Semiquantal dynamics of fluctuations: Ostensible quantum chaos.

The time-dependent variational principle using generalized Gaussian trial functions yields a finite dimensional approximation to the full quantum dynamics and is used in many disciplines. It is shown how these ’semi-quantum’ dynamics may be derived via the Ehrenfest theorem and recast as an extended classical gradient system with the fluctuation variables coupled to the average variables. An extended potential is constructed for a one-dimensional system. The semiquantal behavior is shown to be chaotic even though the system has regular classical behavior and the quantum behavior had been assumed regular. PACS number(s): 05.45.,03.65.S,05.40. There has been substantial effort made, over the years, to understand quantum systems using a system of few classical variables. These can be motivated in the h̄ → 0 limit, such as effective potential techniques[1,2] and semi-classical methods like that of WKB, which lead to the Einstein-Brillouin-Keller (EBK) [3] quantization rules for integrable systems. These also include approximations to the Feynman path-integral formulation [4], used to derive the Periodic-Orbit Trace Formula for chaotic systems [5]. This relates the spectrum of the quantum system to a weighted sum over the unstable periodic orbits of the classical system. They can also arise, as in Quantum Chromo Dynamics for example [6], in the limit of large N (number of degrees of freedom), and a classical phase space can be shown to exist in the N = ∞ limit [7]. Further, there are many equivalent mean-field theories that are used in