Should You Go to the Department of Mathematics or the Department of Physics? -Advice for Prospective Students-

Some prospective students may be struggling with whether they want to study theoretical physics or mathematics at university. While both use mathematical formulas as a language, it can be said that the way they use them is completely different.

In mathematics within a mathematics department, once the premises are set, the goal is a rigorous proof that allows for no exceptions. On the other hand, as the theoretical physicist Wolfgang Pauli once said, many physicists are interested in typical, specific examples of natural phenomena and are not very interested in exceptions that seem unlikely to occur in the natural world.

To give you a sense of the atmosphere of physics, for example, a function that takes a value of 1 for rational numbers and 0 for irrational numbers is often given almost no attention until one encounters a natural phenomenon where it is actually important. Conversely, they might show interest in the properties of smooth functions that can be differentiated any number of times.

Of course, there are researchers in physics who are strong in rigorous mathematics. Please understand that what I am describing here is roughly the median of what I consider to be standard theoretical physicists I have encountered around the world.

With that premise, from a mathematician's point of view, the way physicists use mathematics feels like the "barbaric courage" of a barbarian. To express the physical intuition they have gained, they might even create their own "mathematics" that goes beyond mathematical common sense. From a mathematician's perspective, that has not yet become proper mathematics.

For example, in the early days of quantum mechanics, the theoretical physicist Paul Dirac introduced the delta function, which does not fit into the framework of ordinary functions. It was a concept that could not be described by the mathematics of that time. However, it was later refined into a rigorous mathematical concept called the theory of distributions by mathematicians Mikio Sato and Laurent Schwartz. Such things happen frequently in theoretical physics.

In the actual research field of theoretical physics, there is often a style of "try it first, and if you run into trouble, go back to the logic and think about the mathematics rigorously." When the result obtained seems physically strange, they go back and check the calculations, such as the order of integration and differentiation, or the exchange of integration and limit operations. This is because the work of physics is that of a "detective." Please refer to the article below regarding this.

For example, in quantum field theory in physics, a series of infinitely divergent integrals appears that would make a mathematician faint. Physicists calmly regularize (make finite) those divergent integrals, perform variable transformations and calculations, and then take the limit again. However, through such calculations, experimental results can be explained with astonishing precision. In the calculation of a quantity called the anomalous magnetic moment, it is possible to predict more than 10 digits of experimental data accurately.

Also, thanks to physicists exercising such barbaric courage to perform calculations, they have obtained interesting results, and thinking about them more deeply has sometimes led to important discoveries in physics. The discovery of quantum anomalies in quantum field theory was one of them. An example was found where a symmetry that existed in classical theory is broken quantum-mechanically. That calculation is also one that a proper mathematician would never do. But this quantum anomaly was later actually confirmed in experiments on the decay process of pions.

Furthermore, this research on quantum anomalies in theoretical physics is also linked to pure mathematics, specifically topology. It was shown by theoretical physicists that quantum anomaly terms can be written using topological invariants. Then, the vast amount of research results in topology in the field of mathematics advances theoretical physics all at once. Such things also happen.

There are other famous examples of collaboration between mathematicians and physicists. For example, Albert Einstein built the mathematical framework of general relativity while being taught differential geometry by his mathematician friend Marcel Grossmann. The collaboration between theoretical physicists who use barbaric and powerful mathematics and mathematicians who create delicate and rigorous mathematics has occurred in various situations so far and has greatly advanced physics.

In the 21st century, I personally have high expectations for meaningful collaborative work between physics and mathematics. For that, I believe that mutual understanding, where physicists grasp the values of mathematicians and mathematicians also properly grasp the values of physicists, is necessary to build a relationship of equal trust. I also have such expectations for collaborative work from young physics and mathematics students.

As we have seen so far, there is a big difference in the mathematics used by theoretical physicists and mathematicians. Therefore, for young people who are worried about whether to do theoretical physics or mathematics at university, I recommend that you first look at university physics and mathematics textbooks at a library or bookstore. Just by doing that, I think you will roughly understand which one you want to do. I also recommend that you participate in open campus events or make an appointment to visit a laboratory to hear directly from researchers.