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."
Excerpt, p. 185:
"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]. While a number
of general theorems have been proved by mathematicians in the last decade [7], and
an explicit formula of the desired type was found by Selberg [8] in 1956 in a special
case, nobody had actually used the detailed description of a classical ergodic system
in order to calculate (at least approximate) values for the energy levels. That,
however, is exactly what will be done in this paper for the special example of an
electron moving around a donor impurity in a semiconductor where its effective mass
tensor is anisotropic."
[6] 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 even he gets the energy levels
Only by solving Schrödinger's equation.
[7] The relation between the geodesics and the eigenvalues of the Laplacian
on a Riemannian surface was first studied by Y. Colin de Verdiere,
Compositio Math. 27 (1973) 83, 159 and J. Chazarian, Inventions
Math. 24 (1974) 65.
[8] A. Selberg, J. Indian Math. Soc. 20
(1956) 47-87, of
which a readable account is given by H.P. McKean, Comm. Pure Appl. Math.
25 (1972) 225 adn a very detailed one by D. Hejhal, Lecture Notes
in Mathematics, vol. 548 (Springer, New York, 1976)
The author mentions the Riemann zeta function twice
in this paper, but only indirectly:
p.192: "In order to conform with the nomenclature in the literature on
zeta-functions, e.g. Riemann's zeta-function, the upper half of the complex
E-plane will be transformed into the right-hand half of a complex
s-plane, Re(s) > 0..."
p. 202: "The resulting sum over logarithms can be exponentiated to give
an infinite product which will be called Z(s) because of its
similarity to Riemann's zeta function..."
Excerpts p. 205-206:
"An important, fundamental issue remained unresolved until recently, since
the transition from the classical to quantal mechanics could not be carried out in
the frequent cases where the classical trajectories are ergodic. As the author
pointed out in the earlier report, however, a mathematically completely satisfactory
example could be found among the surfaces of constant negative curvature.
Selberg's
trace formula in this case not only coincides with exactly the formula which the
author proposed in 1970 for general Hamiltonian systems, but it gives an identity
which relates the eigenvalues of the Laplacian to the periodic geodesics.
Selberg's trace formula has never been used to calculate the eigenvalues of the
Laplacian from the complete enumeration of all periodic orbits. Such an enumeration
turns out to be a difficult problem in group theory whose solution is not in sight.
Selberg's trace formula therefore provides an existence proof, but not an explicit
construction, so to speak. Such a construction could be carried out in the anisotropic
Kepler problem, because the enumeration of periodic orbits is relatively easy,
and because, in addition, a very effective approximation for the action integral
of all the periodic orbits could be found, and the summation over them could be
calculated."
"It is clearly desirable to tighten up a number of steps in the whole chain of
reasoning. An error estimate on the trace formula seems a long way off. The one-to-one
correspondence between the trajectories and binary sequences has not been proved
mathematically in spite of certain efforts, although the numerical evidence seems
overwhelming. The very poor treatment of the stability exponent, in contrast to
the action integral, has to be justified. On the other hand, the evaluation of the
sum...over all binary sequences is nothing but the grand canonical partition function
for the one dimensional Ising chain with exponentially decaying interaction. The
calculation in this paper has determined its singularities in the complex temperature
plane, and should be of some interest to statistical mechanics."
"The relation between classical trajectories and quantum mechanical bound states
will not be understood for classically ergodic systems until some effective means of
enumeration and calculation have been found for the periodic orbits at a given energy.
It looks as if the general measure theoretic and topological concepts of stochastic
behaviour are not nearly good enough for this purpose. On the other hand, the anisotropic
Kepler problem gives at least one example where the necessary detailed information
can be put together in a very simple, though approximate but useful form."