A. Douglas Stone

Title

Einstein's unknown insight and the problem of quantizing chaotic motion

Abstract

In 1917 Einstein authored a little-known paper on the problem of generalizing the old quantum theory to problems with several degrees of freedom that are not separable. This paper was his only published work on the correct quantization rule for matter, which was of course not known at that time. His work laid the foundation for a method which is completely correct (within its sphere of applicability), now known as Einstein-Brillouin-Keller quantization, a multi-dimensional generalization of the WKB approximation. However he pointed out that the method fails if there do not exist a number of integrals of motion equal to the number of degrees of freedom, i.e. unless the system is integrable. He suggested that non-integrable classical dynamics is typical and presents an open problem for quantum theory. This brilliant insight was ignored until the late sixties when it became well-known to physicists that partially chaotic motion is indeed generic in classical mechanical systems. The problem noted by Einstein is fundamental and has never been fully overcome; but alternative semiclassical approaches to the quantum mechanics of classically chaotic systems have been developed and applied to interesting problems in atomic, condensed matter and optical physics. I will review Einstein's arguments and place them in a modern context. Then I will mention a few experimental systems to which "quantum chaos theory" can be applied, focusing on the topic of chaotic dielectric microlasers studied at Yale and elsewhere.