On the Mechanism of Quantization of Classical Chaos and Quantization Conditions

Abstract

Since the early stage of the study of Hamilton chaos, semiclassical quantization based on the low‐order Wentzel‐Kramers‐Brillouin (WKB) theory and their variants have been suffering from difficulties such as divergence in the correlation function, non‐convergence in the trace formula, and so on. It is now widely recognized that the essential drawback of these semiclassical theories commonly originates from the erroneous feature of the amplitude factors based on the stability matrix, when applied to classically chaotic dynamics. In this article, we clarify the mechanism of semiclassical quantization of energy spectrum with the Fourier analysis of phase interference in a trajectory‐wise time correlation function. It is accordingly shown that the energy quantization of chaos in semiclassical regime is, in principle, possible in terms of constructive and destructive interference of the phases alone, and the role of the semiclassical amplitude factor is indeed negligibly small.