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M.C. Gutzwiller's number theory-related work

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In 1971, Gutzwiller published his celebrated semiclassical trace formula in "Periodic orbits and classical quantization conditions" (J. Math. Phys. 12 no. 3 (1971)) building on the work of 1967, 1969 and 1971 papers in the same journal. None of these papers involved any number theory.

At some time between 1971 and 1980, he appears to have realised that there was a connection with Selberg's trace formula. This may have been as a result of contact during that period with the following literature:

[S] A. Selberg, "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series", Journal of the Indian Mathematical Society 20 (1956) 47-87.

[FP] L.D Faddeev and B.S. Pavlov, "Scattering theory and automorphic functions", Seminar of Steklov Mathematical Institute of Leningrad 27 (1972) 161-193.

[M] H.P. McKean, "Selberg's trace formula as applied to a compact Riemannian surface", Communications in Pure and Applied Mathematics 25 (1972) 225-246.

[LP1] P.D. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions (Princeton Univ. Press, 1976)

[H1] D. Hejhal, "The Selberg trace formula and the Riemann zeta function", Duke Mathematics Journal 43 (1976) p.459

[H2] D. Hejhal, The Selberg Trace Formula for PSL(2,R), Volume I (Springer Lecture Notes 548, 1976)

M.C. Gutzwiller, "Classical quantization of a Hamiltonian with ergodic behavior", Physical Review Letters 45 no.3 (1980) 150-153

[abstract:] "Conservative Hamiltonian systems with two degrees of freedom are discussed where a typical trajectory fills the whole surface of constant energy. The trace of the quantum mechanical Green's function is approximated by a sum over classical periodic orbits. This leads directly to Selberg's trace formula for the motion of a particle on a surface of constant negative curvature, and, when applied to the anisotropic Kepler problem, yields excellent results for all the energy levels."

This references [S], [M] and [H2].

M.C. Gutzwiller, "The quantization of a classically ergodic system", Physica D 5 (1982) 183-207

[abstract:] "The motion of a charged particle with an anisotropic mass tensor is completely ergodic in two dimensions. Its periodic orbits can be mapped 1-to-1 into the binary sequences of even length. Their action integral is approximated very closely by a quadratic function is a sum over all periodic orbits, and can be evaluated by a transformation which Kac used in the discussion of an Ising lattice with long range interaction. The poles of the trace as a function of the energy can be segregated by their discrete symmetry. The real parts agree very well with the quantum mechanical energy levels, which most of the imaginary parts are small, indicating sharp resonances. This is the first calculation of energy levels on the basis of classically ergodic trajectories."

This references [S], [M] and [H2] above.

M.C. Gutzwiller, "Stochastic behavior in quantum scattering", Physica D: Nonlinear Phenomena 7 (1983) 341-355

[abstract:] "A 2-dimensional smooth orientable, but not compact space of constant negative curvature with the topology of a torus is investigated. It contains an open end, i.e. an exceptional point at infinite distance, through which a particle or a wave can enter or leave, as in the exponential horn of certain antennas or loud-speakers. In the Poincaré model of hyperbolic geometry, the solutions of Schrödinger's equation for the reflection of a particle which enters through the horn are easily constructed. The scattering phase shift as a function of the momentum is essentially given by the phase angle of Riemann's zeta function on the imaginary axis, at a distance of from the famous critical line. This phase shift shows all the features of chaos, namely the ability to mimic any given smooth function, and great difficulty in its effective numerical computation. A plot shows the close connection with the zeros of Riemann's zeta function for low values of the momentum (quantum regime) which gets lost only at exceedingly large momenta (classical regime?) Some generalizations of this approach to chaos are mentioned."

This references [LP] and, for the first time in Gutzwiller's work, directly involves the Riemann zeta function. Note that this is an appearance of the Riemann zeta function in the context of quantum chaology three years before Michael Berry's seminal "Riemann's zeta function: a model for quantum chaos?". Unlike Berry's work, however, this paper doesn't deal with the spectral interpretation, although Gutzwiller does indicate an awareness of the 'Hilbert-Pólya idea' (p.349):

"Riemann's zeta function is mostly known to mathematicians who are interested in number theory. It has not come up so far in any problem with a physical background to my knowledge, although Hilbert seems to have proposed the idea of finding an eigenvalue problem whose spectrum contains the zeros of ".

As with all of the publications listed below, the 'chaotic' implications of Voronin's Universality Theorem and its variants are discussed.

In June 1983, Gutzwiller presented a paper at the NATO Advanced Research Workshop on Quantum Chaos: "Chaotic Behavior in Quantum Systems, Theory and Applications" held in Como, Italy (see proceedings edited by G. Casati (Plenum, 1985) p.149-164). The paper was called "Mild Chaos" and involved the Riemann zeta function (again through the mathematics associated with the motion of a particle on a surface of constant negative curvature).

M. Berry was in attendance, presenting a paper on "Aspects of Degeneracy". This involved random matrix theory, but nothing of a number theoretic nature. It is possible that Gutzwiller's presentation first introduced Berry to the zeta function, although this is presently unclear.

There is also evidence that Berry's quantum chaos research was already proceeding in a number theoretical direction by the summer of 1983.

In Gutzwiller's 1982 paper, p.185-186, we read:

"So far, the relation between classical and quantum mechanics could only be established to the extent that the classical trajectories lie on invariant tori. It is really quite amazing that this situation has prevailed for over 50 years since the discovery of quantum mechanics. The effect of classically ergodic behaviour on the quantum levels has been studied only in a qualitative manner [6]."

Reference [6] reads: "The most specific results have been obtained by M.V. Berry, "Quantizing a classically ergodic system: Sinai's billiard and KKR method", Annals of Physics 131 (1981) 163-216, although he gets the energy levels only by solving Schrödinger's equation."

In Gutzwiller's 1983 paper, he concludes:

"This connection with number theory is inherent in the work of Berry on Sinai's billiard. Instead of a source or sink at the center of a square, an obstacle of circular symmetry is investigated and shown to give an ergodic system. The quantum mechanical bound states are computed with the help ofthe KKR method which uses basically the Mellin transforms of the above-generalized zeta functions. As before, there are two cases to be considered. If the square is replaced by a parallelogram of different shape, there may not be a connection with number theory...The functional equation is still valid, however, and a Riemann-Siegel formula can be derived to faciliate the computations. Again, the physicists have applied special cases...when calculating certain sums in lattices, but they do not seem to have realized the relation to Riemann's zeta function and its chaotic properties."

M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, 1990) deals with the Riemann zeta function in sections 17.9 (p.307-312) and 19.8-9 (p. 374-379). The Selberg Trace Formula plays a central role in Chapter 19, "Motion on a Surface of Constant Negative Curvature".

M.C. Gutzwiller, "Quantum Chaos" - a popular article published in Scientific American (January 1992)