Summary and open problems

Abstract

In the preceding sections hitchhiker’s guide to the architecture of Hilbert spaces was presented. Although the analysis of various graphical representations was restricted to a minimum — as if pictures of buildings were presented without really entering into details of engineering problems — the multitude of spaces used in physics is responsible for this rather long presentation. I realize that many kinds of spaces were omitted and thus description of the architecture of Hilbert spaces given here is far from being complete. The representation of spinor and tensor spaces used in relativistic methods for example was not mentioned. On the other hand spaces most important for molecular, atomic and nuclear physics were all covered. One may draw graphs representing the spaces of a very high dimension, too high to list all paths explicitly, and still find interesting information about the structure of such spaces. Spaces built from a large number of primitive states of the same type have particularly simple structures. The techniques of graphical evaluation of matrix elements are developed in Part II (abelian symmetries) and Part III (non-abelian symmetries). The insight that the graphs give into the structure of model spaces and the graphical methods of matrix element calculation may than be applied to construction of matrix representations of differential operators.