Sculpting Order Out of Quantum Chaos

NA: My father studied at the Tata Institute [TIFR] in Mumbai, and he left for France at the time of starting a Ph.D. with the intention of working under the supervision of Pierre Samuel. My parents met in Paris at the end of the 1960s. My mother still has a letter from her friend, the mathematician Michèle Vergne, who writes in that letter that she wanted to introduce her to a young Indian who had just then arrived in France.

NA: I always understood mathematics very well—but the same was also true for other subjects at school. I remember a few notions that I found difficult to grasp at first (conversion of units in physics, for instance), and on these occasions, I asked for explanations from my parents. But generally, I liked to work on my own and did not need much help.

NA: I grew up in Orléans, a city 120 km south of Paris. I was always ahead of my class. In math and sciences, it required almost no effort. In the humanities, though it required a lot of work, I was amongst the best pupils. In physical education, I was always among the worst: I had the experience there of not managing to progress, and having the impression that if one was not born gifted, there was no method that allowed one to improve. I now realise that many students think the same way about math, and I wish that the school system could provide more methods of progression for those who are in difficulty.

NA: I never participated in math competitions. No one suggested it, and I didn’t even know those competitions existed. I’m not sure I would have liked the idea of “training” for such a competition. My favourite pastime was music and I played the piano, and actually once took part in a piano contest in Paris, at the age of eight. But, I was extremely stressed by the whole experience, and so my parents decided that it was perhaps not a good idea for me to pursue it any further. I have good memories of that even now because of the trip to Paris.

NA: I wanted either to be a pianist or a researcher, but not necessarily in math. Physics, and even biology, also fascinated me then.

NA: I personally think that the years between 12 and 14 are really difficult years for girls. The body undergoes important transformations, and I don’t think one can go through them in full serenity. I think all of my girl friends at that time also felt the same way. I don’t have good memories of those years, as I think of them as some kind of a trial that I had to go through. Afterwards, it gets much easier as one approaches adulthood. I actually started my undergraduate studies at the age of 16 and moved to Paris. The work there was tough, but that is what I had wanted. The years after that were good years, where I learned a lot of mathematics and physics. I learned to struggle with difficult problems, and I actually liked that struggle. I gained a lot of confidence.

NA: I was 16. Going to Paris to study was my dream really, and to me, it was the best place ever to study. There was no reason or desire to go elsewhere. Not many parents accept the idea of sending a girl to live and study alone in another city, and that too at the age of 16! I think my parents too were not completely sure that it was a good idea, but I myself had no doubts and managed to convince them.

NA: I wanted a subject at the crossroads of several branches of mathematics, and this is an inclination that I have always had. I had really liked the ergodic theory course, where J.P. Thouvenot had introduced the basic concepts of ergodic theory and had also shown how these were used in Hillel Furstenberg’s proof of Szemeredi’s theorem in number theory. I also knew from François

Labourie’s course and the ergodic theory seminar that hyperbolic geometry was deeply related to the dynamics of the so-called “geodesic flow”. I think, around this time, someone advised me to meet François Ledrappier. I didn’t really know at that time that he was a very well-recognised mathematician. He first gave me a bunch of papers to read about random walks on hyperbolic groups. Later on, he asked me to understand a preprint by Dmitry Dolgopyat on the exponential mixing of geodesic flows. He always let me decide by myself the ideas that I wanted to pursue, and the questions I wanted to study. This was rather unsettling at first, but I am very grateful to him for this.

NA: Honestly, I met him a little by chance, following the advice of another professor who knew my tastes. I appreciated his modesty and his gentleness from the start.

NA: I liked theoretical physics and studied it together with math while I was an undergraduate. But some visits to an experimental lab convinced me that I actually preferred math. When I looked for a Ph.D. subject, I did not particularly want it to be at the interface with physics, and in fact, it was not. Indeed, my Ph.D. was in the field of dynamical systems, where I worked on understanding the fine topological features of geodesics on negatively curved manifolds. It is only in 2012, when I won the Poincaré prize, that the words “mathematical physics” became attached to my work. I am not always comfortable with this, as I do not have a particular wish to always stay and work at the interface with physics.

NA: Imagine a ball rolling on a billiard table and bouncing off the edges. In the idealised case where there is no friction, this motion will go on forever. This is the epitome of a “dynamical system”. This example seems a little frivolous, but it actually serves as a paradigm for the study of more relevant dynamical systems occurring in nature (the solar system for instance, where the planets could be considered as billiard balls subjected to the gravitational attraction of the Sun). The main questions here that mathematicians have to consider are: Can one solve the equations of motion explicitly, and with the right choice of coordinates? Does the system have a periodic behaviour? Is the motion stable or unstable, and in the latter case, how can one understand chaotic behaviour? Is there a phenomenon of convergence to equilibrium? And so on…

NA: Well, clearly, the motion of the billiard ball depends a lot on the geometry of the billiard table. For a table with a smooth, convex boundary (like a circular billiard table), KAM (Kolmogorov–Arnold–Moser) theory tells us that there are many trajectories that behave in a quasi-periodic way. On the other hand, if the boundary of the billiard table is piecewise concave, the motion is chaotic and becomes highly unpredictable.

Poincaré was one of the first to develop a clear picture of the relation between geometry and dynamics, and it was he who first perceived the mathematical meaning of “chaos” in his work on the three-body problem. He often intertwined physical intuition with mathematical reasoning. Jacques Hadamard and George David Birkhoff’s works are also fundamental in order to understand the interplay between geometry and dynamics.

NA: Integrable means that many physical variables are preserved by the dynamics; so that, after a change of coordinates, the other variables (called angle variables) evolve in a quasi-periodic way. In other words, integrable means that the motion is completely explicit and predictable. Circular or rectangular billiards are integrable; the simple pendulum or the two-body problem (one planet gravitating around the Sun) is integrable too.

Chaotic, on the other hand, means that the motion behaves like a random system. This can happen even if the equations of motion are deterministic! If there is even a tiny uncertainty about the knowledge of the initial conditions, one cannot predict the long-term behaviour of the system any better than, say, by actually throwing dice. Counter-intuitive as it may seem, there are simple physical systems, such as the double pendulum, that are chaotic. The three-body problem is chaotic in a certain regime of parameters.

The existence of chaotic behaviour came as a surprise in Poincaré’s work on the three-body problem, at the end of the 19th century. But it was realised little by little that chaotic behaviour is actually quite generic in dynamical systems. Mathematicians developed a theory of chaos all through the 20th century.

The problem of non-integrability appeared to be a crucial one in quantum mechanics from the beginning. In 1913, Bohr formulated his famous “quantisation conditions” to describe the spectrum of the hydrogen atom. The “classical” image of a hydrogen atom is an electron gravitating around the nucleus. This is similar to the two-body problem. Einstein realised that the fact that this is completely integrable is fundamental for Bohr’s conditions.

When physicists tried to formulate quantisation conditions for the helium atom (two electrons gravitating around the nucleus), they ran into deep problems due to the fact that the three-body problem is not completely integrable. They read Poincaré’s work on celestial mechanics with the hope that it could be of use for the helium atom, but later realised that it was necessary to start afresh with a new theory. This was the beginning of modern quantum mechanics.

NA: This is a little technical to explain, and in fact, the statement is also not exactly true. It is supposed to mean that you get the same answer if you either start with a classical physical quantity (say, the position of an electron), transform it into a “quantum observable” by the mathematical operation called “quantisation”, and then let it evolve under the laws of quantum mechanics. Or, first, evolve the classical physical quantity under the laws of classical mechanics, and then apply the operation of “quantisation”.

This is only an approximation, and is called the “semi-classical approximation”, valid in a certain regime of parameters: the wavelength of the wave function should be small enough, and the time of observation should not be too long. This is formally equivalent to saying that “the Planck constant h goes to 0”. In this regime, it is almost correct to imagine that waves move according to the classical laws of motion. This is what we commonly do, for instance, when we think of light moving in straight rays, as though it was a particle, completely forgetting its undulatory (quantum) nature. But in other parameter regimes, we observe interference and diffraction, which are actually non-classical phenomena.

NA: As just explained, the word “semi-classical” stands for a vision of things where the classical and undulatory aspects are mixed. Although we are studying a wave, we use notions from classical mechanics to understand it better, because classical notions are simpler to grasp.

The notion of an invariant measure is absolutely central to the classical theory of dynamical systems and is in fact at the very heart of ergodic theory. An invariant measure is a probability measure on the set of initial conditions, and which remains invariant under time evolution. It is also a potential candidate for the notion of an “equilibrium state” of the system.

In quantum mechanics, there is also a notion of invariant states which are eigenfunctions of the evolution operator. In the semi-classical regime, the quantum invariant states approach classical invariant measures. The latter are called semi-classical measures. In a nutshell, semi-classical measures are a special subclass of invariant measures, those that approximate the quantum invariant states.

NA: Ergodic theory studies the evolution of dynamical systems, from the point of view of measure theory, the central object being the notion of an invariant measure. A dynamical system with an invariant measure is said to be ergodic if it is irreducible in the measure-theoretical sense: that is, if it cannot be decomposed into two invariant sub-pieces of positive measure. By Birkhoff’s ergodic theorem, this is equivalent to saying that the time-average of any physical quantity, over a randomly chosen trajectory, coincides with its average over the whole phase space (thus making the crucial connection with Boltzmann’s hypothesis, as in the kinetic theory of gases).

Poincaré perceived the mathematical meaning of “chaos” in his work on the three-body problem

There are not-so-many known techniques to prove ergodicity. Most often, one starts by proving that there is exponential sensitivity to initial conditions. This is the property called “hyperbolicity”, a precise mathematical formulation of the concept of “chaos”—a word which is itself not used as much by mathematicians, except perhaps to please journalists!

NA: At that time (1917), the description of quantum mechanics was still mostly classical. Niels Bohr started from the classical planetary model of the hydrogen atom and added the axiom that the kinetic momentum should be “quantised”, that is, could only take values in a discrete set. This explained the discrete spectrum of the hydrogen atom, and the next task was to find similar quantisation rules for systems with larger atoms. Einstein found some general quantisation rules for Hamiltonian systems, but he simultaneously noted that these rules also almost implicitly assumed that the system was completely integrable and thus, could not really apply to ergodic systems. It is not clear to me whether he found this question important, or if it was just an incidental remark.

NA: To perform the operation of “quantisation”, we first need to consider a symplectic dynamical system on a symplectic manifold. Arnold’s cat map—a linear map on the two-dimensional torus—is a simple example of such a map. To quantise it, we need to build a Hilbert space and a unitary map, whose iteration constitutes the “quantum dynamics”. This also needs to satisfy the principle that “quantisation commutes with evolution” asymptotically in the limit \hbar \to 0. For the cat map, quantisation can be done either via the Weyl formalism (using the fact that the cat map corresponds to the time t=1 of a linear Hamiltonian flow on {\mathbb R}^2, and using the corresponding Schrödinger equation) or by the Berezin–Toeplitz formalism. In both cases, the mathematical object that we get is a unitary matrix acting on a finite-dimensional Hilbert space, whose dimension is N=1/h, and this goes to infinity in the semi-classical limit.

NA: It is believed that one can classify Hamiltonian systems into “universality classes”, families that share a number of common features. The two universality classes that are well-identified are, on the one hand, completely integrable systems, and on the other hand, chaotic systems. In between, there are intermediate classes that are not so well-identified. For completely integrable systems, it is believed that the quantum spectrum should obey a Poissonian statistics and that the eigenfunctions of the Schrödinger operator be localised. For chaotic systems, the quantum spectrum should obey the Wigner statistics, and the eigenfunctions should be delocalised. These ideas formed little by little, by observing specific examples. Unfortunately, we are very far from being able to give rigorous mathematical proofs of these beliefs, and it is also very difficult to study most examples other than through numerical methods.

NA: In my papers with Stéphane Nonnenmacher, where we used the notion of entropy to measure the fact that eigenfunctions of Schrödinger operators have a disordered behaviour, we used the word “delocalisation”. When writing this, I didn’t have in mind any connection with Anderson localisation. Anderson localisation is a property of infinite quantum systems, and the expression refers to the fact that in certain circumstances, the eigenfunctions decay very fast at infinity, and thus are essentially localised in a finite portion of space (meaning that the system is an insulator). In my own work though, I was studying a compact phase space, and by “delocalisation” I meant that the eigenfunctions could not be localised in a tiny portion of phase space, or equivalently, that they possess a very small dimension.

It is only recently, after my own work on graphs, that I clearly realised that the two questions could actually fit within a common framework: namely, that in the study of Anderson (de)localisation, one can approximate an infinite system by a sequence of finite systems, each of size n growing to infinity; and this actually constitutes a sort of a semi-classical approximation.

In the study of the Anderson model, one observes that in the universality classes I mentioned before, the localised phase corresponds to Poisson statistics, and the delocalised one corresponds to Wigner statistics. The latter effect is only observed numerically; also, whereas the localised phase is by now well-understood, the delocalised one still resists the efforts of mathematicians.

NA: Mathematicians and physicists don’t have the same definition of “scars”: for mathematicians, the definition is more precise and also more restrictive. Physicists speak of a scar as soon as they see a wave function that is enhanced in the vicinity of a classical unstable periodic trajectory. This is mostly a visual definition, something you see on pictures coming from numerical simulations. Mathematicians, on the other hand, have a more precise definition (“strong scars”). Having a precise definition is necessary to prove or disprove the very existence of scars: the probability density defined by eigenfunctions should have a positive mass in the vicinity of a periodic orbit, and this should also persist, even in the semi-classical limit.

According to the Shnirelman theorem (or the quantum ergodicity theorem), if the system is classically ergodic, then most eigenfunctions have a uniform distribution over phase space, and hence do not have strong scars. Thus, the scarred eigenfunctions, if they exist, form a sparse family.

According to the quantum unique ergodicity conjecture (still unproven except in the so-called “arithmetic” case), if we are on a negatively curved manifold, then scarred eigenfunctions should be absent; in fact, the conjecture predicts that the full family of eigenfunctions becomes uniformly distributed over phase space in the semi-classical limit.

NA: The motivation was mostly theoretical and came through discussions with colleagues. Working with discrete models is not necessarily simpler than with continuous models, but the change of perspectives brings in a lot of new ideas, especially with the possibility of working with random geometric models.

NA: Ramanujan graphs are at the crossroads of several streams of ideas, related to number theory, graph theory, probability, etc. They are optimal graphs in terms of the rate of mixing of the simple random walk, and I believe that they should also have optimal properties in terms of the delocalisation of eigenfunctions.

NA: Non-backtracking paths on a graph may be thought of as local geodesics, so a non-backtracking random walk may be seen as a discrete analogue of a geodesic flow. It can also be used to define a kind of discrete analogue of the wave equation.

NA: I must say that administration is taking more and more of my time away from math, and so I don’t really need any distraction from research at all. My favourite pastime is playing the piano, i.e, western classical music. I have actually never stopped practising since I was 7, except at the birth of my son. Even now, I am trying to find more time for it. I mostly like the German composers from the 19th century, and Beethoven above all. I do read a lot, and nowadays actually enjoy sculpting clay, more and more.