Modern Mathematical Physics: What It Should Be

When somebody asks me, what I do in science, I call myself a specialist in mathematical physics. As I have been there for more than 40 years, I have some definite interpretation of this combination of words: " mathematical physics. " Cynics or purists can insist that this is neither mathematics nor physics, adding comments with a different degree of malice. Naturally, this calls for an answer, and in this short essay I want to explain briefly my understanding of the subject. It can be considered as my contribution to the discussion about the origin and role of mathematical physics and thus to be relevant for this volume. The matter is complicated by the fact that the term " mathematical physics " (often abbreviated by MP in what follows) is used in different senses and can have rather different content. This content changes with time, place and person. I did not study properly the history of science; however, it is my impression that, in the beginning of the twentieth century, the term MP was practically equivalent to the concept of theoretical physics. Not only Henri Poincaré, but also Albert Einstein, were called mathematical physicists. Newly established theoretical chairs were called chairs of mathematical physics. It follows from the documents in the archives of the Nobel Committee that MP had a right to appear both in the nominations and discussion of the candidates for the Nobel Prize in physics [1]. Roughly speaking, the concept of MP covered theoretical papers where mathematical formulae were used. However, during an unprecedented bloom of theoretical physics in the 20s and 30s, an essential separation of the terms " theoretical " and " mathematical " occurred. For many people, MP was reduced to the important but auxiliary course " Methods of Mathematical Physics " including a set of useful mathematical tools. The monograph of P. Morse and H. Feshbach [2] is a classical example of such a course, addressed to a wide circle of physicists and engineers. On the other hand, MP in the mathematical interpretation appeared as a theory of partial differential equations and variational calculus. The monographs of R. Courant and D. Hilbert [3] and S. Sobolev [4] are outstanding illustrations of this development. The theorems of existence and uniqueness based on the variational principles, a priori estimates, and imbedding theorems for functional spaces comprise the main content of this direction. As a student …