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Berry, Michael. “Chaos and the Semiclassical Limit of Quantum Mechanics (Is the Moon There When Somebody Looks?)"

Michael Berry’s essay addresses the problematic relation between the presence of chaos in classical mechanics and its absence in quantum mechanics. If classical mechanics is the limit of quantum mechanics when Planck’s constant h can be ignored, why does a system appear nonchaotic according to quantum mechanics and yet chaotic when we set h = 0? Moreover, if all systems obey quantum mechanics, including macroscopic ones like the moon, why do they evolve chaotically? Berry’s approach is to locate this problem within a larger one: namely the mathematical reduction of one theory to another. His claim is that many of the problems associated with reduction arise because of singular limits, which both obstruct the smooth reduction of theories and point to rich “borderland physics” between theories. The limit h _ 0 is one such singular limit, and this fact sheds light on the problem of reduction in several ways. First of all, nonclassical phenomena will emerge as h _ 0. Secondly, the limit of long times (t _ ñ), which are required for chaos to emerge in classical mechanics, and the limit h _ 0, do not commute, creating further difficulties.

To illustrate the role of singularities in the semiclassical limit, Berry first considers a simple example: two incident beams of coherent light. Quantum mechanics predicts interference fringes, and these fringes persist as h _ 0 due to the singularity in the quantum treatment. But in the geometrical-optics form of classical physics (where the wave-like nature of light is ignored) there are no fringes, only the simple addition of two light sources. To regain the correspondence principle between classical and quantum mechanics we must first average over phase-scrambling effects due to the influence of the physical environment in a process called “decoherence.”

A second, more complex, example illustrates the relation between these singularities and chaos. Berry describes the chaotic rotational motion of Hyperion, a satellite of Saturn. Regarded as a quantum object, Hyperion’s chaotic behavior should rapidly be suppressed. Remarkably, however, the suppression is itself suppressed due to decoherence: even the “kicks” from photons from the sun on Hyperion are enough to induce decoherence. This means that, while it is true that chaos magnifies any uncertainty, in the quantum case the magnification would wind up suppressing chaos if this suppression were not itself suppressed by decoherence induced by interactions with the environment.

Finally, Berry turns to emergent semiclassical phenomena. These phenomena do not involve chaos, and unlike more familiar examples of macroscopic quantum phenomena such as superfluidity, their detection requires magnification. His first example is the focusing of a family of light trajectories, such as rainbows or light patterns in a swimming pool. These patterns, or caustics, are singularities in geometrical optics. But upon microscopic examination, caustics dissolve into intricate interference patterns which catastrophe theory describes as emergent semiclassical phenomena called diffraction catastrophes. His second example is spectral universality: if we consider quantum systems whose classical mechanical treatment is chaotic, we find that the statistics of the spectra of all such systems is the same. Spectral universality is nonclassical, because it is a property of discrete energy levels, and it is semiclassically emergent because the number of levels increases in the classical limit, h _ 0. Berry’s conclusion is that, as we generalize to a deeper theory, the singularities of the old theory are dissolved and replaced by new ones.

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