Signatures of Randomness in Quantum Chaos

We investigate toy dynamical models of energy-level repulsion in quantum eigenvalue sequences. We focus on parametric (with respect to a running coupling or ”complexity” parameter) stochastic processes that are capable of relaxing towards a stationary regime (e. g. equilibrium, invariant asymptotic measure). In view of ergodic property, that makes them appropriate for the study of short-range fluctuations in any disordered, randomly-looking spectral sequence (as exemplified e. g. by empirical nearest-neighbor spacings histograms of various quantum systems). The pertinent Markov diffusion-type processes (with values in the space of spacings) share a general form of forward drifts b(x) = N−1 2x −x, where x > 0 stands for the spacing value. Here N = 2, 3, 5 correspond to the familiar (generic) random-matrix theory inspired cases, based on the exploitation of the Wigner surmise (usually regarded as an approximate formula). N = 4 corresponds to the (non-generic) non-Hermitian Ginibre ensemble. The result appears to be exact in the context of 2 × 2 random matrices and indicates a potential validity of other non-generic N > 5 level repulsion laws. PACS Numbers: 02.50, 03.65, 05.45 1 Regular versus irregular in quantum chaos The vague notion of so-called quantum chaos, normally arising in conjunction with semiclassical quantum mechanics of chaotic dynamical systems [1], currently stands for a key-word capturing continued efforts to give a proper account to what extent quantization destroys, preserves, or qualitatively reproduces major features of classical chaos. There is no general agreement about what actually is to be interpreted as ”quantum chaos” or its definite manifestations. That in part derives from an inherent ambiguity of quantization schemes for nonlinear, possibly nonconservative, driven and damped classical problems, and is intrinsically entangled with a delicate reverse problem of a reliable (semi)classical limit procedure for once given quantum system. Other origins of this elusiveness seem to be rooted Presented at the XIV Marian Smoluchowski Symposium on Statistical Physics, Zakopane, Poland, September 9-14, 2001