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We speak to the eminent researcher after whom the ‘Pancharatnam-Berry phase’ is named.

BY DEBDUTTA PAUL

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Michael Berry is Melville Wills Professor of Physics (Emeritus) at the University of Bristol, UK. He is a theoretical physicist renowned for his work in the intersections between classical and quantum physics. He is best known for discovering the geometric phase, often referred to as the ‘Pancharatnam-Berry phase’, when a system is subjected to cyclic adiabatic processes. The geometric phase has applications across various fields of wave physics. His awards include the Maxwell Medal and the Dirac Medal from the Institute of Physics, the Royal Medal from the Royal Society, the Pólya Prize from the London Mathematical Society, the Wolf Prize, and the Lorentz Medal. He received a knighthood in 1996. Prof. Michael Berry (MB) spoke to Debdutta Paul (DP) on his recent visit to ICTS-TIFR for the ‘A Hundred Years of Quantum Mechanics’ program.

The full text of the interview is reproduced below. The answers are lightly edited, and long paragraphs have been split for readability. The questions and initials are in bold.

DP: What is the status of the foundational questions in quantum mechanics now?

MB: I have no idea, I don’t work on them. I have a slightly negative view about attempts to, as people say, “interpret” or “understand” quantum mechanics. There are different formulations: you have the Schrödinger equation, you have the Heisenberg’s matrices, you have Feynman’s path integrals, you have Wigner functions, and then you have Many-worlds, Copenhagen, and so on. Now, I think the fact that quantum mechanics agrees so well with experiments means we already, in a sense, understand it.

Transport the question back to classical mechanics. Two points. Is Newton’s equation more fundamental than Hamiltonian’s? Philosophers could argue about it. In fact, Newton’s equations are more general, that’s another matter. But basically, it’s an idle question: in relevant examples, they’re equivalent. Nevertheless, classical mechanics is deeply mysterious. You ask somebody in the street, they would say: Why should the motion of planets and particles in mechanics be governed by second-order differential equations? It seems very abstract. People say the same about quantum mechanics. The mathematics is slightly less familiar, but now we understand it pretty well. So, it’s a matter of what you get used to. There’s a paper by Christopher Fuchs and Asher Peres with the wonderful title Quantum Theory Needs No ‘Interpretation’.

There is a positive way to think of some of these investigations. Quantum mechanics is certainly false. It’s absurd to think that in a few hundred years, a recently evolved species on a little planet somewhere, a long way from the centre of a galaxy, would have found the secret of the universe. It’s nonsense. It is, of course, a very good theory, but it will surely one day be revealed as a special case of something deeper. Just as, for example, classical mechanics is a special case of quantum mechanics with appropriate subtleties. So, of course, one can then ask: What’s a promising way to understand what lies beneath quantum mechanics?

The situation is very different from a hundred years ago when people were understanding what is beneath classical mechanics — that the answer is quantum. Different because then there were experiments all the time. I recently read a biography of Heisenberg. Almost every month, he learned about a new experiment and factored it into the matrix mechanics he was developing. Unfortunately, now there are no experiments that show that quantum mechanics is wrong. We have ideas where they might be wrong, like black holes, but we can’t do experiments. So, the situation is different now. One day, I’m sure there’ll be something fundamentally incompatible with quantum mechanics. It may have to do with its discordance with gravity. So, the positive aspect is getting beneath quantum mechanics. The negative aspect is we don’t have experimental guidance.

DP: Over the last couple of days, we have been hearing about quantum computations and quantum information. Will quantum computers be a functional reality soon?

MB: I don’t think so. I don’t work in this area, but my colleagues in quantum information do. We heard here that quantum information is much broader than a quantum computer. There are already applications of quantum information, even though we don’t have a quantum computer. As I’ve learned here, there will be quantum computers sometime, but the problems of scientific interest that quantum computers might solve are, at the moment, not very extensive. Quantum information is, however, a wonderful subject. I would put it this way: we know very little about the Hilbert space of more than two particles. That’s incredible! There are huge depths in quantum mechanics yet to be mined and understood. This understanding will transform the world in the way that quantum mechanics has already transformed it.

DP: Can you tell us about your work on the geometric phase in quantum mechanics? How do you see its significance evolving in modern physics and technology?

MB: When I wrote my paper, I was ignorant of many relevant earlier works. I had more or less the complete story, but there were many precursors. In India, of course, there is S. Pancharatnam, who I admire enormously. I had never heard of him. I knew all the other members of the brilliant Raman family, except Raman himself, who had died, and Pancharatnam, who had died. When I found the geometric phase, I thought it was some interesting little tweaky corner of quantum mechanics, but my colleague John Hannay said, “No, this is going to be important.” I didn’t understand that, and I haven’t followed all the applications because my interests have shifted. I’m somebody who, having started something — it’s happened several times in my career — having started something and more or less laid the foundations, I’m not very interested in pursuing it. Of course, I’m pleased to see these advances, especially with the geometric phase curvature, because I regarded that as the most interesting part of what I did in the 1980s. I’m pleased to see this playing a role in condensed matter physics — topological matter has curvatures appearing everywhere — but it wasn’t something I anticipated.

DP: What do you think will shape the future of quantum mechanics in the next century?

MB: Well, that’s an interesting question. Of course, there’s this vast Hilbert space, but there is an elephant in the room, and this is a good opportunity to speak about it.

In the 1920s, Dirac said we understand the whole basis of chemistry. Most chemists, even today, don’t use quantum mechanics, but quantum chemistry is still a thriving subject, so Dirac was surely right. Now, we physicists like to boast that quantum mechanics explains all matter, at least the kind of matter that we see on this planet, at these temperatures and pressures and so on. Think of the Quantum Hall Effect, topological matter, and quantum phase transitions — so many things understood. But the elephant in the room, and I mean this literally, is that on this planet, a lot of matter is alive.

Where is aliveness in quantum mechanics? I don’t speak about quantum biology, a beautiful subject concerning specific biological processes — photosynthesis, where we harvest sunshine; magnets in the heads of birds that help them navigate. Do these depend on quantum mechanics for their existence? Does quantum mechanics enhance them over what you would find classically? Does quantum mechanics inhibit them? These are wonderful questions, but the fundamental question is aliveness. Where is it in the gigantic Hilbert space of even the smallest living organisms, plus rules for interacting with the environment? It has to be there somewhere. Schrödinger, of course, raised the question and made some attempts, which accelerated modern molecular biology, but not the question as I’m asking it. Some very subtle correlation between different bits of the Hilbert space? We just don’t know. So, I think that’s an elephant in the room. I don’t doubt that it’s there. I’m not a vitalist, so this means there is a lot of matter that we don’t understand. In the coming century, I see this coming into sharper focus.

DP: Right, you’re talking about the phenomenon of emergence?

MB: Yes. There are many emergent phenomena, and a lot of my work has been concerned with this aspect. Now, there are aspects of emergence that are endlessly discussed by philosophers. How can one theory contain another at a different level of description? They miss a central mathematical point. Very often, certainly in physics, this is when some parameter gets small or large. For example, the number of degrees of freedom gets large, Planck’s constant gets small, and in fluid mechanics, viscosity goes to zero: it’s a singular limit because when the viscosity gets small, you get turbulence, which is unanticipated. Many phenomena in quantum mechanics involve singular limits: as you go towards a classical limit (I spoke about it), there is the random-matrix distribution of energy levels and so on. The point is that interesting limits in physics are singular. It means that living in the borderlands are phenomena that were not envisaged. This is emergence. It’s a mathematical problem. Different techniques are being developed in different areas. Asymptotic semi-classical techniques, renormalization in thermodynamics and statistical mechanics, where the number of particles is large — and so on. Life is, of course, an emergent phenomenon, but that doesn’t mean we can’t understand it. We need some analogue of renormalization or asymptotics, as yet unimagined.

DP: Coming to the random-matrix theory, why are the predictions of random-matrix theory so robust in spite of the microscopic world, the physical systems in that world and their interactions being so very different from random matrices?

MB: Well, exactly, and that’s work we did also in the 1980s (around the same time as the geometric phase). Universality in random-matrix theory, as applied to quantum systems with chaotic classical counterparts, comes from classical universality. That’s the origin of it.

And that classical universality concerns the distribution of very long periodic orbits, as I described in my talk. There’s a sum rule from John Hannay and Ozorio de Almeida which has this intuition underlying it:

You take a given energy and one long periodic orbit; it will wind in a complicated way around the energy surface. Take all of them. There are exponentially many. They will cover the energy surface with the slightest coarse-graining, uniformly with a microcanonical distribution. And that classical sum-rule underlies the universality of random-matrix theory when applied to Gutzwiller’s trace formula for the density of energy levels of a quantum system, which is chaotic. It’s a sum over the periodic orbits of the system, and it’s the long orbits that determine random-matrix theory. And that’s why it’s robust: because it’s universal.

Random-matrix theory has its limits. Over long ranges of correlations, random-matrix theory is false, not universal because short periodic orbits are different from system to system. They don’t cover the energy surface uniformly. So, you can’t use that sum rule. We understand this now. It’s a question of more refined asymptotics. If you look at nearby levels over a range (and we know exactly in terms of Planck’s constant, how big that range can be), random-matrix theory works; it’s marvellous and universal. But if you go outside that range and are still semi-classical, still small compared to where you are in the spectrum, then the correlations which are sensitive to long ranges, like the pair correlation and so on, are not universal. They differ from system to system.

DP: Is there any deeper reason for the universality to hold for long periodic orbits?

MB: Yes, it’s the Heisenberg uncertainty principle. It’s the rule that large energy ranges correspond to short orbits, which are different from one system to another. So you don’t have the astonishing unanticipated fact that at shorter ranges, you get universality.

DP: What can the modern point-of-view of doing quantum many-body physics and quantum information science using gate-based circuits learn from classic random-matrix theory?

MB: I don’t know. I don’t know how to talk about things that I don’t know about.

DP: Do you have any advice for people who work in this field or who aspire to work in this field?

MB: Yes. I have two contradictory pieces of advice for people who ask me for career advice.

The first piece of advice is: don’t take advice.

But, if pressed, I would say that if I were starting out, I would probably work on quantum information. Probably, though I can’t tell — this is what philosophers call counterfactual history. So I would say: work on quantum information. There are so many riches to be uncovered there to do with these big Hilbert spaces, even with a modest number of particles. So that’s what I would say.

DP: Thank you so much for your time, Prof. Berry.

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The author thanks Ananya Dasgupta, Spenta Wadia, and Sthitadhi Roy for inputs and suggestions.

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Header photograph by Debdutta Paul.