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.