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 Gutzwiller Trace Formula ___  [link]   ___ Selberg Trace Formula | | [link] | | | | [link] | | Riemann's zeta function ___  [link]   ___ Riemann-Weil Explicit Formula | | [link] | | | | [link] | | Connes'(hypothetical) trace formula ___  [link]   ___ (Deninger's) Lefschetz Trace Formulae

"The classical periodic orbits are a crucial stepping stone in the understanding of quantum mechanics, in particular when then classical system is chaotic. This situation is very satisfying when one thinks of Poincaré who emphasized the importance of periodic orbits in classical mechanics, but could not have had any idea of what they could mean for quantum mechanics. The set of energy levels and the set of periodic orbits are complementary to each other since they are essentially related through a Fourier transform. Such a relation had been found earlier by the mathematicians in the study of the Laplacian operator on Riemannian surfaces with constant negative curvature. This led to Selberg's trace formula in 1956 which has exactly the same form, but happens to be exact. The mathematical proof, however, is based on the high degree of symmetry of these surfaces which can be compared to the sphere, although the negative curvature allows for many more different shapes."

M.C. Gutzwiller, "Chaos in Quantum Mechanics" (1998 lecture notes)

In the excellent and thorough review article:

A. Voros and N.L. Balasz, "Chaos on the pseudosphere", Physics Reports 143 no. 3.

the authors interpret the Selberg trace formula in terms of dynamical systems, and thereby make explicit a connection between the Selberg trace formula and the Gutzwiller trace formula. Recall that the latter relates the trace of an operator associated with a chaotic system to a sum over periodic orbits:

"On a compact (i.e. closed and bounded) two-dimensional surface of negative Gaussian curvature the classical motion [of a point mass] takes place on the geodesics, and it is as chaotic and nonintegrable as possible (being Bernoullian). On this surface there exists also a well-defined quantum dynamics, where the Laplace-Beltrami operator (the invarian Laplacian) acts as the Hamiltonian in the Schrödinger equation. A limiting procedure, exactly parallel to the semiclassical tradition in ordinary quantum mechanics, takes the quantum theory into the classical one when the energy E becomes large, E-1/2 playing the role of Planck's constant...(We note that the general "inverse problem" of quantising classical mechanics on a curved space, which presents its own difficulties of another order, does not arise here.) If in addition the curvature is constant, this semiclassical transition is even understood in a certain sense, exemplified by the Selberg trace formula [5]. This formula, which was motivated by Riemann's zeta function, relates in an exact way the spectrum of the quantal motion on compact surfaces of negative curvature to the classical motion. The so-resulting mathematical literature [6] has deep connections with manifold theory, automorphic functions, number theory, etc.; however, it does not address itself explicitly to questions which are of crucial interest to the physicist, i.e., the detailed properties of the discrete energy spectrum and associated eigenfunctions, and their relation to questions of quantum ergodicity and quantal chaos.

Physicists were not much concerned by the purely classical aspects of this model. For them ergodicity and mixing were the consequence of forces of interaction between particles, and not of physically nonrealisable constraint forces. However the situation becomes different if we view this as an exactly soluble classical model to be quantised, in order to study quantum-mechanically the manifestation of chaoticity. Nevertheless, very little use has been made of this model. In effect, only two pioneering studies have appeared in this direction until now. Gutzwiller [7] has drawn attention to the relation between Selberg's trace formula and the semiclassical expansion of Green's function described by a path integral. He also studied [8] the scattering on a compact surface of constant negative curvature using the work of Lax and Phillips [9]."

A. Voros and N.L. Balasz, "Chaos on the pseudosphere", Physics Reports 143 no. 3, p. 112.

Recall also that the resemblance of the Selberg Trace Formula and the Riemann-Weil explicit formula was the first historical appearance of 'evidence' for the Hilbert-Pólya conjecture, the zeta zeros in the latter corresponding to eigenvalues in the former.

Michael Berry, coming from a quantum chaology background, and hence familiar with the Gutzwiller Trace Formula, seems to have tied all this up by hypothesising that Hilbert and Pólya were right - the nontrivial Riemann zeros are eigenvalues of an operator, and that operator is associated with some as-yet-undiscovered quantum chaological system.

A connection between STF and RZF goes back to the 1950's, and the GTF was linked to the RZF shortly after its discovery. So this gives us an indirect link between the GTF and STF, but one which doesn't really involve dynamical systems, just a mutual resemblance of explicit formulae for the Riemann zeta function. The GTF relates to both the STF and the RZF, but I'm not sure which of these relationships was noted first.

Both the STF and the GTF are mentioned in Keating's article "Physics and the Queen of mathematics". Keating explains, helpfully, that Gutzwiller's 'response' function, which sums Dirac delta functions over each energy level, is the density of states function.

He points out that the GTF is problematic in three ways: (1) It's not exact, but is only valid in the semiclassical limit (so it's a kind of approximation). (2) You need to know all periodic orbits, and they're notoriously difficult to compute. (3) Even if you did know them all, the sum won't necessarily converge.

Keating points out that for 'cat maps', GTF is exact, not just a semiclassical approximation.

"There are a number of other chaotic systems for which Gutzwiller's formula is exact for all values of Planck's constant. One family of such systems correspond to constrained motion on surfaces of constant negative curvature (these surfaces are rather difficult to visualise - they are the converse of spheres which are surfaces of constant positive curvature). For these, the low quantum energy levels and the short classical periodic orbits were computed by Frank Steiner and co-workers and used to study the convergence properties of the appropriate Gutzwiller formula. In this case, however, the relationship between eigenvalues and periodic orbits carries a different name: it is actually known as the Selberg trace formula, because it was first written down by Atle Selberg, a mathematician, over 20 years before physicists became interested in the quantisation of chaotic systems. Selberg had an entirely different motivation. He was making an attack on what is generally regarded as he most important problem in number theory: the Riemann Hypothesis."

Here are Gutzwiller's words on the matter. He compares the two formulae, but doesn't really explain the connection:

"The [Gutzwiller trace formula] gets its name from a spiritual ancestor, the Selberg Trace Formula, which was discovered by the Norwegian mathematicain Atle Selberg in the 1950s. Whereas the original trace formula claims the numerical equality of two rather unequal-looking functions, the descendant makes this claim only in the limit of Planck's quantum going to zero, or equivalently, in the limit of a wavelength small compared to the size of the container. While the degree of validity for the new formula has been reduced, its domain of application has been greatly enlarged.

Instead of a well-defined and crisp formula, we now have a basic approach to making a connection between a function of time or of energy computed in quantum mechanics, on the left-hand side of the ~ sign, and another such function calculated in classical mechanics on the right-hand side. The exact formal expressions on the right depend somewhat on the behavior of the classical dynamical system, whether regular, softly, or harshly chaotic; but the left side is independent of this classical predicament. the main idea is to use the knowledge of the classical behavior to compute the right-hand side, and then switch sides so as to draw conclusions about the quantum mechanics of the system.

The author was the first to devise this general scheme as a way to answer Einstein's question: how can classical mechanics give us any hints about the quantum-mechanical energy levels when the classical system in ergodic?"

On p. 297 of his book we get a more useful explanation:

"The Selberg Trace Formula is essentially equivalent to the statement that gc(E) as given by the Gutzwiller trace formula, is equal to the trace g(E) given above for the motion of a particle on a compact surface of constant negative curvature."

g(E) is a response function defined to be the sum of all 1/(E - Ej) for energy levels Ej. This function has singularities at each energy level, and a classical approximation gc(E)