Lectures in mathematical physics. Vol. II. (English) Zbl 0228.00002
Mathematics Lecture Note Series. Reading, Mass.: W. A. Benjamin, Inc. xviii, 749 p. $ 6.95 (1972).
This book is the second volume of a text presenting several topics in mathematical physics. The first volume (1970; Zbl 0206.26501) presented “material in a lecture-note format that could be used in an undergraduate course …and that would provide a more challenging and contemporary outlook than the standard textbooks”. In this book, the aim of the author is similar. It is divided into eight more or less independent parts.
The first one (“Introduction to Quantum Mechanics”) is an exposure of the process of quantization and of the main ideas of quantum mechanics, with an interesting parallel with classical mechanics. The second chapter (“Dirac spaces and generalized functions”) is more related to the first volume which already contains notions on Dirac spaces (Ch. IV) and generalized functions (Ch. V).
The main chapter of this book is the third one (“Groups and their representations”). On one side it can be considered as “an independent treatise on group theory”, and on the other side it gives a certain unity to this textbook. Of course this chapter is not complete, and is centered on Lie groups, and is written in a style similar to the one encountered in other books of the author on the same subject.
Chapter IV presents “The Feynman path-integral approach to quantum mechanics”, essentially to pursue the relation and parallelism between classical and quantum mechanics. “Scattering theory” is the subject of chapter V. The material presented here is the classical one, with, however, some hints into the direction of more modern works on the scattering operator on Hilbert space (Kato, etc. …). Written in the same spirit, we find a sixth chapter on statistical mechanics.
The last chapter is devoted to a brief “Introduction to quantum theory of fields”, leading to the introduction of Feynman rules. This part is probably weaker than the others if compared with the announced goal. Its shortness and simplicity, with only allusions to the fundamental problems, will probably convince a mathematician that the situation is hopeless, in contradiction with the actual efferverscence of the constructive field theory.
The book ends with a comprehensive appendix on measure theory.
The first one (“Introduction to Quantum Mechanics”) is an exposure of the process of quantization and of the main ideas of quantum mechanics, with an interesting parallel with classical mechanics. The second chapter (“Dirac spaces and generalized functions”) is more related to the first volume which already contains notions on Dirac spaces (Ch. IV) and generalized functions (Ch. V).
The main chapter of this book is the third one (“Groups and their representations”). On one side it can be considered as “an independent treatise on group theory”, and on the other side it gives a certain unity to this textbook. Of course this chapter is not complete, and is centered on Lie groups, and is written in a style similar to the one encountered in other books of the author on the same subject.
Chapter IV presents “The Feynman path-integral approach to quantum mechanics”, essentially to pursue the relation and parallelism between classical and quantum mechanics. “Scattering theory” is the subject of chapter V. The material presented here is the classical one, with, however, some hints into the direction of more modern works on the scattering operator on Hilbert space (Kato, etc. …). Written in the same spirit, we find a sixth chapter on statistical mechanics.
The last chapter is devoted to a brief “Introduction to quantum theory of fields”, leading to the introduction of Feynman rules. This part is probably weaker than the others if compared with the announced goal. Its shortness and simplicity, with only allusions to the fundamental problems, will probably convince a mathematician that the situation is hopeless, in contradiction with the actual efferverscence of the constructive field theory.
The book ends with a comprehensive appendix on measure theory.
Reviewer: G. Loupias (Montpellier)
MSC:
| 81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |
| 82-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics |
| 81Rxx | Groups and algebras in quantum theory |
| 81Txx | Quantum field theory; related classical field theories |
| 46N50 | Applications of functional analysis in quantum physics |
| 47N50 | Applications of operator theory in the physical sciences |
