Mathematical Physics: A Modern Introduction to Its Foundations 2nd Edition by Sadri Hassani (PDF)

Description

The goal of this book is to expose the reader to the indispensable role that mathematics plays in modern physics. Starting with the notion of vector spaces, the first half of the book develops topics as diverse as algebras, classical orthogonal polynomials, Fourier analysis, complex analysis, differential and integral equations, operator theory, and multi-dimensional Green’s functions. The second half of the book introduces groups, manifolds, Lie groups and their representations, Clifford algebras and their representations, and fibre bundles and their applications to differential geometry and gauge theories.This second edition is a substantial revision with a complete rewriting of many chapters and the addition of new ones, including chapters on algebras, representation of Clifford algebras, fibre bundles, and gauge theories. The spirit of the first edition, namely the balance between rigour and physical application, has been maintained, as is the abundance of historical notes and worked out examples that demonstrate the “unreasonable effectiveness of mathematics” in modern physics.

User’s Reviews

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

⭐This product, a 1200- page encyclopedia of graduate-level mathematical physics, is a sequel to the author’s “Mathematical Methods for Students of Physics,” a book not only written with undergraduates in mind instead, but one that also carries subjects like Complex Analysis, Differential Equations and Tensors nearly seamlessly with this volume, making these two works a two-thousand page masterpiece deserving of its place by Gravitation and Classical Physics on my shelf.Starting with finite-dimensional vector spaces (ending all the way over in the Polar Decomposition, meaning this is very nearly a self-enclosed course in applied Linear Algebra including operator theory!), the book then addresses infinite-dimensional vector spaces like Hilbert Spaces, orthogonal polynomials and everyone’s favorite Fourier approximation methods. The third subect continues the earlier volume’s Complex Analysis work, ending in advanced topics like the all-important meromorphic functions, but reviewing a bit more from the earlier volume with other topics like the Cauchy integral (while adding critical methods like the Principal Value, as well).After continuing the Differential Equations sections from the earlier volume, capping off with a treatment of Sturm-Liouville, the volume continues with group and Lie theory before moving on to the section I’ve gone in depth on the most – tensors and differential geometry. This is a well-done section respectful of the mathematics behind the treatment. Gauges and Calculus of Variations are also included here. Although Hassani is an admitted “diffy-phile,” I think this part – from Group/Lie to Riemannian/Gauge – is the best part of this amazing tome. This will give me years of learning and re-learning.If you are a mathematical physicist or are interested in it as a possible area of exploration as a critical link between these two regal disciplines, buy this book, and buy its prequel, Mathematcal Methods for Students of Physics. These combine to form two thousand pages of a classic in the making for this field.The only critique – apart from occasional grammar and other typos bespotting Springer publications lately – is a heavier preference for mathematcial notation over the home team notation for physics. I suppose I could also throw in a weakness with Probability and Statistics – which is only really covered briefly at the end of the preceding volume. These are nothing to the value the book gives, however. Get both in this set!

⭐I cannot recommend this text to anyone attempting to learn Mathematical Physics. As other reviewers mentioned there isn’t enough depth to learn from the text. It’s an admirable attempt, but virtually all topics require much more elaboration for a student that has never seen the information before.Reiterating what others have said, this book is a GREAT reference. I can only recommend it as a reference. I will probably return it as I have no use for a $100 reference book that has information I can find on the internet for free.

⭐I have owned this book for a month or so now and have been using it primarily for the algebra chapters. So far, it has exceeded my expectations.Depth. This book is pretty thick (3″, actually). There are 37 chapters grouped into 10 parts: Finite Vector Spaces, Infinite Vector Spaces, Complex Analysis, Differential Equations, Operators, Green’s Functions, Groups/Representations, Lie Groups, and Fiber Bundles. Hassani actually renders quite a few of my other books unnecessary.Clarity. Concepts are explained clearly and the author really tries to help you understand the abstract concepts that are presented within. Take, for instance, the section on group actions. Following five definitions thrown at you in a single paragraph, you are presented with Remark 23.3.1:”… If you think of G_m [the stabilizer of m] as those elements of G that are confined to (stuck, or imprisoned at) m, then a “free” action of G does not allow any point of M to imprison any subset of G. …”Referencability. Typically textbooks have to make tradeoffs between being good learning material and being a good reference. Hassani I would say does both well. While being very clear and readable, the book is also arranged very nicely for anyone looking for a reference textbook. Important definitions and theorems are boxed and key points are summarized in the margins.In addition, each section contains historical notes about various mathematicians and physicists that are quite interesting to read. I do still have a couple of small complaints, though. The first is the size of the book. The thickness of the pages renders the book very large which makes it difficult to transport and to read on the bus. Given the range of topics covered, I would prefer the book be split into multiple volumes. Secondly, though I did mention clarity as a strength of this book, this does vary somewhat from chapter to chapter and some topics (the geometry chapters in particular, Clifford algebras especially) seem to be following a define-first-motivate-later approach.

⭐This book is an invaluable collection of mathematical principles required for advanced physics. But it should not be intended for instruction. This book educates little and is simply a massive reference to useful mathematical principles. If you have already been through a course like this and have your doctorate, then this is likely your favorite reference. If you are in an advanced mathematical physics course and this is the text, you are basically at the mercy of the talents and abilities of the instructor to teach you the concepts you have yet to learn, while this book will be a great reference when doing the homework. As well as any other math phys book, countless websites on the topics, and all of the peers in your course.

⭐Very good reference but doesn’t expound on the topics enough to be used for learning them the first time around. You need a familiarity with most of the things shown before this is very useful.It is an excellent compendium on mathematical techniques and topics needed for modern physics. For what its purpose is though I was expecting more applications outside the exercises.

⭐Im angloamerikanischen Sprachraum scheint es ja einigermaßen üblich zu sein, Bücher über mathematische Methoden der Physik als “Mathematical Physics” zu betiteln. Das rechtfertigt es noch lange nicht. Im Vorwort wird ja völlig korrekt erklärt, was mathematische Physik ist. Da wird Hermann Weyl zitiert! Das Buch ist nicht schlecht, wenn man zu einem Thema einen ersten Einstieg sucht. Im Vergleich zu anderen Büchern über mathematische Methoden kann sich dieses Buch durchaus sehen lassen. Leider werden die meisten Themen aber äußerst oberflächlich behandelt, wie z.B. Fouriertransformation, Distributionen oder Mannigfaltigkeiten. Man darf nicht erwarten, irgendwo zu finden, was eine Schwartfunktion ist oder was das Paley-Wiener-Theorem besagt. Eine Mannigfaltigkeit wird oberflächlich definiert als eine Menge von Punkten, die sich auf glatte Weise lokal euklidisch aussieht. Die Begriffe “topologischer Raum”, “hausdorffsch”, “zweites Abzählbarkeitsaxiom” fallen nirgends. Man darf von diesem Buch auch nicht erwarten, den Unterschied zwischen einer Hamel- und einer Schauderbasis zu erfahren.Ich empfehle die beiden Bände “Moderne mathematische Methoden der Physik” von Goldhorn, Heinz und Kraus als deutlich überlegene Alternative! Dort werden nicht so viele Themen diskutiert, die ganze lineare Algebra und Analysis/Funktionentheorie wird vorausgesetzt (Maßtheorie wird in einem Kapitel behandelt – im Gegensatz zu dem hier besprochenen Buch, das Maßtheorie weder behandelt, noch voraussetzt oder erwähnt; dass das L im L² für Lebesgue steht, wird in einer biographischen Notiz erwähnt, dass Lebesgue die Riemannsche “notion” für “highly discontinuous functions” verallgemeinert hat). Die behandelten Themen werden aber qualitativ deutlich besser und tiefer diskutiert und das mit weniger Seiten und zu einem günstigeren Preis.Ich will das Buch hier nicht schlechtreden. Auch, wenn Vieles nicht rigoros gemacht wird, erfährt man hier interessante Dinge. Aber wer wirklich an mathematischer Physik interessiert ist, sollte sich an lieber an oben genanntes Buch halten sowie für die mathematischen Aspekte der QM das vierbändige Standardwerk “Methods of Modern Mathematical Physics” von Michael Reed und Barry Simon und für die physikalischen Aspekte das vierbändige Standardwerk “Lehrbuch der mathematischen Physik” von Walter Thirring. Als goldener Mittelweg existiert eine vielversprechende geplante Lehrbuchreihe von Andreas Knauf (die bis jetzt jedenfalls einen ausgezeichneten Mechanik-Band enthält!). “Mathematical Methods in Quantum Mechanics” von Gerald Teschl kann ich ebenfalls sehr empfehlen, ebenso das Buch “Curvature in Mathematics and Physics” von Shlomo Sternberg. In diesen Büchern wird wirklich mathematische Physik gemacht. In diesem nicht!Da das Buch den Anspruch erhebt, rigoros zu sein, halte ich diese mittelmäßige Bewertung für angemessen, denn das ist irreführend.

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⭐Ich kann die Rezension von “Mathematik- und Physikstudent” nicht nachvollziehen, ich fand das Buch inhaltlich ausgezeichnet.Nach dem üblichen Grundcurriculum in theoretischer Physik gibt es auch fortgeschrittene Theorie-Vorlesungen, für die weitergehende Methoden benötigt werden. In der theoretischen Physik im Rahmen einer Promotion sowieso. Als Anwender der Mathematik ist es sinnvoll, ein systematisches Grundverständnis für die mathematische Theorie zu haben und genau das leistet Hassani’s Mathematical Physics in meinen Augen. Dazu tragen auch die vielen (nicht immer ganz trivialen) Übungsaufgaben bei, die für ein anwendungsorientes Verständnis hilfreich sind. [Anm.: Ich kann nur über den Differentialgeometrie und Green’s Funktionen Part genauer berichten, aber diese Aufgaben waren eigentlich durch die Bank weg gut bearbeitbar und auch der Zusammenhang zum Lehrtext war ersichtlich.]Es ist ersichtlich, dass der Autor des Buches einen großen Schwerpunkt auf die Theorie und Anwendung der Green’schen Funktionen legt – Dreh- und Angelpunkt der Exposition sind die (gewöhnlich und partiell) Differentialgleichungen, wie sie in der theoretischen Physik gerne vorkommen. Die mit den Differentialgleichungen assoziierbaren Integralgleichungen werden ebenfalls in einführender Manier behandelt, wobei eine detailliertere Diskussion des DuHamel’schen Prinzips für lineare Differentialgleichungen wünschenswert gewesen wäre. Differenzengleichungen und speziellere Themen wie Delay-Differentialgleichungen finden leider wenig Eingang.Sehr gut gefallen, und auch in Teilen in Tutorien zum Einsatz gekommen, hat mir die Darstellung zur Konstruktion eichinvarianter Lagrange-Dichten, der Groschen ist auf diese Weise bei vielen Teilnehmern gefallen.Wer mehr Wert auf die mathematische Terminologie legt, ist mit Texten der reinen Mathematiker deutlich besser beraten oder Thirring.

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⭐Take a look at the dimensions before buying this book; it’s one of the biggest books I’ve ever seen! Personally, I think it should have been split up into 2 or 3 volumes, just to make it a bit easier to carry around and handle. I may even look into splitting it up and re-binding it myself.The kindle edition might seem like a less unwieldy way to go, but it has it’s own problems. I actually returned the kindle edition and bought the hard copy because the kindle edition had really poor formatting of mathematical expressions and symbols (seems to be a common problem with kindle books). On my tablet, the book was unreadable in some places because some of the symbols were unrecognized. I can tolerate math formatting issues when a book is cheap, but for $80, I expect some half-decent math typesetting…On the plus side, this book looks great content-wise (though I haven’t read much yet). I’m still glad I made the purchase (hence the 4 star rating); I just wish it came in an easier-to-use format.

⭐Whatever Problem you’re facing, the book has a Chapter about it. There are Definitions with explanations of them, which is very usefull. Also Examples and problems for your own turn. I bought myself a lot of Mathbooks, some of them i regret, but i didn’t regret the buy of this one.

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⭐Un ottimo testo. Tanto vasto quanto ben scritto. Eccellente per l’insegnamento “Metodi Matematici per la Fisica” del secondo anno del corso di Laurea in Fisica.

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