User talk:Martin Gutzwiller

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    Answer to the reviewer C for the article Quantum Chaos

    1. Figure 4 is a plot for obtaining the spectrum of the anisoscopic Kepler problem (AKP) where a function of the energy is plotted. The intersections of this function with the axis yield the bound states once with the help of 8 and once with 71 periodic orbits (PO), with obvious improvements. Figure 5 is a comparable table of numerical results, a) the simple quantum calculation of 1969, b) the first of all times calculation with periodic orbits, including 3 more levels, of 1980, c) the first modern quantum calculation of 1988. Neither of these details are hard to understand, and they are valuable for the reader in showing the kind of precision one can achieve.

    2. The idea of "resurgence spectroscopy" for me is a spectacular new way to move from the experimental spectrum of the energies for the bound states to the times of the PO's in the classical mechanics. That is more than a "technical" achievement, and many readers will appreciate it.

    Notice that I would like to mention not only the grand ideas, but also the detailed outcome in specific simple cases. Number 1. and 2. are obviously necessary in order to convey to the reader the technical importance of our understanding of PO's. That depends on what the reader wants to find, but I am speaking for those like myself who need such clear demonstrations of a great idea in physics.

    3. Two important directions I have not persued:

    A) The work of Michael Berry on the "meaning of the trace formula" has not been mentioned. First, I have not mentioned anybody at all, particularly, any of my many colleagues (including myself) in connection with any detail. I could mention Pouncare, Dirac, Feynman, and Selberg, because they are no longer alive, and their work is well before and above the level of Quantum Chaos. More specifically, Michael Berry took almost 10 years to believe the trace formula, and there is an article in 1976 of Proc.Roy.Soc. where I am negatively quoted for promoting the trace formula (TF). He did eventually come around to recognize the TF, and we always were on good terms. So I decided to add a famous talk he gave on "Quantum Chaology" in 1987 to Some Reading.

    B) The work on the time dependence in Quantum Chaos, which was discussed first by Joseph Ford of Atlanta, and his friends in Como, Italy, and in Novosibirsk, Russia, is indeed not mentioned, except somewhat casually among "concert halls, drums, church bells, tsunamis, etc." I could make an effort in that direction, but I would not feel very competent about it. Maybe somebody can write an appendix, or a separate article.

    Comments on reviewer B:

    He rightly insists on precise formulations and avoidance of "intuitive' explanations, because many readers have not as yet developed a thorough acquaintance with the topic to be explained. Therefore, the recommendations of reviewer B should be accepted. 1. "small systems like atoms and molecules" instead "atoms and molecules". 2. The physical length of any arbitrary trajectory is obtained by summing for each time interval the difference of the kinetic energy minus the potential energy. This measure was worked out 1760 by Lagrange as a young man in his native Torino (Italy). I cheerfully admit that, in contrast to the total energy = kinetic energy + potential energy, the difference of the energies does not appeal to my intuition. But Lagrange's idea is fundamental in mechanics. 3. My effort to provide some intuitive interpretation of the trace formula is not successful. It may be sufficient to point out the difference between the resonances in QM and the lengths of the periodic orbits in CM, which contribute some classical waves running around the the PO. I still think that this approximation between spectrum and PO's represents some real physics. 4. The curvature of any 2-d surface has a profound effect on the stability of the geodesics, as Hadamard discovered at the end of the 19-th century. The ideal case are the surfaces where the Gaussian curvature is the same everywhere. Poincare among others worked on this topic; they are some sort of ideal chaos for their geodesics. But their connection with quantum chaos comes through the trace formula, which happens to be exact on such surfaces. Selberg hoped that the Riemann zeta function would be a special case, and could be proved in this manneer. 5. The spectrum in figure 6 is not random, because of its relation to the classical PO's. But could have explained the correlation functions before the trace formula. 6. Figure 12 shows a nuclear spectrum in the neutron region. The lines are very sharp, and the measurements were done in the 1950's without explanation, or relation to some other physics. 7. Nuclear physicists have every reason to be unhappy compared to atomic and molecular physicists, because the forces among the particles in a nucleus are not understood, even now!

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