Thoughts on Mathematical Physics

Yesterday I had a discussion with my student regarding the difference between theoretical physics and mathematical physics. A short answer to this will be the rigour, with mathematical physics being the more rigorous of the two. Also, one can say that theoretical physics tend to dwell on conceptual ideas that are needed to model the physical system. Why this discussion arises is due to the matter of selecting the appropriate journal to publish one's research. Sending to a mathematical physics journal requires one to be more mathematically rigorous in writing out the results. There is very little room for hand-waving arguments. The degree of rigour can be pretty subjective but a rule of thumb that one can use is the more familiar a topic or a technique is to the community of theorists, then the more rigour one needs to put in.

What is mathematical physics? From wikipedia, it is the development of mathematical methods (hence the rigour) to problems in physics. I remember a younger colleague dismisses it as applied mathematics, to which I rebuke, then why would Cambridge University have the Department of Applied Mathematics and Theoretical Physics (DAMTP). Let me just mention another instance of degrading the subject of mathematical physics; it is said that mathematical physics label lacks attraction and the label of mathematical engineering was adopted instead. Another case was the suggestion of physical mathematics to replace mathematical physics, which I guess is for asserting that mathematics is the primary subject. One could argue endlessly on these labels and in my view it is counterproductive. Let me just say then the following: what has been accepted at the international level? Indeed, like in this wikipedia article on classification of subjects of mathematics, one could group several topics of mathematical/theoretical physics under the broad umbrella of applied mathematics. But this also applies to other forms of applications e.g. probability, operations research and computational mathematics. If the idea is to refine the classification, then mathematical physics would lump all theoretical aspects of physics into mathematical physics (see here). For recent classification made by Zentralblatt Math, one can use this document or visit their site. Indeed, if one studies the history of classifying mathematical subjects, mathematical physics has always been part of mathematics. For instance, the International Congress of Mathematicians, they have one prize dedicated for mathematical physics.

My own view of what mathematical physics is, is dependent on the period when it is considered. Certain established techniques will either be taught at a more basic level or even phased out. For instance, older books of mathematical physics tend to emphasize solutions of differential equations or integral equations. More modern mathematical physics books tend to include more abstract structures that have gone in vogue for theoretical physics. This is why when I was teaching the graduate course, Advanced Mathematical Physics last semester, I use the book of Szekeres and explore more structures than what is often taught. I was hoping to include these in the revised curriculum but unfortunately this was not accepted. In the context of DAMTP, the topics that involve differential equations and numerics tend to be grouped differently from the group that has quantum field theory, relativity and particle physics to which more abstract structures are used.

From the link of the mathematical physics prize for ICM, it is good for us to be aware about the forthcoming congress in St. Petersburg in 2022. INSPEM was involved in the congress in Seoul in 2014 (unfortunately, I did not get to go). The congress is held every four years, and the one in 2018 was held in Rio de Janeiro. I'm not aware of anyone from INSPEM went to the congress in 2018 but I know Prof. Teo Lee Peng went to the congress. At some point, I think there were interests in bidding for the congress to be in Malaysia but I heard we just do not have a sizable community and active participation in such international events. For those interested what are discussed at ICM, the proceedings are available here. Just around the corner is the International Congress on Mathematical Physics, scheduled on 2-7 August 2021 in Geneva. Mode of the conference seems to be both physical and online. The online mode has a slightly cheaper registration fee (but still expensive for us normal people). I do hope that they will post up videos online for us to watch.