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Methods of Modern Mathematical Physics #2
II: Fourier Analysis, Self-Adjointness: Fourier Analysis, Self-Adjointness
This volume will serve several purposes: to provide an introduction for graduate students not previously acquainted with the material, to serve as a reference for mathematical physicists already working in the field, and to provide an introduction to various advanced topics which are difficult to understand in the literature. Not all the techniques and application are treated in the same depth. In general, we give a very thorough discussion of the mathematical techniques and applications in quatum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations. Finally, some of the material developed in this volume will not find applications until Volume III. For all these reasons, this volume contains a great variety of subject matter. To help the reader select which material is important for him, we have provided a "Reader's Guide" at the end of each chapter.
361 pages, ebook
First published September 28, 1975
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Displaying 1 - 3 of 3 reviews
May 22, 2026
The second volume of Michael Reed and Barry Simon’s Methods of Modern Mathematical Physics teaches the elements of two problem domains of crucial relevance in many, if not most physical applications, viz., Fourier analysis and the theory of self-adjoint operators (we have just reviewed Vol. I here). Before turning to these, however, let us linger over a remark the authors make in their preface.
Given how functional analysis, in particular appears to revolve around large spaces, often of high cardinality, one might well wonder whether it surpasses the capacity of the human mind to understand. Indeed, L.E.J. Brouwer (1881-1966; cf. the entry on him in the Oxford Companion to Philosophy, ed. Ted Honderich) says essentially this, but he is mistaken. For the key is to tease out adequate finiteness conditions which can then be relaxed in a controlled manner so as to solve the original problem in the large, if possible. The most outstanding such condition would be that of compactness, whether in its primary denotation in topology or in its derived connotation in operator theory.
Intuitionism fails to recognize that the heart of the matter lies in how to relax. An obvious way to do this, given say a vector φ that sits in the span of a finite-dimensional subspace, would be to use the norm and admit any other vector ψ in the encompassing space such that ‖ψ – φ‖ < ε for any ε > 0. Functional analysts, however (unlike physicists) have discovered many topologies other than the norm topology that could well prove useful in a setting such as this.
A good illustration would be in Chap. IX on Fourier analysis. In a beginning textbook such as that of Elias M. Stein and Rami Shakarchi, Fourier Analysis: An Introduction (Princeton University Press, 2003, reviewed by us here), the Fourier transform is studied on a circle or on the real line. Mightn’t one want to take the Fourier transform on other, more complicated spaces? Yes, we can if we avail ourselves of Laurent Schwartz’ distributions. A distribution, in this sense, is always just an element of the dual space to a given space of functions, where the choice of topology on the latter is of the essence. What Schwartz discovered around 1950 is that the natural function space on which the Fourier transform lives is not, say, L², but the space of functions of rapid decrease at infinity. The latter is to be defined by a system of seminorms involving polynomials times the partial derivatives of a given function with respect to any multi-index.
Therefore, the Schwartz space 𝒮(R) and its analogues have a special role to play in Fourier analysis. Reed-Simon treat basic properties, then dive into details on some special topics, such as how to restrict the domain to a submanifold, the usual properties with respect to convolution, abstract interpolation, fundamental solutions to differential equations with constant coefficients; elliptic regularity and the Gårding-Wightman axioms of relativistic quantum field theory. In an appendix, products of distributions, wave front sets and oscillatory integrals.
The technical level of Reed-Simon’s exposition is very high, of course, as one has come to expect. Almost all of the 78 problems to Chap. IX are useful, as are the extensive end-notes and reader’s guide. Unless one wants to go into hard-core analysis and quantum field theory, however, all this material is ancillary. Chap. X, on the other hand, amounts to anything but this. For quantum mechanics to be a useful, i.e., predictive theory, one has to establish the existence of dynamics and study the impact of pertubations. In either case, it comes down to the properties of unbounded operators and their self-adjointness.
At 216 pp, the quite extended Chap. X descends into all that it can be desired to know about extensions of symmetric operators (esp. von Neumann’s classification); self-adjointness; perturbations; positivity; semigroups and their generators etc. Special topics include the axioms of free quantum fields, the Feynman-Kac formula, classical nonlinear wave equations and the Hilbert space approach to classical mechanics (what constitutes an interesting perspective, without apparent purpose). Another 81 most interesting homework exercises, by the way. Beware, Chap. X is rather technical and challenging throughout!
Five stars, gets to even better mathematical physics than in vol. I. Here is a good place to learn about the semigroups of linear operators beyond Hilbert space which play such an important role in modern probability theory.
Given how functional analysis, in particular appears to revolve around large spaces, often of high cardinality, one might well wonder whether it surpasses the capacity of the human mind to understand. Indeed, L.E.J. Brouwer (1881-1966; cf. the entry on him in the Oxford Companion to Philosophy, ed. Ted Honderich) says essentially this, but he is mistaken. For the key is to tease out adequate finiteness conditions which can then be relaxed in a controlled manner so as to solve the original problem in the large, if possible. The most outstanding such condition would be that of compactness, whether in its primary denotation in topology or in its derived connotation in operator theory.
Intuitionism fails to recognize that the heart of the matter lies in how to relax. An obvious way to do this, given say a vector φ that sits in the span of a finite-dimensional subspace, would be to use the norm and admit any other vector ψ in the encompassing space such that ‖ψ – φ‖ < ε for any ε > 0. Functional analysts, however (unlike physicists) have discovered many topologies other than the norm topology that could well prove useful in a setting such as this.
A good illustration would be in Chap. IX on Fourier analysis. In a beginning textbook such as that of Elias M. Stein and Rami Shakarchi, Fourier Analysis: An Introduction (Princeton University Press, 2003, reviewed by us here), the Fourier transform is studied on a circle or on the real line. Mightn’t one want to take the Fourier transform on other, more complicated spaces? Yes, we can if we avail ourselves of Laurent Schwartz’ distributions. A distribution, in this sense, is always just an element of the dual space to a given space of functions, where the choice of topology on the latter is of the essence. What Schwartz discovered around 1950 is that the natural function space on which the Fourier transform lives is not, say, L², but the space of functions of rapid decrease at infinity. The latter is to be defined by a system of seminorms involving polynomials times the partial derivatives of a given function with respect to any multi-index.
Therefore, the Schwartz space 𝒮(R) and its analogues have a special role to play in Fourier analysis. Reed-Simon treat basic properties, then dive into details on some special topics, such as how to restrict the domain to a submanifold, the usual properties with respect to convolution, abstract interpolation, fundamental solutions to differential equations with constant coefficients; elliptic regularity and the Gårding-Wightman axioms of relativistic quantum field theory. In an appendix, products of distributions, wave front sets and oscillatory integrals.
The technical level of Reed-Simon’s exposition is very high, of course, as one has come to expect. Almost all of the 78 problems to Chap. IX are useful, as are the extensive end-notes and reader’s guide. Unless one wants to go into hard-core analysis and quantum field theory, however, all this material is ancillary. Chap. X, on the other hand, amounts to anything but this. For quantum mechanics to be a useful, i.e., predictive theory, one has to establish the existence of dynamics and study the impact of pertubations. In either case, it comes down to the properties of unbounded operators and their self-adjointness.
At 216 pp, the quite extended Chap. X descends into all that it can be desired to know about extensions of symmetric operators (esp. von Neumann’s classification); self-adjointness; perturbations; positivity; semigroups and their generators etc. Special topics include the axioms of free quantum fields, the Feynman-Kac formula, classical nonlinear wave equations and the Hilbert space approach to classical mechanics (what constitutes an interesting perspective, without apparent purpose). Another 81 most interesting homework exercises, by the way. Beware, Chap. X is rather technical and challenging throughout!
Five stars, gets to even better mathematical physics than in vol. I. Here is a good place to learn about the semigroups of linear operators beyond Hilbert space which play such an important role in modern probability theory.
Want to Read
January 9, 2024IX.10 Distributions, wave front sets, and oscillatory integrals
The product of two tempered distributions is not a tempered distribution!
(See theorem IX.42)
The product of two tempered distributions is not a tempered distribution!
(See theorem IX.42)
Want to Read
January 27, 2019Analysis and its applications to quantum physics
Displaying 1 - 3 of 3 reviews