The particles and the whole

As a thought experiment, picture a shoe box. Inside the box, tiny pellets zip around and interact with each other. Let’s say that they repel each other, like electrons in metal. The closer they are packed together, the stronger the repulsion, to the point at which the motion of a single pellet affects the motion of all the others: “Thus, every particle affects the whole,” says Christian Hainzl from the Department of Mathematics at LMU. The Professor of Mathematical Physics tries to tame the behavior of huge ensembles – so-called quantum mechanical many-body systems – by means of mathematical formulas.

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In fact, many interesting effects arise in quantum mechanics from the collective behavior of an extremely large number of interacting particles. Take electrons, for instance, which act as an ensemble under certain conditions and become a superconductor, through which electricity flows without resistance. In the future, such high-temperature superconductors could revolutionize the energy industry and electrical engineering. Or take gas particles, which become a Bose-Einstein condensate at very low temperatures, where individual particles can no longer be distinguished or localized. The atoms or molecules form a unit; they are effectively everywhere at once and move in lockstep.

“Once we depart from the microscopic scale, the properties of such quantum mechanical many-body systems become scarcely comprehensible,” says Hainzl. Although there are mathematical models which precisely formulate quantum mechanical effects at the microscopic level, they would not furnish any usable information for the macroscopic behavior of the ensemble. Nevertheless, in theory at least, the behavior of the system as a whole should be inferable from all the positions, states, and interactions of the individual particles.

In 2023, Hainzl was awarded German Research Foundation (DFG) funding through a Collaborative Research Centre (CRC) for the work he has been single-mindedly pursuing for many years. Hainzl is spokesperson of the CRC, which was founded to research “the various facets and types of correlations and entanglements in many-body quantum systems” from different mathematical perspectives. The scientists intend in this manner to further develop mathematical methods and ultimately contribute to the advancement of quantum technology.

Describing reality with an equation

As with all physical systems, scientists seeking to describe a particular many-body quantum system look for a formula that describes the energy development of the ensemble over time – or the Hamiltonian function as it is called. For quantum mechanical systems, you can use the Hamiltonian to derive the famous Schrödinger equation. This partial differential equation describes the change of a physical, non-relativistic state according to the rules of quantum mechanics. For physicists, the Schrödinger equation is one of the most important formulas there is.

A fundamental component is the Hamiltonian operator – a matrix that, when suitably simplified, reveals the energy values of the system. For example, the operator supplies the energy levels of electrons in atoms or molecules. “As soon as correlations are in play, however, the states of each Hamiltonian operator of a many-body system are very difficult to determine,“ says Hainzl. When the number of particles gets extremely large, it actually becomes impossible. That being said, sometimes there is a workaround: You have to ignore all the interactions that ultimately do not affect the collective behavior. With a little luck, the unbelievably complex system becomes simple enough to be mathematically described.

And how would one go about doing this? As physicists do, Hainzl defines the types of interaction of the particles and the potential – that is, so to speak, the effect of a force field on the masses and charges of the particles. For his deliberations, he also uses the hypothetical box mentioned at the outset: To model the real situation as well as possible, he lets the density of the particles contained within tend to a limiting case – in many instances, this is infinite, such as for electrons in metal. Now, although the complexity of the system increases astronomically, some terms of the equation of state tend to zero and drop out. With a bit of skill, such maneuvers can be used to describe the system mathematically in the end.

Not just conjectures, but proof!

Physicists are also wont to disregard certain factors, such as interactions, that affect the overall picture only marginally or not at all. In many cases, this enables them to capture the observations made in reality with relatively high precision and predict the outcome of experiments. But mathematicians like Hainzl are not satisfied with this approach: “The state descriptions of physicists are mostly just a form of conjecture from the mathematical viewpoint,” he says. “Our job is to prove that you can really omit certain terms to get a correct description of the system. Or to show for which cases the simplifications undertaken are actually valid.”

To determine the correlation energy of electrons in metal, for example, physicists have simplified the interaction term to such a degree that only the interaction of individual pairs bound together in a certain way remain. They treat these pairs as bosons, which has the effect of simplifying the Hamiltonian operator so much that they can derive the energy values from it. “Mathematicians like myself are now proving that the terms they discard actually do tend to zero at high particle densities.” The idea that these pairs behave effectively like bosons also needs to be demonstrated. “Proving such things with due mathematical rigor is very difficult,” emphasizes Hainzl.

It might sound clichéd, but in his daily work, Hainzl scribbles rows of mathematical constraints, interactions, summation formulas, integrals, and boundary conditions on paper or his whiteboard, crosses things out, corrects them, and starts again. “Many of the mathematical methods required for proofs do not yet exist and have to be developed first,” says Hainzl.

He regularly exchanges ideas with colleagues or stands with them in front of a whiteboard puzzling over equations which read like gobbledygook to non-mathematicians. This thought process can drag out over months or even years. “Heaps of balled-up paper fill the wastepaper basket during this time,” recounts Hainzl. Again and again, he thinks he is close to his goal only to realize the next day that he made a mistake in the proof and that a term cannot be ignored as he had supposed. “As a mathematician, you need a high frustration threshold,” he says.

And when you’ve finally come up with a rock-solid proof, that’s usually not the end of the work: “Often the corresponding formulas are very complicated and several pages long. Bit by bit, you have to go about simplifying them,” he points out. After all, only when a proof has been wrestled into relatively compact and tidy form will peers take the time to check it. All those years of thinking and solving should not end up as a journal article that nobody reads, Hainzl observes. Rather, the hard work is done so that the ideas and approaches are picked up and developed by the mathematical community.

Physics fertilizes mathematics

Physical problems have enriched mathematics on various occasions in the past. For example, important inequalities of analysis, which have since obtained classic status, were first discovered in the context of quantum physics. The mathematical proof that systems of atomic nuclei and electrons do not collapse gave rise, for instance, to the inequality for estimating the sum of all eigenvalues of a matrix. This was a big deal for mathematics.

In addition, mathematical results on the nonlinear Schrödinger equation had a major influence on other areas of partial differential equations and on harmonic analysis. And mathematics and quantum mechanics have long been fertilizing each other, with each domain adopting perspectives and methods from the other. The theory of random matrices, for instance, which has applications that range from number theory to the theoretical neurosciences, originated from the modeling of the complex structure of the energy levels of excited atomic nuclei.

Something similar could happen with the mathematical approximations of collective quantum phenomena: “In seeking to describe the underlying correlation structures of many-body systems, we want to advance existing mathematics and develop new methods,” says Hainzl. Because the mathematics of many-body quantum systems is multifaceted, a fertile exchange arises between various fields of mathematics.

In the long term, the proofs and state equations could also help physicists understand quantum mechanical processes in nature and technology better than before. “We hope that people will translate our approaches into numerical analysis and build on this to develop computer algorithms that researchers can apply with relative ease.” This could lead to progress in technologies such as quantum computers or high-temperature superconductors. Even though these kinds of advances are no doubt part of the motivation for mathematicians, you will rarely find them making such claims, says Hainzl. “We’re very realistic and cautious when it comes to the goals of our work.” Here, too, the golden rule is: Conjectures be damned; we want proof!

Text: Janosch Deeg

Prof. Dr. Christian Hainzl has been Chair of Mathematical Physics at LMU’s Department of Mathematics since 2019. Born in 1972, Hainzl studied mathematics at Vienna University of Technology and physics at the University of Vienna. He also did his doctorate at the University of Vienna. After post-doc stints in Vienna, at LMU, at Paris Dauphine University, and in Copenhagen, Hainzl was Assistant Professor and Associate Professor at the University of Alabama in Birmingham, before taking up the Chair of Mathematical Methods in the Natural Sciences in Tübingen.

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