Mathematical Problems in Classical Physics

Abstract

In Mathematics is the name for those domains of theoretical physics that are temporarily unfashionable. The advantage of being unfashionable is that it presents the possibility of the rigorous and deep investigation of well-established mathematical models. A sleeping "physical theory" can be formulated as a chain of statements having exact mathematical meaning of mathematical conjectures. Such conjectures can then be proved or dis-proved. In many cases the mathematical problems arising this way are very difficult, and progress is rather slow. It is much easier to obtain a new re-sult in an unexplored domain. Hence, most researchers carefully avoid any thinking on the old classical problems. On the other hand, most of the new developments in physics are due to the exploitation by physicists of the theories developed by the mathemati-cians in previously unfashionable domains. Thus, it is useful to compile from time to time the lists of sleeping problems in unfashionable domains-just to know that the problems are still open. The inclusion of a problem in the list that follows does not reflect its objective importance; rather, the choice is based on my personal taste. 1.1 Differential Invariants and Functional Moduli Consider any local classification problem: We classify some objects (for example, functions, fields, varieties, or mappings), and we call two objects equivalent if one can be reduced to the other by a clever change of variables. Example 1. The classification of the Riemannian (or Einstein) metrics f at a neighborhood of a point of the space up to the local diffeomorphism of the space preserving the point. Example 4. The classification of the Hamilton vector fields f at a neigh-borhood of a zero point of the Hamilton field up to the local symplecto-morphisms preserving this point. Example 5. The local classification of differential equations y" = f(x, y, y'). Example 6. The classification of the germs of hyperkahlerian structures on a manifold of dimension 4n up to the local diffeomorphisms. The classical normal forms, to which one can reduce these objects by the action of the corresponding infinite-dimensional group, contain arbitrary functions. In most cases these functions depend on fewer variables than the initial objects. However, these descriptions seem to depend on the special choice of the algorithms of the reduction. Is there any intrinsic, coordinate-free meaning in the assertion that the answer "depends on m arbitrary functions of n variables"? The arbitrary functions, intrinsically associated to the objects of our classification, are called the functional moduli. The problem might be formulated formally as follows [1]. We start with some functional space of objects that we wish to classify (or of their Taylor series at a point). A group of diffeomorphisms acts on this space. We fix an integer k and we consider the k-jets (the Taylor polynomials of degree k) of our objects. They form a finite-dimensional manifold. The ac-tion of the group of diffeomorphisms defines the action of the corresponding finite-dimensional Lie group on the finite-dimensional space of jets. Consider the dimension of the orbits space at the k-jet of a given object f. We call this dimension of the moduli space of the k-jet of f the moduli number of the k-jet. We denote it by m(k). To describe all these moduli numbers together we form the Poincare series of moduli numbers 00