This blog contains thoughts and observations on a wide range of physics related topics that I find interesting and which I believe are worthwhile sharing. You will find that I have strong opinions on some topics. I'm always happy to discuss them in more depth. Feel free to send me an email for further discussion: stefan.kehrein@theorie.physik.uni-goettingen.de

Following up on my previous post: we have most certainly crossed a watershed, and the other side of it is a very steep descent into terra incognita for how we are going to pursue the natural sciences in the future.
In short: it took a publicly available Claude model only a little more than two weeks after Navier-Stokes to solve a frontier calculation in theoretical physics, namely a nine-loop calculation in N=4 super Yang-Mills theory. This is a highly nontrivial accomplishment, beyond what humans had achieved before. What I find amazing is the simplicity of the prompt: “The problem is to compute the Six-particle (hexagon) amplitude in planar N=4 SYM at nine loops.” That’s it, plus some supervisory messages and enough credits to use about 1,000-2,000 USD worth of tokens.
Then, on October 6, came the Mathocalypse, a dump of 722 (!) manuscripts in 372 families of results on mathematics and computer science by OpenAI containing some really important results. Sure, this was done on an internal model, but with computing time averaging only roughly 3 hours per problem. You can (and should) read more about this on Scott Aaronson’s blog.
All of this is a lot to digest. There is huge potential here, also in theoretical physics, but for me it is overshadowed by concern about whether this new way of doing science will be sustainable. Specifically, how do we keep the next generation of scientists trained and motivated? I do not think we can or should contemplate a future in which scientists regress in their intellectual abilities compared to now and science becomes utterly dependent on AI.

For me, the past two weeks have been a watershed moment for science. On September 8, OpenAI announced that one of its advanced models had produced a proof of the Navier-Stokes Millennium Prize Problem by showing the existence of a flow singularity. This is not some uninteresting side problem, but one of six remaining Millennium Problems that the Clay Mathematics Institute posed in 2000 for their fundamental importance to mathematics. Many top mathematicians have been working on this problem for decades, and OpenAI's model arrived at a solution in about 88 hours of intensive computation spread over roughly a week, by throwing incredible computing resources (10,000 autonomous AI agents) worth millions of USD at the problem. It's worth noting that, as of this writing, the result has been formally checked in the proof assistant Lean but has not yet been independently verified by the Clay Mathematics Institute or the broader mathematical community.

What does this imply for how science will be done a few years from now? Mathematics is particularly susceptible to AI since math proofs can be formalized and verified by AI itself, which greatly reduces the risk that a hallucinated or flawed logical step slips through unnoticed. The only truly limiting factor seems to be computing resources (which equals money) — a challenging thought and realization for mathematicians.

What about other sciences, like my field, physics? How will experimental and especially theoretical physics research look in five years, given the rapid increase in sophistication of AI models? The only thing I'm sure about is that it will be very different from today.

Last but not least, it should be mentioned that there is a quite a bit of controversy regarding the OpenAI result, because a day earlier, two mathematicians, Tristan Buckmaster and Levent Alpöge, announced that they had solved a closely related problem (Euler equation singularity). They employed AI tools but not a swarm of autonomous AI agents. It seems that OpenAI became aware that Buckmaster and Alpöge were close to solving a Millennium-type problem and decided to throw their computing resources at Navier-Stokes. Additionally, there is the question of whether OpenAI's solution unintentionally profited from Buckmaster's and Alpöge's use of OpenAI models; OpenAI addressed this directly in a September 10 update, stating it had not accessed the pair's specific work or user data, while acknowledging that de-identified product-usage data more broadly could have contributed to its models' training.

Leaving aside these important ethical questions, a watershed moment this is. You can find more information in two Quanta Magazine articles — one on the OpenAI result, and one with reflections on the role of AI in physics. I also highly recommend the thoughts of Fields medalist Terence Tao on AI in mathematics (and beyond!).

I recently came upon an intriguing debate about the existence of Glauber states in quantum optics. In 1997 Klaus Mølmer published a paper with the title "Optical coherence: a convenient fiction" that challenged the textbook wisdom that the laser field above threshold is a Glauber coherent state. His reasoning is essentially that the state of the electromagnetic field should be thought of as the reduced density operator after tracing out the atoms in the gain medium. This gives rise to a density operator that is diagonal in the number of photons basis with identical diagonal elements as the Glauber state, but without off-diagonal coherences. The notion of such coherences plays an important role in how one usually understands experiments like homodyne detection, or in different contexts BEC interference experiments or BCS ground states. This 1997 paper ignited (or re-ignited) a lively debate that led to countless publications, nicely summarised in a paper entitled "Dialogue Concerning Two Views on Quantum Coherence: Factist or Fictionist" by S. Bartlett, T. Rudolph and R. Spekkens. Their conclusion is that neither side wins the debate, both viewpoints are possible and internally consistent. One key ingredient is that the number diagonal mixed state is equal to the incoherent superposition of Glauber states when averaging over the unknown phase of the Glauber state. Going through the dialogue between the "factist" and the "fictionist" is fun (disclaimer: for physicists) and points to interesting and fundamental questions.

Matter consists mainly of vacuum. So even in matter we would expect electromagnetic waves to propagate with the vacuum speed of light c nearly everywhere. However, we know from Physics 101 that this is wrong. Electromagnetic waves in matter have a velocity c/n, where n is the refractive index (for simplicity let me only consider non-dispersive media). How does this come about? What is the microscopic mechanism behind generating an electromagnetic wave with velocity c/n? The answer to this rather obvious question is definitely not addressed in Physics 101, and as a matter of fact not widely known in the first place. It is given by the Ewald-Oseen extinction theorem dating from 1915! It demonstrates a remarkable and intricate interplay between electric dipoles, that get excited by the incident wave and then in turn produce fields that act back on the dipoles themselves. The solution to this self-consistency problem shows that the incident wave becomes extinct and is replaced by a wave with wave vector k=n 𝜔/c. It is a fun linear algebra programming exercise that gives real insight. For more details check out The Ewald–Oseen extinction theorem and extinction lengths and An extinction theorem for electromagnetic waves in a point dipole model.

Thanks to the International Quantum Year there is a lot of interest in quantum mechanics. While no one doubts its correctness, one often encounters the statement that quantum mechanics is counterintuitive and cannot be understood. My take on that has been unchanged since I first taught quantum mechanics. Reader warning: Strong opinion ahead.

People who express that quantum mechanics is counterintuitive and cannot be understood participate in a modern day version of the Copernican revolution, that is the shift from the Ptolemaic world view with Earth being in the center of the Universe to the Copernican world view with Earth being just another planet revolving around the Sun. Except they participate in this debate by supporting the wrong side. Just replace Earth by a classical world view that we consider normal and intuitive because this is what we experience as effectively classical beings. We naturally define everything in relation to that classical world view, therefore one it tempted to put that classical world view into the center of our mental universe. However, this is just another case of taking ourselves too seriously, of favoring a certain frame of reference because it is “ours”.

There is absolutely no reason why the laws of Nature should care about what we macroscopic beings consider normal and intuitive. So feel free to say that quantum mechanics is counterintuitive, but my recommendation would be to get over it and to move on.

Yes, you have read correctly. Magnetic field lines are not always closed. If you are surprised about this then you fell prone to a common misconception that is perpetuated by many textbooks (Sommerfeld, Griffiths, and many more) and instructors. This includes me until recently a student asked about this and I actually had a closer look.

The first thing to realize is that closed magnetic field lines are a sufficient condition for the validity of Maxwell's equation div B=0. But closedness is not a necessary condition. A simple example for this observation is a pair of identical Helmholtz coils with a common axis and equal but opposite currents. The magnetic field at the symmetry point between the two coils vanishes. The magnetic field lines approaching this symmetry point along the axis from either of the coils therefore have to terminate at the symmetry point. On the other hand, there are magnetic field lines starting at the symmetry point and running perpendicular to the axis. The overall effect is that the flux through a small sphere around the symmetry point vanishes, consistent with a divergenceless magnetic field. A second more generic situation are magnetic field lines that continue forever without beginning or end, but also without being closed. Imagine the closed field lines around a single Helmholtz coil, which you then perturb by a current running through some wire nearby. For a weak perturbation the field lines still wrap around the Helmholtz coil, but in general they will do so in an incommensurate manner without being closed. As a matter of fact, closed magnetic field lines are a set of measure zero in the space of current geometries. However, many textbook examples consider specific simple geometries leading to closed lines, which one then believes to always be the case. Putting it otherwise: Simply solvable cases can really give you the wrong idea about what is generic.

All of this is perfectly well known since at least 1929 (textbook by I. Y. Tamm), but the wrong picture is prevalent. Just go around and ask your fellow students/colleagues, the reactions can be quite entertaining. For more information including lots of references and the two examples above (plus many other examples) I recommend the following two articles: The Misconception of Closed Magnetic Flux Lines and A taxonomy of magnetostatic field lines (from 2024!).