Methods of modern mathematical physics. 1: Functional analysis. (English) Zbl 0242.46001

New York-London: Academic Press, Inc. xvii, 325 p. $ 12.50 (1972).
Summary: There are a few theoretical physicists who have recently expressed concern about the parting ways of mathematics and physics. F. J. Dyson [Bull. Am. Math. Soc. 78, 635–651 (1972)] speaks of the missed opportunities in the past and at present, by mathematicians to develop new mathematics arising out of new physical theories. On the other hand, R. Hermann [Physics Today (December 1972)] deplores the level of mathematical knowledge of theoretical physicists and very correctly points out how it has been very fruitful for the engineers to have updated their mathematical learning. In the past centuries, physics and mathematics enjoyed an extremely fruitful relationship. Mathematics in this century has made great progress; on the other hand, theoretical physics in the past twenty years has been left alone and has become less innovative. There is clearly a need for the physicists to develop the much needed friendship and collaboration with mathematicians and this can only be achieved by developing the proper language to communicate with each other. Even now physicists learn their mathematics from the books of Courant and Hilbert and Morse and Feshbach in their graduate classes whereas it appears fairly obvious that these methods should be taught in undergraduate courses! It is abundantly clear that more modern methods must be taught in its place. It would then be possible to teach graduate quantum mechanics and statistical mechanics at a much more satisfying level using the methods of functional analysis and \(C^*\)-algebras.
The book under review is the first volume of a three volume series on methods of modern mathematical physics, and is perhaps the first of its kind to try to fill the gap that exists presently between the present day mathematics and theoretical physics. Professor Hermann has also published a few lecture notes on some methods of theoretical physics recently, but the topics covered by him are slightly different from those covered in the present series. This volume is very well written and I am sure it will become a standard text book within the next few years. It has been carefully prepared with enough problems for the student as well as the teacher. The one chief complaint I have is that the authors could have included many more examples from physics so that the book could also have been used with equal profit by mathematicians.
The first volume contains an exposition of functional analysis. It begins with a brief chapter on various definitions. Chapters II to V contain a very readable ac-count of Hilbert, Banach, topological, and locally convex spaces. All the important theorems are stated and proved with care, either in the text or in separate appendices. In Chapter V there are a few examples from physics briefly discussed. After a discussion of fixed point theorems, a separate subsection is devoted to its applications to prove existence of local solutions of ordinary differential equations, proving consistency of the bootstrap equations of particle physics (existence of Mandelstam representation, etc.) as well as uniqueness and existence of the phase of the scattering amplitude, and existence of correlation functions at low density in statistical mechanics in the limit of infinite volume. Chapter VI contains an exposition of bounded opera-tors. The spectral theorem is discussed in Chapter VII. The applications of this theory to quantum mechanics are so many that an entirely new volume can be written on it! Useful notes are given at the end of Chapter VII concerning various aspects of this. The last chapter discusses unbounded operators. This study is motivated by first describing two well known examples from quantum mechanics (the position operator and the harmonic oscillator problem). The spectral theorem and Stone’s theorem are both discussed fully as their implications to quantum mechanics are so fundamental. There are also subsections on quadratic forms, convergence of unbounded operators, and tensor products of operators in Hilbert spaces. This chapter, and the volume ends with a brief discussion of three mathematical problems of quantum mechanics, viz., choice of proper “self-adjointness” of operators to represent physical situations, investigation of spectra of observables and the behaviour of a quantum mechanical system for large times (scattering theory). A few pages are devoted as Notes developing the mathematical ideas of quantum mechanics. The detailed exposition of these questions and other topics are left to second and third volumes.
I heartily recommend this book for all those interested in theoretical physics. I eagerly look forward to the appearance of the two volumes which the authors promise to bring out in 1973. It may well be that by 1984, we will have changed the situation that now exists between theoretical physicists and mathematicians, if only books of this kind become graduate level textbooks all around the world!
Reviewer: A. K. Rajagopal


46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
46Axx Topological linear spaces and related structures
46Bxx Normed linear spaces and Banach spaces; Banach lattices
46Cxx Inner product spaces and their generalizations, Hilbert spaces
47E05 General theory of ordinary differential operators
47F05 General theory of partial differential operators
47H10 Fixed-point theorems
47Axx General theory of linear operators
47Bxx Special classes of linear operators
46Mxx Methods of category theory in functional analysis