## Connecting the micro and the macro

Matsui, an associate professor at the Department of Mathematics at the Faculty of Science, teaches linear algebra to her students, while her research theme descriptions are dotted with terms like spin and quantum fields, giving off physics vibes. Is Matsui a mathematician? Or is she a physicist?

“I consider myself a physicist. My research field is on the boundary between mathematics and physics, however. So new mathematics may be born as we use our physical intuition to advance our research.”

Matsui smiles as she says this, but the listener's mystery only deepens. The boundary between mathematics and physics - what kind of world could that be?

Who triggered her interest in science in the first place was her fifth-grade homeroom teacher with his brilliant scientific explanations.

“He was the kind of teacher who taught us various things beyond the limits of elementary school and used scientific theories to explain everyday phenomena. I began to feel that science was interesting. In junior high and high school, I thought that if I studied physics properly, I would be able to understand the other scientific fields as well.”

Underlying this, of course, was a simple and unquenchable passion to comprehend the universe and uncover its mysteries.

She entered the Faculty of Science at the University of Tokyo to study physics, but “the undergraduate classes were very difficult. Especially when the explanations were simply given by the phrase “because of the principles of physics.” So, I could not grasp the material well,” she says. Matsui was particularly attracted to statistical mechanics, which she decided to study in graduate school.

Statistical mechanics may be called the link between the microscopic and macroscopic worlds. Or perhaps it can be said that statistical mechanics explores how the phenomena that occur in the microscopic world appear in the macroscopic world.

“From a microscopic point of view, each particle moves according to Schrödinger's or Newton's equations. For example, the equations can be solved for the motion of a single particle, but when the number of particles is extremely large, such as Avogadro's number, which is 10 to the 23rd power, it is impossible to solve the equations for all their motions. However, statistical mechanics is a way to use statistical knowledge to extract from the laws of the microscopic world the properties that emerge only when there are many particles gathered together in a so-called many-body system.”

In other words, how do particles in the microscopic world that move according to the laws of quantum mechanics behave? What kind of phenomena do they cause when they are in a cluster of tremendous numbers? Statistical mechanics is the study of the properties of such a macroscopic world using the concepts of probability and statistics.

Matsui continued to study and research statistical mechanics in graduate school, but her interest eventually shifted more towards mathematics.

“Quantum mechanics and statistical mechanics are the two major fundamental theories in physics, and both theories reach a point where they must be worked out mathematically. However, there are various ways of “working them out”: some are mathematically rigorous, and some are not rigorous but can be approximated. Some are not even approximations, but numerical calculations. Among these, I am very interested in the mathematically rigorous ones. I wonder what kind of classes can be treated rigorously and what properties emerge only because they can be treated rigorously. That is how my interests shifted, and why I moved to the Department of Mathematics.”

What on earth are these things that can be treated mathematically rigorously? The mystery deepens…

## Exactly solvable models?

One of Matsui's major research themes is "solvable models.” It seems that these can be “treated rigorously.” But first, let us ask her to explain what a model is.

“In physics, when you have a certain phenomenon or substance that you want to observe, you don't just represent it in its entirety in a mathematical equation. You try to make it so that only the phenomena or properties you want to focus on stand out. By doing so, it may not be possible to describe other properties, but it will be possible to successfully describe the part that you want to extract. I think that organizing in such a way is ‘modeling.’”

In other words, to understand a certain physical phenomenon a mathematical framework is constructed to describe it without contradictions. That is what “solvable” stands for in “solvable models”- that is, the model that is capable of “solving” a problem in a mathematically rigorous manner (to be precise, it is a generic term for multiple models).

“There are many different types of methods for studying models in statistical mechanics. Among them, when it comes to solvable models - models that can be solved rigorously and systems in which various physical quantities can be calculated rigorously - the relationship with mathematics is very deep. I wanted to focus my research on such models in particular.”

When asked to give an example of what she found interesting about solvable models, Matsui replies like this.

“For now, statistical mechanics can only describe systems in equilibrium, where there is no change on a macroscopic scale. There are several principles in statistical mechanics. For example, when a state consisting of many particles (e.g., gas molecules) in a box is in an equilibrium. If you observe the microstates in the box, there are many patterns. However, if these many microstates are equal in energy, they are almost indistinguishable from each other on a macroscopic scale. This is called typicality. The principle of equal a priori probabilities states that all microstates of equal energy have the same probability of being realized, and that even if some microstates differ from the equilibrium state, their weight in the total is so small that the exceptions can be ignored. Statistical mechanics is a description based on these statistical ideas."

To put it another way, for example, the boiling water in a kettle that has been turned on is in a non-equilibrium state. However, once the kettle is switched off, and the water cooled down to room temperature, it is considered to be in an equilibrium state. Typicality means that the water molecules in the kettle can be in various states of motion at various speeds, but in most cases, the macroscopic quantities take on the same value. The principle of equal a priori probabilities means that if the overall energy in the kettle is the same, each water molecule can be in a state of motion at different speeds. It means that all states of motion have the same probability of occurring.

“In the description of this equilibrium system, if we consider solvable models, like the ones I am working with, we have to replace the simplicity of “equal energy.” Besides energy, the other macroscopic quantities called “conserved quantities,” must take the same values for all microscopic states. The number of conserved quantities increases as the system becomes large.with “In this case, the number of micro energy states corresponding to a certain equilibrium is small, so the statistical approach cannot be used. Being solvable means that the symmetry becomes very high, which means that there are many conserved quantities. When that happens, various conditions need to be considered. We cannot collect enough number of macroscopically indistinguishable microscopic states to be able to use a statistical approach. Therefore, there is talk that a system like this may be unexplainable by the conventional ideas of thermal equilibrium. This research area has been booming recently, and I find it very interesting.”

Matsui also says that there is another interesting aspect of the project when we move away from the viewpoint of statistical mechanics.

“A solvable model is a rather abstract version of the physical model. It is as if the mathematical structure behind a physical phenomenon had been extracted. For example, a phenomenon in nanophysics can be described by the same solvable model as a phenomenon on the cosmic scale (high-energy physics, string theory, etc.). Two seemingly unrelated physical phenomena can share a common mathematical structure. This is another very interesting aspect of solvable model research. I sometimes wonder if someday we will be able to understand everything from the microscopic (atomic level) to the macroscopic (cosmic level) level all at once through the study of solvable models. Of course, the real world is much more complicated than that, so we can only extract the essence of it by abstraction.”

Matsui is smiling happily as she is saying this. But the mystery remains a mystery.

## This is the boundary area between mathematics and physics

Matsui, who considers herself a “physicist,” explains why.

“For example, even if the research target is the same, there is a difference in motivation between mathematicians and physicists, in my opinion. When I hear about physics, I reflexively respond in a way that is linked to physical intuition or physical phenomena, such as “Oh, what would happen if we transformed it into this kind of model,” or “What would happen if the result was in this kind of quantity?” I think like a physicist. On the other hand, I have the impression that those who are interested in what kind of mathematical extensions are possible beyond that, and whether new mathematics will emerge, are the mathematicians. Even though I am a “physicist,” I hope I can contribute to both fields.”

Matsui describes her contribution so far as follows.

“I am studying solvable models that can be solved exactly. There will always be situations where new mathematics will be needed, even if we extend the research from the point of view of physics. I don't just mean mathematics that is completely new, but also mathematics that is already known in a completely different context. There are many such cases. For example, in my recent work on the spin model (a model in which atoms interact with each other according to the direction of their rotation), I asked a question that had been unsolved for a long time: “Can you detect a spin current when you first disturb the spin configuration and then wait long enough?” It turned out to be a mathematical problem. Others had previously proven the existence of a non-zero spin current. By using a certain mathematical quantity, I was able to show the saturation of the inequality. The method had been discovered in a completely different field of mathematics and no one expected it to be used in this spin model. The mathematicians themselves who had developed the method had no idea that it would be used in this way. I think that being able to find and use mathematics in this way would not have been possible, had I done solely physics or solely mathematics research. This crossover is an important aspect of doing both mathematics and physics, and I think that is what I can contribute to the field.”

Does this mean that this is the boundary between mathematics and physics?

“My research field is what is called integrable systems, and recently more and more people are becoming interested in this field. There used to be a time when it was shunned because people said it was too close to mathematics and unrealistic. Integrable models sometimes include unrealistic interactions to maintain mathematical order, and this was said to be arbitrary and not conducive to experimentation. However, as experimental techniques have improved, the interactions that were said to be unrealistic could be reproduced experimentally, and the unique properties of integrable systems have become clear. In the past, when I gave a presentation, the room was quiet, but now I get a lot of questions.”

Matsui continues to take on challenges even now. Her dream problem is still out there, waiting to be solved.

“The solvability (whether the behavior of many particles can be treated mathematically rigorously) of closed systems has been fairly explored already. However, much of the solvability of open systems, i.e., systems that interact with the outside world, is not yet known. There are examples of solvable models that are “miraculously” solvable: that is, we do not know why they are solvable. I want to formulate them mathematically. This is what I have set my sights on. Because the classical system is basically a limit case of the quantum system, my approach is to create a theory starting close to the classical system, using it as an entry point. At least that is what I believe and that is why I keep toiling away. It’s not going that smoothly, however.”

Still, to continue research while constantly having to go back and forth between mathematics and physics must take a tremendous amount of effort.

“No, I enjoy it very much. But I do feel a sense of guilt and anxiety that I am not doing justice to either discipline.”

## A physicist's intuition and inspiration

Matsui is the mother of two young children. Her partner is also a researcher, specializing in superconductivity and non-equilibrium systems. Do the two of them, researchers in the same field of physics, ever get into arguments at home about their fields?

“We do (laughs). We do argue a lot.”

…As expected.

“When I have discussions with people outside my family, I listen carefully to their opinions and purposefully try to understand what they are saying. However, when I argue with those close to me, I can get impatient and talk back saying “I know that already” (laughs). But it is actually very enjoyable to have someone at home with whom I can discuss things at any time. My husband picks up our children from daycare, prepares meals, etc. We share the responsibilities of taking care of our family.”

Matsui, herself a “model” female researcher in a way, has this message for women who want to become researchers.

“A world of fun awaits you if you follow your interests. I do hope you take the opportunity to step into this world.”

As for mathematics, one of the “fun worlds” for Matsui, she describes its appeal as follows.

“The main attraction of mathematics is that it may allow us to understand in our own way how the world came to be. Plus, I am not very good at languages, but mathematics is a universal language. Although it is difficult, it is written in a way that, if followed carefully, can be understood by anyone, regardless of culture or language. I think that is what makes it so appealing.”

That being said, it is still a line of unfamiliar symbols…

“How about thinking like this? The reason why mathematics seems so difficult is because we tried to write it in a way that everyone could understand, and it became difficult. That is why it should not be so difficult in the first place. However, because the formulas are complicated and the symbols used are special, people tend to shy away from it. But it is just written according to certain rules. Once you’re familiar with them, you will not perceive many things as difficult as you previously did. By the way, there are symbols that I still am not familiar with (laughs).”

Matsui says she prefers to do calculations “by hand” rather than on a computer. She used to calculate by writing on paper, but now she prefers using a tablet. She likes the fact that she can erase a mistake right away. However, she says that the feeling of despair she gets when she finds out that all of her lengthy calculations are actually wrong is “unbearable.”

“I have a physics background, so I sometimes have a intuition of how things should be. When that intuition turns out to be true and a problem that could not be solved before can be shown with a mathematical formula, it is profound and inspirational. It's like, “I knew it was going to be like this! I knew it!” However, that doesn't happen that often. If it did, I would be writing many more papers.”

Matsui grins with delight again.

※Year of interview: 2023

Interview/Text: Minoru Ota

Photography: Junichi Kaizuka