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The sensitivity of a wave field's evolution to small perturbations is of fundamental interest. For chaotic systems, there are two distinct regimes of either exponential or Gaussian overlap decay in time. We develop a semiclassical approach for understanding both regimes and give a simple expression for the crossover time between the regimes. The wave field's evolution is considerably more stable than the exponential instability of chaotic trajectories seems to suggest. The resolution of this paradox lies in the collective behavior of the appropriate set of trajectories. Results are given for the standard map.

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Chaos introduces essential complications into semiclassical mechanics and the conventional wisdom maintains that the semiclassical time-dependent Green's function fails to describe the quantum dynamics once the underlying chaos has had time to develop much finer structure than a quantum cell ih). We develop a method to evaluate the semiclassical approximation and test it for the first time under these circumstances. The comparison of the quantum and semiclassical dynamics of the stadium billiard shows remarkable agreement despite the very intricate underlying classical dynamics.PACS numbers: 03.65. Sq, 03.40.Kf, 05.45.+b Semiclassical mechanics has a long, illustrious history of providing both physical insight and approximations to a wide array of quantum-mechanical problems. Being a wave mechanics based solely on input of a classical nature, it focuses all the attention on the properties of a quantum system's classical analog. If the underlying classical dynamics are integrable, the theory is fairly well understood. On the other hand, if the underlying dynamics are chaotic, a number of basic difficulties exist. Indeed, for this situation it was recognized many years ago during the period of the "old quantum theory" that essential complications arose in attempting to approximate stationary quantum solutions [1]. Recently, a great deal of research has been directed toward resolving these problems and yet surprisingly, prior to this work ih^ fundamental semiclassical approximation (by which we mean the semiclassical Green's function in the time domain) has never truly been tested for a chaotic system-its validity is quite unknown. One reason for this gap in understanding is a deep rooted, intuitive pessimism about the approximation's applicability since the nonzero size of Planck's constant must be responsible for some kind of smoothing over the intricate complexity that is the essence of chaotic dynamics. This intuition would suggest that phase-space structures on a scale much finer than h cannot be relevant. A typical argument stresses, for example, that it is easy to find a perturbation which, though barely affecting the quantum system, radically alters each of the classical trajectories-the correspondence principle is failing. A further explanation of this gap lies in the technical difficulty of just evaluating the formal semiclassical expression. In this Letter, we shall outline a method that we have developed for perforrning the evaluation and demonstrate the astonishingly quantitative agreement existing between the quantum and semiclassical dynamics for a chaotic system, the stadium billiard. This agreement extends well past the time when classical structure far finer than a quantum cell is put into the semiclassical mechanics.Our starting point is to consider the quantum-mechanical time-dependent Green's function in a coordinate representation. It is denoted G(q,q';r)=

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Postmodern movements are well known in the arts. After a major artistic revolution, and after the “modern” innovations have been assimilated, the threads of premodern thought are always reconsidered. Much of value may be rediscovered and put to new use. The modern context casts new light on premodern thought, which in turn shades perspectives on modernism.

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