Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2009-12-23
Physics
High Energy Physics
High Energy Physics - Theory
38 pages, 17figures
Scientific paper
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.
Bender Carl M.
Hook Daniel W.
Meisinger Peter N.
Wang Qing-hai
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