Techniques and Applications of Path Integration (Dover Books on Physics) by L. S. Schulman (PDF)

Description

A book of techniques and applications, this text defines the path integral and illustrates its uses by example. It is suitable for advanced undergraduates and graduate students in physics; its sole prerequisite is a first course in quantum mechanics. For applications requiring specialized knowledge, the author supplies background material.The first part of the book develops the techniques of path integration. Topics include probability amplitudes for paths and the correspondence limit for the path integral; vector potentials; the Ito integral and gauge transformations; free particle and quadratic Lagrangians; properties of Green’s functions and the Feynman-Kac formula; functional derivatives and commutation relations; Brownian motion and the Wiener integral; and perturbation theory and Feynman diagrams.The second part, dealing with applications, covers asymptotic analysis and the calculus of variations; the WKB approximation and near caustics; the phase of the semiclassical amplitude; scattering theory; and geometrical optics. Additional topics include the polaron; path integrals for multiply connected spaces; quantum mechanics on curved spaces; relativistic propagators and black holes; applications to statistical mechanics; systems with random impurities; instantons and metastability; renormalization and scaling for critical phenomena; and the phase space path integral.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The first line of chapter one reads, “The best place to find out about path integrals is in Feynman’s paper.” This is the paper “Space-time Approach to Non-relativistic Quantum Mechanics” (Rev. Mod. Phys. 20, p. 367 (1948)),which is included in “Selected Papers on Quantum Electrodynamics” edited by Julian Schwinger, also reprinted by Dover. So you may want to start there first.The current book has many short, readable chapters on various applications of path integration, each with copious references. The author’s bias is toward non-QFT applications, which is fine as there are plenty of books on QFT.This Dover reprint includes a 60 page supplement with notes, errata, and further references.

⭐This book is accessible and helpful. Rigorous proofs are in there but the emphasis is on explaining ideas clearly and simply. Am still reading this but highly recommend to anyone who has worked through the basics of QM and wants to get a good introduction to path integrals.

⭐I am very pleased with my purchase, I got the book as promised

⭐The author collected many branches of modern physics abouut path integral.It included many references for advanced reading.Anyone wants to go into Feynman’s path integral should read it.

⭐First note: the fifty page supplement can be downloaded from Schulman’s academic page. If you have the hardcover edition, then the supplement will update the book (errata, references, notes, Dover paperback).(1) Return to 1981 (original publication year). At that time, there was very little available if you desired an introductory account of path integrals. First, there was Feynman and Hibbs (1965), later there was an article, The Feynman Integral (1975, American Mathematical Monthly, Keller and McLaughlin). Those two publications are not really introductory. Shankar’s Principles of Quantum Mechanics (1980) textbook fit the bill as an introduction (chapter eight, ten pages), writing of the path integral: “its omission from so many books is hard to understand.” April 1980 saw a journal review article, Path Integrals in Quantum Theory (Physics Reports, Marinov, 57 pages), it is advanced.Reiterating: Introductions were difficult to come by.(2) Now, Schulman: “The level is such that anyone with a reasonable first course in quantum mechanics should not find difficulty.” (preface). I am uncertain if that is really realistic (as, Shankar’s 1980 text was a graduate-level introduction). Despite my suspicions, Schulman has presented a thoughtful book. It was one of the few expositions to combine both techniques and applications of path integration.For my tastes, chapter two, probability and probability amplitudes, is too brief.(3) The highlight of the book is the interplay between mathematics and physics. The mathematics is of two varieties: mundane and sophisticated. Of the mundane, Taylor series expansions are utilized often (as in most such expositions). An appendix lists integral formulas (page 20), a pity that the derivations were not presented on same page. A side note: I am still uncertain as to how to justify equation #5.13 (page 29). A pedagogic note, read sooner, rather than later: Section 32.2, “Un-completing the Square,” as it is too important to be relegated to the end of the book (pages 325-332). Some material, such as Sigma-algebras and cylinder functions will be unfamiliar to a beginner (page 57).(4) WKB approximation, remember that ? You will revisit that topic multiple times throughout the book. Read: “There is a WKB description of barrier penetration, what can path integration say ? “ (page 155). Schulman proceeds to say it. Before reading that, review Shankar (second edition, pages 429-450). Martin Gutzwiller’s book, Chaos in Classical and Quantum Mechanics, is a nice supplement to this material.(5) There are two Feynman diagrams (pages 65-69), look elsewhere for a detailed treatment. Writes Schulman: “as the reader may have guessed, quantum field theory is not my favorite arena for path integration (supplement, page 397).(6) You got scaffolding in the initial seventy pages, applications are next (Part Two, page 71).Chapters 11 and 12 will be mathematical reviews: asymptotic analysis (nicely done) and calculus of variations (page 72-91). Sidney Coleman is referenced (page 106 and 288), continue with his book, Aspects of Symmetry.(7) I highlight Chapter 22: Spin. The discussion is lucid, though brief. Spin is “separated” from external degrees of freedom (page 184, referencing WKB approximation). Recall time-ordered products (page 183).(8) Of applications: Chapter twenty-five (pages 225-236) describes relativistic propagators and connections to black-hole physics (which, for 1981, was another distinction for this book). The so-called Planck length is briefly alluded to (bibliographic note). Of prime interest (at least to my mind) is reference to appendix A of Feynman’s 1950 paper (find it in the Schwinger anthology: Selected Papers on Quantum Electrodynamics), as it deserves separate, detailed, study.(9) Concluding: I have only scratched the surface of Schulman’s thoughtful monograph. It includes exercises situated within sections, plentiful notes and bibliographic references. There is much to assimilate (for instance, Van Vleck determinants). Keeping abreast of the cited journal literature is an interesting part of this enterprise, and Schulman has given much guidance. I supplement Schulman with Shankar (read it first), Feynman’s own writings (especially his Papers) and the later-published Path Integrals and Quantum Processes (1992, Swanson).

⭐This is a book about Feynman’s interpretation of Quantum Mechanics as a `sum of all possible paths’. Feynman’s own book,

⭐, now also published by Dover is probably the easiest and nicest introduction to the topic. However, this book is a superb one since it collects all possible applications of Path Integrals in Quantum Mechanics. Divided in 32 relatively small chapters, you can choose which chapters to read and which to omit. Therefore, it can be used as an excellence reference looking up topics only when needed.There is no bias towards non-Quantum Field Theory (QFT) applications; this book is about applications in Quantum Mechanics, not applications in QFT. If you wish, you can see Quantum Mechanics as a 0+1 (zero space and one time dimension) QFT but this is not necessary. In Chapter 32, entitled `Omissions, Miscellany and Prejudices’, the author explains what topics are not covered in the book. Not surprisingly, his first topic is QFT. But this is another topic requiring another volume. I think that the choice of topics is very good and they all fit within the Quantum Mechanics framework in which the author chose to stay.This should be the second book after Feynman’s book that you should have in your bookcase for Path Integrals.

⭐This book can certainly be recommended. It is very good in giving concrete examples of applications of path integrals that go beyond the content of any standard textbook. But this book should not be used as an introduction into path integrals, since I would say it is more useful for readers with a basic knowledge in that field. Each chapter is short and provides enough references.For me, definetly a must-have.

⭐Sans prétention, simple et très clair. Je recommande vivement, de très haute qualitépar rapport à ses concurrents, hormis les deux livres de Feynman lui-même.

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⭐量子力学の表現形態の一つである経路積分を、幅広く適用できる事を示した、経路積分愛好家の為の本。これを読むと経路積分が原子核物理から宇宙物理、物性、散乱理論、統計力学まで適用できる事が解り、ファインマンの発想の素晴らしさを知る事ができる。幾何光学にまで経路積分を使って説明している所は著者の「意地」を感じる。（笑）この本を読めば、経路積分がミクロとマクロの世界をつなぐ理論である事がよく分かるので、読んでいて面白かった。計算は荒削りなので、しっかり理解する為には自分でゴリゴリ計算しなければいけない。分野は幅広すぎるので、全て丁寧に読み続けるのは時間の無駄かもしれない。自分が研究している分野で経路積分が使えそうな所があるならば、この本を読んで自分の考察対象に近い箇所をリファレンス的になぞりながら理解を深めるといい。

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