equation for complex trajectories and unifying quantum and classical probabilities within this framework.

Contribution

It introduces a complex stochastic differential equation for quantum particles and demonstrates how quantum and classical probabilities can be unified through complex probability distributions.

Findings

Complex quantum trajectories are modeled using stochastic differential equations.

The complex probability distribution is verified via the complex Fokker-Planck equation.

Quantum and classical probabilities are shown to be integrable within the complex probability framework.

Abstract

Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle random motion in the complex plane. The probability distribution of the particle position over the complex plane is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, this probability distribution is verified by the solution of the complex Fokker Planck equation. It is shown that quantum probability and classical probability can be integrated under the framework of complex probability, such that they can both be derived from the same probability distribution by different statistical ways of collecting spatial points.