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 geometricaloptics form of classical physics (where the wavelike 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 phasescrambling 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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