Probability Density in the Complex Plane

TL;DR

This paper introduces a local quantum probability density in the complex plane and demonstrates the existence of contours where it behaves like a classical probability, advancing the understanding of quantum-classical correspondence in complex domains.

Contribution

It establishes the existence of complex contours where quantum probability density is real and positive, bridging complex quantum and classical mechanics.

Findings

Existence of infinitely many complex contours with positive probability density

Finite probability integral along these contours

Mathematical analysis involves asymptotics beyond all orders

Abstract

The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density $\rho(z)$ in the complex plane. It is shown that there exist infinitely many complex contours $C$ of infinite length on which $\rho(z) dz$ is real and positive. Furthermore, the probability integral $\int_C\rho(z) dz$ is finite. Demonstrating the existence of such contours is the essential element in establishing the correspondence between complex quantum and classical mechanics. The mathematics needed to analyze these contours is subtle and involves the use of asymptotics beyond all orders.