After graduating from Exeter and St Andrews, Michael Berry entered Bristol University, where he has been for considerably longer than he has not. He is a physicist, focusing on the physics of the mathematics…of the physics. Applications include the geometry of singularities (caustics on large scales, vortices on fine scales) in optics and other waves, the connection between classical and quantum physics, and the physical asymptotics of divergent series. He delights in finding the arcane in the mundane – abstract and subtle concepts in familiar or dramatic phenomena:

  • Singularities of smooth gradient maps in rainbows and tsunamis;
  • The Laplace operator in oriental magic mirrors;
  • Elliptic integrals in the polarization pattern of the clear blue sky;
  • Geometry of twists and turns in quantum indistinguishability;
  • Matrix degeneracies in overhead-projector transparencies;
  • Gauss sums in the light beyond a humble diffraction grating.

Martin Gutzwiller and his periodic orbits

Michael Berry, H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK

 

In the 1970s, physicists were made aware, largely through the efforts of the late Joseph Ford, that classical hamiltonian mechanics was enjoying a quiet revolution. The traditional emphasis had been on exactly solvable models, with as many conserved quantities as degrees of freedom, in which the motion was integrable and predictable. Examples are the Kepler ellipses of planetary motion, and the simple pendulum: 'as regular as clockwork'. The new research, incorporating Russian analytical mechanics and computer simulations inspired by statistical mechanics, revealed that most (technically, 'almost all') dynamical systems behave very differently. There are few conserved quantities, and motion, in part or all of the phase space, is nonseparable and unpredictable, that is, unstable: initially neighbouring orbits diverge exponentially. This is classical chaos.
It was quickly realised that this classical behaviour must have implications for quantum physics, especially semiclassical physics, e.g. for the arrangement of high-lying energy levels and the morphology of eigenfunctions. The study of these implications became what is now called quantum chaos (though I prefer the term quantum chaology). This is an area of research in which Martin Gutzwiller made a seminal contribution, described in the following, which I have adapted from a speech honouring his 70th birthday. Since a substantial part of my own scientific life has been devoted to the development and application to Martin's ideas, I won’t attempt to be detached.

Martin published the last of his series of four papers [1-4] on periodic orbits exactly forty years ago. I encountered them at that time, while Kate Mount and I were writing our review of semiclassical mechanics. That was prehistoric semiclassical mechanics: before catastrophe theory demystified caustics, before asymptotics beyond all orders lifted divergent series to new levels of precision, and above all before we knew about classical chaos.

Of Martin's series of papers, the most influential was the last one [4], containing the celebrated 'Gutzwiller trace formula'. That was a tricky calculation, based on the Van Vleck formula for the semiclassical propagator, giving the density of quantum states (actually the trace of the resolvent operator) as a sum over classical periodic orbits. In particular, Martin calculated the contribution from an individual unstable periodic orbit. Nowadays we can see this as one of the 'atomic concepts' of quantum chaology, but in those days chaos was not appreciated. But he emphasized the essential novelty of his calculation in a similar way: it applies even when the classical dynamics is nonseparable. I'm rather proud of what we wrote at the beginning of 1972, as the last sentence of our review:

"Finally, the difficulties raised by Gutzwiller's (1971) theory of quantization, which is perhaps the most exciting recent development in semiclassical mechanics, should be studied deeply in order to provide insight into the properties of quantum states in those systems, previously almost intractable, where no separation of variables is possible."

The trace formula could be approximated by taking just one periodic orbit and its repetitions. This led to an approximate 'quantization formula' that gave good results when applied to the lowest states of an electron in a semiconductor, whose mass depended on direction. I am referring to the birth of Martin's treatment of the anisotropic Kepler problem [5].

For a few years, his calculation was widely misinterpreted (among the ignorant it is misinterpreted even today) as implying a relation between the individual energy levels and individual periodic orbits of chaotic systems. One might call this the 'De Broglie interpretation' of the trace formula: that there is a level at each energy for which the action of a periodic orbit is a multiple of Planck. This is nonsense: the simplest calculation shows that the number of levels is hopelessly overestimated – in a billiard, for example, there is an 'infra-red catastrophe', that is, the prediction of levels at arbitrarily low energies.

Martin's papers quickly inspired others. In 1974, Jacques Chazarain showed that the trace formula could be operated 'in reverse', so that a sum over energy levels generated a function whose singularities were the actions of periodic orbits. This was exact, not semiclassical, and led (often unacknowledged) to what later came to be called 'inverse quantum chaology' and 'quantum recurrence spectroscopy'. In 1975 Michael Tabor and I generalized some of the results in the first of Martin's semiclassical papers [1] to get the general trace formula for integrable systems, where the periodic orbits are not isolated but fill tori. In nuclear physics, similar formulas had been obtained by Strutinsky in the context of the shell model. Tabor and I used our result to show that the level statistics in integrable systems are Poissonian - more about that later. William Miller and André Voros resolved a puzzle about the application of the trace formula for a stable orbit: by properly quantizing transverse to the orbit, they restored the missing quantum numbers; then Martin's single-orbit quantization rule makes sense, as the 'thin-torus' limit of Bohr-Sommerfeld quantization.

Probably Martin didn't realize that his formula was so fashionable at that time that it induced a certain hysteria. Michael Tabor and I were quietly finishing the work I just described when we learned that William Miller wanted to visit us in Bristol, to talk about his new work on periodic orbits. We convinced ourselves that this must be the same as ours, and laboured day and night (up a ladder, actually, because Michael was helping me paint my new house) to get our paper written and submitted before he arrived. We were foolish to panic, because William's work was completely different.

An awkward feature of stable orbits, recognized clearly by Martin in those early days, was that focusing occurs along them, leading for certain repetition numbers and stability indices to divergences of the contributions he calculated, associated with bifurcations. That awkwardness was removed in 1985 by Alfredo Ozorio de Almeida and John Hannay, who applied ideas from catastrophe theory that had come into semiclassical mechanics in the 1970s. Their development of Martin's formula became popular much later, when the features they predicted could be detected numerically.

In the early 1970s, Ian Percival made us aware of the amazing developments in classical mechanics by Arnold and Sinai, before chaos became popular. Percival insisted that semiclassical mechanics must take account of chaos. Later, we learned more about chaos from Joseph Ford. Of course Martin had paved the way with his trace formula for unstable orbits.

A persistent question was whether the formula could generate asymptotically high levels for a chaotic system. My opinions fluctuated. In 1976 I thought it could not, arguing that long orbits - required to generate the high levels - were so unstable that the Van Vleck propagator would not be valid for them. Instead, I thought (using ideas developed by Balian and Bloch) that periodic orbits could at best describe spectra smoothed on scales that were large compared with the mean spacing – but still classically small, so that some detail beyond the Weyl rule was accessible, though still not individual levels. This question is still not settled definitively, but my pessimistic opinion was changed by two developments.

The first was energy level statistics. In the 1970s, following a suggestion from Balazs Gyorffy, I imported from nuclear physics the idea that random matrices could be relevant in the quantum mechanics of chaos. The first application of this suggestion was not to chaotic systems at all, but to integrable systems, where it was shown – as I just mentioned – that the levels are not distributed according to random-matrix theory. That work inspired Allan Kaufman and Steven McDonald to the first calculation of level spacings for a chaotic system: the stadium. Then I did the same for Sinai's billiard. In those days we were fixated on the spacings distribution. My way of deriving level repulsion was a generalization of Wigner's: through the codimension of degeneracies. This gave the same result as random-matrix theory for small spacings, and explained the differences between the different ensembles, but gave no clue as to why random-matrix theory worked for all spacings, and why it was connected with classical chaos.

Then came Oriol Bohigas and Marie-Joya Giannoni and Charles Schmit. What they did, in the early 1980s, was simple but very important. They repeated the calculations that Kaufman and McDonald and I had done, for the same systems and using the same numerical methods, but instead of focusing on the one statistic of the level spacing they appreciated that the random-matrix analogy is much broader: it predicts all the spectral statistics, in particular long-range ones. They calculated one of these: the spectral rigidity (equivalent to the number variance).

Their observation was enormously influential. In particular, it was central to my construction in 1985 of the beginnings of the semiclassical theory of spectral statistics from Martin's atoms: the periodic orbits. Another crucial ingredient in this was also a development of periodic-orbit theory: the inspired realization by John Hannay and Alfredo Ozorio de Almeida that the Gutzwiller contributions of long orbits obey a sum rule whose origin is classical and whose structure is universal - that is, independent of details. Pure mathematicians (Margulis, Parry, Pollicott) had found similar rules - more general in that they applied to dissipative as well as hamiltonian systems, but also more restricted in that Hannay and Ozorio's theory applied also to integrable systems (where Tabor and I had found their particular result in 1977 but failed to appreciate its general significance). Thus periodic orbits were able to reproduce key formulas from random-matrix theory, and random-matrix universality found a natural explanation as the inheritance by quantum mechanics of the classical universality of long orbits. There was more: the periodic orbit theory of spectral statistics showed clearly and simply why and how random-matrix theory must break down for correlations involving sufficiently many levels. There were misty mathematical aspects – now being clarified – of those arguments, but the formulas were not misty, and were the first step in convincing me that long orbits in Martin's trace formula were meaningful.

Quantum spectral determinant (zeta function) for a particle confined between branches of a hyperbola, calculated exactly (dashed curve) and from a renormalized version [9,10] of Gutzwiller's sum over the unstable classical periodic orbits (full curve); the energy levels are the zeros, indicated by stars. Reproduced from [10], with permission.

The second step sprang from the realization - increasingly urgent in the early 1980s - that the series of periodic orbits in the trace formula does not converge. The cause was realized by Martin in 1971 [4]:

"Even more serious is the fact that there is usually more than a countable number of orbits in a mechanical system, whereas the bound states of a Hamiltonian are countable."

The failure of the trace formula to converge was emphasized especially by André Voros, who pointed out that this defect is shared by the formally exact counterpart of the formula for billiards with constant negative curvature, namely the Selberg trace formula. And later Frank Steiner taught us that trace formulas can sometimes converge conditionally, in ways depending delicately on the topology of the orbits (expressed as Maslov phases). Eventually these concerns about convergence led naturally to the study of zeta functions. The idea there is to find a function where the energy levels are zeros, rather than steps or spikes as in the density of states. The grandparent of all these objects is Riemann's zeta function of number theory. I learned its possible relevance to quantum chaology from Oriol Bohigas, and also from Martin's semiclassical interpretation of the Faddeev-Pavlov scattering billiard, where Riemann's zeta function gives the phase shifts [6, 7]. It is amazing that Martin had already realized the connection with zeta functions in his 1971 paper. He wrote:

"This response function is remarkably similar to the so-called zeta functions which mathematicians have invented in order to survey and classify the periodic orbits of abstract mechanical systems."

(He cited Smale). And in 1982 Martin explicitly wrote a semiclassical zeta function of the kind we consider today, and used it in conjunction with some tricks from statistical mechanics to sum the periodic orbits for the anisotropic Kepler system [7, 8].

A crucial ingredient turned out to be the Riemann-Siegel formula, that makes the sum over integers for the Riemann zeta function converge. I realized this in 1986, and later developed the idea with Jon Keating [9]; we were helped by André Voros's precise definitions of the regularized products in these zeta functions. The result was an adaptation of the trace formula to give a convergent sum over periodic orbits, soon employed to good effect by Keating and Martin Sieber [10] (see the figure). A related idea was the invention of cycle expansions by Predrag Cvitanovic and Bruno Eckhardt; in these, essential use is made of symbolic dynamics to speed the convergence of the sum over orbits. This application of coding to semiclassical mechanics was also originally Martin's idea: he used it in the 1970s and early 1980s to classify and then estimate the sum over the orbits, again for the anisotropic Kepler problem [7, 8].

The two applications of Martin's periodic-orbit ideas that I have just described, to spectral statistics and to zeta functions, were combined by Eugene Bogomolny and Jonathan Keating. This development, and more recent insights from Martin Sieber, Fritz Haake and Sebastian Müller, are taking the derivation of random-matrix formulas from quantum chaology to new levels of sophistication and refinement.

In the mid-1980s, Eric Heller discovered that for some chaotic systems the wavefunctions of individual states are scarred by individual short periodic orbits, in ways that depend on how unstable these are. From this came further extensions of Martin's ideas, to new sorts of spectral series of periodic orbits, not involving traces, and for Wigner functions as well as wavefunctions.

In spite of all this progress, we are still unable to answer definitively and rigorously the central question Martin posed in 1971 [4]:

"What is the relation between the periodic orbits in the classical system and the energy levels of the corresponding quantum system?"

Of course the trace formula itself is one such relation, but I am sure that what Martin meant is: how can periodic orbits be used for effective calculations of individual levels. For the lowest levels there is no problem, but – and again I quote from Martin’s 1971 paper -
"the semiclassical approach to quantum mechanics is supposed to be better the larger the quantum number"
and to reproduce the spectrum for high levels, using even the convergent versions of the trace formula that are now available, requires an exponentially large number of periodic orbits. This is a gross degree of redundancy unacceptable to anybody who appreciates the spectacular power of asymptotics elsewhere. Martin's old ideas continue to challenge us.

A few years ago, I refereed an application for research funding for a German-British collaboration. This required me to comment on the applicants' "timetable for research" and their "list of deliverables". I wrote "In science there are no deliverables; researches are not potatoes". Martin Gutzwiller ignored these toxic fashions. What makes him so attractive as a scientist is that he refuses to follow any fashion; instead, he generates ideas that become the fashion.

 

References

[1] Gutzwiller, M. C.,1967, The Phase Integral Approximation in Momentum Space and the Bound States of an Atom J. Math. Phys. 8, 1979-2000
[2] Gutzwiller, M. C.,1969, The Phase Integral Approximation in Momentum Space and the Bound States of an Atom II J. Math. Phys. 10, 1004-1020
[3] Gutzwiller, M. C.,1970, The Energy Spectrum According to Classical Mechanics J. Math. Phys. 11, 1791-1806
[4] Gutzwiller, M. C.,1971, Periodic orbits and classical quantization conditions J. Math. Phys. 12, 343-358
[5] Gutzwiller, M. C.,1973, The Anistropic Kepler Problem in Two Dimensions J. Math. Phys. 14, 139-152
[6] Gutzwiller, M. C.,1983, Stochastic behavior in quantum scattering Physica D 7, 341-355
[7] Gutzwiller, M. C.,1982, The Quantization of a Classically Ergodic System Physica D 5, 183-207
[8] Gutzwiller, M. C.,1977, Bernoulli Sequences and Trajectories in the Anisotropic Kepler Problem J. Math. Phys. 18, 806-823
[9] Berry, M. V. & Keating, J. P.,1992, A new approximation for zeta(1/2 +it) and quantum spectral determinants Proc. Roy. Soc. Lond. A437, 151-173
[10] Keating, J. P. & Sieber, M.,1994, Calculation of spectral determinants Proc. Roy. Soc. Lond. A447, 413-437

 

[Released: May 2012]