The resolution of this "paradox" is at the heart of "Quantum Chaos", the study of the relationship between classical chaos and quantum mechanics. In a chaotic

The resolution of this "paradox" is at the heart of "Quantum Chaos", the study of the relationship between classical chaos and quantum mechanics. In a chaotic system, minute differences in initial conditions are quickly amplified, making long time predictions impossible. A striking example of this is the potential instability of the solar system; we cannot rule out that a planet (earth!?) shoots off into space long before the sun turns into a red giant. On the other hand, in the wavelike world of quantum mechanics, trajectories are ``smeared out'' because of the uncertainty principle, and this seems to preclude the complicated picture of interlocking, yet divergent trajectories present in chaotic systems. Thus, a fundamental (and somewhat philosophical) question is: how can macroscopic chaos arise in a world governed by quantum mechanics? The answer to this has been the subject of debate, but from the point of view of mathematical physics, a more concrete question is at the center: in what ways does chaos in classical dynamical systems manifest itself quantum mechanically, in particular in terms of spectral properties of "quantized Hamiltonians". There is a fascinating connection between quantum chaos and number theory (the Riemann zeta function): the distribution of (normalized) gaps between zeros of the Riemann zeta function appears to be the "same" as the distribution of gaps between eigenvalues of quantized Hamiltonians associated to classically chaotic systems. There is a fun story of how H. Montgomery had (partially) computed the "pair correlation" ofthe gaps between Riemann zeta zeros, and at tea time at IAS he mentioned this to F. Dyson, who immeditely told him that this is the same as for random matrices, more details can be found in http://www.bourbaphy.fr/keating.pdf http://www.bourbaphy.fr/keating.pdf Some further info about quantum chaos: http://www.ams.org/notices/200801/tx080100032p.pdf http://www.ams.org/notices/200801/tx080100032p.pdf https://en.wikipedia.org/wiki/Quantum_chaos https://en.wikipedia.org/wiki/Quantum_chaos https://www.scientificamerican.com/article/quantum-chaos-subatomic-worlds/ https://www.scientificamerican.com/article/quantum-chaos-sub... http://www.scholarpedia.org/article/Quantum_chaos http://www.scholarpedia.org/article/Quantum_chaos