Mathematical Physics: A Modern Introduction to Its Foundations (英語) ハードカバー – 1998/12/21
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PAGEOPH [Pure and Applied Geophysics]
Review by Daniel Wojcik, University of Maryland
"This volume should be a welcome addition to any collection. The book is well written and explanations are usually clear. Lives of famous mathematicians and physicists are scattered within the book. They are quite extended, often amusing, making nice interludes. Numerous exercises help the student practice the methods introduced. … I have recently been using this book for an extended time and acquired a liking for it. Among all the available books treating mathematical methods of physics this one certainly stands out and assuredly it would suit the needs of many physics readers."
Review by G.Roepstorff, University of Aachen, Germany
"… Unlike most existing texts with the same emphasis and audience, which are merely collections of facts and formulas, the present book is more systematic, self-contained, with a level of presentation that tends to be more formal and abstract. This entails proving a large number of theorems, lemmas, and corollaries, deferring most of the applications that physics students might be interested in to the example sections in small print. Indeed, there are 350 worked-out examples and about 850 problems. … A very nice feature is the way the author intertwines the formalism with the life stories and anecdotes of some mathematicians and physicists, leading at their times. As is often the case, the historical view point helps to understand and appreciate the ideas presented in the text. … For the physics student in the middle of his training, it will certainly prove to be extremely useful."
Review by Paul Davies, Orion Productions, Adelaide, Australia
"I am pleased to have so many topics collected in a single volume. All the tricks are there of course, but supported by sufficient rigour and substantiation to make the dedicated mathematical physicist sigh with delight."
EMS [EUROPEAN MATHEMATICAL SOCIETY] NEWSLETTER
"This book is a condensed exposition of the mathematics that is met in most parts of physics. The presentation attains a very good balance between the formal introduction of concepts, theorems and proofs on one hand, and the applied approach on the other, with many examples, fully or partially solved problems, and historical remarks. An impressive amount of mathematics is covered. … This book can be warmly recommended as a basic source for the study of mathematics for advanced undergraduates or beginning graduate students in physics and applied mathematics, and also as a reference book for all working mathematicians and physicists."
Sadri Hassani, Department of Physics, Illinois State University, USA
Amazon.com で最も参考になったカスタマーレビュー (beta)
The book is divided into eight parts, each comprising three or four chapters, on: Finite-dimensional Vector Spaces, Infinite-dimensional Vector Spaces, Complex Analysis, Differential Equations, Operators on Hilbert Spaces, Green's Functions, Groups and Manifolds, Lie Groups and Applications. Fear not: although it isn't designed for freshmen, it emphatically isn't the sort of math book where you have to crack the code to get any benefit.
The layout is excellent, there are many, many worked examples, and I found very few slips or typos. One black mark, the reason I don't give it 5 stars: although there are a massive 850 problems, there are no solutions (just like Byron & Fuller). Unless you're confident in your mathematical ability, you may find that a drawback for self-study. Finally, a word to the wise: check out this title at amazon.co.uk (provided you aren't in a hurry).
The book structures itself around the concept of a vector space, and the author does not shy away from abstractions which involve mathematical structures such as fields, algebras, groups, etc. as well as the topological concepts like completeness, compactness, bounded operators and so on. In this respect, the view toward math is modern and stresses the mutualism between physics and math in the advancement of both.
In general, there is a noticeable trend shift in writing of mathematical physics texts, which was inaugurated by Dennery and Krzywicki`s text and Walter Thirring`s two volume classic as opposed to the Morse and Feshbach variety which mostly focuses on the detailed solutions of certain problems of interest in physics. In this respect, this book is a nice complement and update to both. The quality of writing is reflected on the references and again culling the best of two worlds. Indeed, the references span numerous important fields and approaches to math (Rudin`s Functional Analysis, Bott & Tu`s Differential Forms in Algebraic Topology, Barut & Raczka`s Group Representation Theory...), and this more than compensates the absence or truncation of any ideas/concepts.
The topics that are absent from the text include measure theory, a possible prologue to algebraic geometry, hyper-complex analysis and geometric algebra (except for a short digression on quaternions). However, this does not devalue the book owing to the reasons presented above. I highly recommend this text for anyone who has an appreciation of the strong link between the physical world and its description via mathematical constructions.
Given all of this, I highly recommend a supplementary text for those using this book to learn new material - Arfken and "Mathematics of Classic and Quantum Physics" by Byron and Fuller are both very readable. Those using it as a refresher or reference will find a thorough, compact, and consistent book.
Hassani first introduces the concept of a vector space and gives numerous examples including the less "intuitive" function spaces and matrix spaces. He quickly builds upon this idea to encompass linear operators, algebras and functions defined in terms of them. Hassani's sweep of basic concepts is comprehensive and thorough while seamlessly weaving in ideas from many different branches of mathematics that provide the edifice for much of modern physics. The side notes enable one to browse through the book and find a particular topic. Interspersing the text with short biographical sketches of mathematicians who made important contributions to the field within the past three centuries adds further interest to the book.
The author has devoted much of the book to those special functions that emerge as solutions to the prototype differential equations of Physics. These functions are presented at a more generalized and rigorous mathematical setting than in many Mathematical Physics books aimed at beginning graduate students while sparing the more tedious proofs all too common in books on Functional Analysis, for example. In particular, his exposition of Green functions and Operator Theory are much more comprehensive and easy to follow than comparable treatment in other texts targeting the same readership.
Hassani continues on through Lie Groups to end this tome with a look at symmetries and conservation laws. The latter are especially relevant in Quantum Field Theory and HEP. Topics given cursory treatment in texts on these subfields of Physics are presented by Hassani in greater detail, again sparing the reader the mathematics that often serve more to sidetrack that edify.
Given the subject matter, it is no easy task to produce such a user-friendly tome, but Hassani has admirably risen to the task. I look forward to more texts by the author, perhaps one with more emphasis on Measure Theoretic approaches?