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M.C. Gutzwiller, "Stochastic behavior in quantum scattering"

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."
 

The following excerpts (with accompanying notes) outline the number theoretical content of this article:

1. Introduction

"...in quantum mechanics: closed systems have a discrete spectrum, while open systems have a continuous one. In the first case, all stationary states are bound, i.e. localized, while in the second case, a wave comes in from infinity and is scattered back out again. This occurs for all energies above a certain threshold, and the quantity of interest is the phase shift between incoming and outgoing wave as a function of momentum (or energy) and direction (if there is a choice)."

The author then describes a particular open system which is "put together from a piece of Lobachevski space, i.e. a Riemannian space of constant negative curvature, where certain boundaries are glued together (identified) so as to obtain a very smooth structure."

"The incoming and outgoing waves are very simple function of the coordinates in this model, and their phase can be calculated directly by taking advantage of the symmetries, exactly as in the traditional closed flat boxes. The phase shift is given essentially by the imaginary part of the logarithm (i.e. what is usually called the phase angle) for Riemann's zeta function. This neat and unexpectedly simple result could have been stated as a theorem without proof, or even as a postulate to be realized by some unknown physical system, but it is very pleasing to know its origin to be rather direct, and on a geometrical-physical foundation.

The stochastic elements in this system are, therefore, all hidden in this famous transcendental function, with its connections to prime numbers. Two of its features will be discussed in some detail, its ability to mimic other functions, and the difficulty of its computation. These two traits will be called its "chaos" and its "complexity". There are indications that Riemann's zeta function and its several cousins are present in many other scattering problems, whose classical analogs show stochastic behavior."
 

2. The Poincaré model

This outlines the basic theory of the upper-half plane model of the hyperbolic plane, and the Projective Special Linear group of 2 by 2 matrices with real elements PSL(2,R).
 

3. Plane waves

Schrödinger's equation is derived for spaces with hyperbolic metric.
 

4. The leaky box

A particular space of constant negative curvature is introduced, likened by the author to a torus-shaped box with an infinite 'horn': "The leaky torus is topologically different from the ordinary torus which physicists like to use in Euclidean space. The four corners of the domain D [a "square" cut out of the hyperbolic plane] become one point as usual, but this point is now infinitely removed." This 'box' is where the scattering in question is to occur.

The space is then related to an arithmetic subgroup of the matrix group PSL(2,R). The author acknowledges that "This group has been studied extensively by Harvey Cohn in connection with some problems in number theory, cf. Discontinuous Groups and Riemann Surfaces, L. Greenberg, ed. Annals of Mathematics Studies No. 79 (Princeton Univ. Press, 1974) p. 81-98."
 

5. The incoming and the outgoing waves

Due to the definition of the group , relatively prime matrix entries become central to this model.

[Expression (20):
$\phi = \alpha^{iw}{y^{1/2-iw} + \Sigma'_{(c,d)=1} \frac{y^{1/2-iw}}{|cz + d|^{1-2iw}}}$]

"The geometry of the domain D (or D') has now been turned into an expression of pure arithmetic."

[Expression (21):
$\phi = \alpha^{iw}{y^{1/2-iw} + \Sigma'_{\gamma \in \Gamma} \frac{y^{1/2-iw}}{|cz + d|^{1-2iw}}}$]

The phase shift (related to the wave function) is calculated, which (as with the expressions (20) and (21)) involves a sum over relatively prime integers. The author acknowledges that "The calculation in this section is an obvious modification of a similar calculation which can be found in P.D. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions (Princeton Univ. Press, 1976) p.170". It's worth noting that this book included some remarks on the Riemann zeta function in the context of scattering theory.

In his paper "An adelic causality problem related to abelian L-functions" (J. Number Theory 87 Série I (2001) 423-428) J.-F. Burnol writes: "The study of connections between the Riemann zeta function and scattering theory is at least thirty years old. In particular the Faddeev-Pavlov study of scattering for automorphic functions [FP] further developed by Lax and Phillips in [LP] has attracted widespread attention. In their approach the scattering matrix is directly related to the values taken by the Riemann zeta function on the line Re[s] = 1, and the Riemann Hypothesis itself is equivalent to some decay properties of scattering waves."

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

[LP] P. Lax and R.S. Phillips, Scattering Theory (1st edition, 1967) revised edition, Pure and Applied Mathematics 26 (Academic Press, 1989)
 

6. Some elementary number theory

The Euler totient function is introduced, and the Riemann zeta function defined:

[Expression (28): $\zeta(s) := \Sigma_{n=1}^{\infty} \frac{1}{n^s} = \Pi_{p} (1- \frac{1}{p^s})^{-1}$]

It is then explained: "The final expression for the reflected wave can be written compactly in terms of the combination

[Expression (29): $Z(s) = \Gamma(s/2)\pi^{s/2}\zeta(s)$]

which satisfies the crucial functional equation of Riemann

[Expression (30): $Z(1 - s) = Z(s)$]

The phase shift is then re-expressed in terms of the ratio Z(1 + 2iw)/Z(1 - 2iw)

[Expression (31): $\Int^{6}_{0} dx \phi = 6y^{1/2}{(\frac{y}{\alpha})^{-iw} + \alpha^{+2iw}\frac{Z(1+2iw)}{Z(1-2iw)} x (\frac{y}{\alpha})^{+iw}}$]
 

7. The scattering phase shift

"...The Hilbert space of all wave functions decays into two independent subspaces corresponding to scattering and to bound states, with one special state in between, so to speak, namely $\phi$ = const. whose square can be integrated since the total area is finite, but whose amplitude does not decay exponentially. Very little is known about the bound state spectrum, nor is it clear to what extent the wave function contains any bound states. Selberg's trace formula can be extended so as to establish a direct relation between the discrete spectrum and the periodic geodesics, but the mathematical expressions are not as elegant as in the case of compact surfaces of constant negative curvatures... The phase can now be written in detail as

[expression (32): $\beta = 2w\log\alpha + \frac{1}{i}\log[\frac{\Gamma(1/2 + iw}{\Gamma(1/2-iw}\pi^{-2iw}] + \frac{1}{i}log\frac{\zeta(1+2iw}{\zeta(1-2iw)}$]

The first term simply indicates the need for the wave to cover the distance $\log\alpha$ before it gets reflected. The second term can easily be evaluated with the help of Stirling's formula...This asymptotic formula for large w gives essentially correct values for w as small as 1, and contributes a smooth, monotonic dependence of $\beta$ on w. The only sign of stochastic behaviour comes from the last term in (32) which will be discussed for the rest of this paper as if it were the only important contribution to $\beta$.

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 ."
 

8. The Riemann zeta function

An interesting historical observation: "Two years [after proving the Prime Number Theorem] Hadamard gave the first discussion of geodesics on a surface of negative curvature as an ergodic system, but he did not hint at any connection with the zeta function."

[See J. Hadamard, "Sur le billiard non Euclidean", Soc. Sci. Bordeaux, Proc. Verbaux 1898 (1898) 147; J. Math Pure Appl. 4 (1898) 27.]

The last term of expression (32) is considered: "If Euler's product is inserted without worrying about convergence...[each] term in [the resulting] sum is a real periodic function of w with the frequency 2logp. Since p > 2, one can even expand the two logarithms and find [a] Fourier series...

Wintner has shown that these expansions are legitimate...

[See A. Wintner, Duke Math. J. 10 (1943) 429-440.]

Wintner's result is important in view of the discussion in the next section. The zeta function will be shown to be about as chaotic as one might wish any smooth function to be. On the other hand, only a discrete set of frequencies suffices to bring about such behaviour. But these frequencies are linearly independent. No linear combination of logarithms of prime numbers with integer coefficients can ever vanish. Otherwise the representation of an arbitrary integer as a product of powers of primes would no longer be unique. All such linear combinations actually occur as frequencies in the definition of if one writes each term in the sum as $n^{-\sigma}\exp(-it\logn)$, for constant $\sigma$ and varying t."

Note that the author here comes close to the interpretation of the Riemann zeta function as a partition function of statistical mechanics (Mackey, Julia, Spector).
 

9. Chaos in the Riemann zeta function

Since has no singularities in any vertical strip above the real axis, the phase shift $\beta(w)$ is a smooth function of w for arbitrary large w. The absolute value of grows exceedingly slowly, essentially as log log t. nothing dramatic can possibly happen, although $\beta(w)$ never settles down to any kind of simple behaviour never contains an infinity of linearly independent frequencies.

The question then arises: How unruly is in the critical strip between Re s = 0 and Re s = 1? The answer was given recently by S.M. Voronin, and then made explicitly by Axel Reich..."

[See A. Reich, "Werteverteilung von zetafunktionen", Arch. Math. 34 (1980) 440-451.]
 

10. The complexity of Riemann's zeta function

"The stochastic behavior of any mathematical object can be recognized in the difficulty of its compuation. The phase shift $\beta$ as a function of the velocity w requires the calculation of on the line s = 1 + it, at the border of the critical strip 0 < Re s < 1. How easy is it to do the computations there?"

The Riemann-Siegel formula is discussed, and the relative difficulty of computing .

[Expression (40): $\Sigma_{1}^{N} \frac{1}{N^s} + \chi(s)\Sigma_{1}^{N} \frac{1}{n^{1-s}} + (-1)^{N-1}C(s)S_{N}$]

"...Clearly, the Riemann zeta function is much more difficult to evaluate than any of the standard functions of a complex variable which one encounters in mathematical physics."
 

11. Discussion

"...Does one always run into the Riemann zeta function for the scattering phase shift? The answer is clearly negative, unless the group of symmetries belongs to a special class called the arithmetic groups. So what remains of our analysis? Formula (20) remains valid, but not (21). One has now to deal with so-called Eisenstein series, and finds that many of the results can be generalized, but nothing as elegant as the phase shift in (31) is obtained.

By way of speculation, one can finally ask whether the Riemann zeta function shows up in other scattering problems. The answer is a resounding yes, although the connection is not directly in the phase shift. Consider the flat unit square with opposite sides identified and some kind of exponential horn attached in the middle through which a particle can enter or exit exactly as in the leaky box. The spherical waves from the center are now matched by as many copies in the whole square lattice, so that the boundary conditions on the unit square are satisfied. These waves are some kind of Bessel functions, of course, but their Mellin transforms with respect to the momentum are essentially sums over (m² + n²)^-s for m and n running through the integers. Expressions of this kind occur in the theory of algebraic numbers and lead eventually to modifications of Riemann's zeta function which are due to Dedekind and Hecke. Their behaviour is just like , including formula like (28) and (30).

This connection with number theory is inherent in the work of Berry on Sinai's billiard. [See M.V. Berry, "Quantizing a classically ergodic system: Sinnai's billiard and KKR method", Annals of Physics 131 (1981) 163-216.] Instead of a source or sink at the center of the 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 of the 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, and formula (28) does not arise. The function equation (30) is still valid, however, and a Riemann-Siegel formula (40) can be derived to facilitate the compuation. Again, the physicists have applied special cases of (40) 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."