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Since the early stage of the study of Hamilton chaos, semiclassical quantization based on the low-order Wentzel-Kramers-Brillouin (WKB) theory, the primitive semiclassical approximation to the Feynman path integrals (or the so-called Van Vleck propagator), and their variants have been suffering from difficulties such as divergence in the correlation function, and so on. Even the celebrated Gutzwiller trace formula is not an exception. It is widely recognized that the essential drawback of these semiclassical theories commonly originates from the erroneous feature of the amplitude factors in their applications to classically chaotic systems. This makes a clear contrast to the success of the Einstein-Brillouin-Keller quantization condition for regular (integrable) systems.
We show, in the first half of my talk, that energy quantization of chaos in semiclassical regime (characterized with a small Planck constant) is, in principle, possible in terms of constructive and destructive interference of phases alone, and the role of the semiclassical amplitude factor is indeed negligibly small, as long as it is not highly oscillatory [1]. To do so, we first sketch the mechanism of semiclassical quantization of energy spectrum with the Fourier analysis of phase interference in a time correlation function, from which the amplitude factor is practically factored out due to its slowly varying nature [2]. Then we present numerical evidences that chaos can be indeed quantized by means of amplitude-free quasi-correlation functions. This is called phase quantization [1]. At the same time, we note that the phase quantization naturally breaks down when the oscillatory nature of the amplitude factor is comparable to that of the phases. Such a case generally appears when the Planck constant of a large magnitude pushes the dynamics out of the semiclassical regime.
In the second half, we will propose a theory that goes beyond the Van-Vleck determinant (the widely used WKB theory) without resorting to the Planck constant expansion or the hierarchical series of transport equations like the Maslov theory. No expansion with respect to the order of the potential function is made either. [1] Takahashi S and Takatsuka K 2007 J. Chem. Phys. 127, 084112. [2] Ushiyama H and Takatsuka, K 2005 J. Chem. Phys. 122, 224112. |
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