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I just saw a post like this one, but particularly for statistical mechanics, I thought I'd ask the question in general.

Where does a mathematically trained person go to learn mathematical physics? By that I mean, what books or manuscripts are demanding in the area of mathematical maturity but not particularly demanding in the area of physics knowledge (physics maturity I guess, idk if they use that word in physics?). I myself am particularly interested in computational fluid dynamics and other kinds of computational physics, but I want to keep this general to help as many people as possible. Also, if someone knows a good book for mathematicians to help with one of the biggest difficulties I've found "Physics INTUITION" that'd be helpful.

As usual, one answer per post so votes can be tabulated well.

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7 Answers 7

up vote 2 down vote accepted

A classic reference is Courant and Hilbert (volume 1, volume 2).

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Ahh yes, Courant. I remember my father (a chemist) always saying, whenever you need to know a formula, check Courant, then you'll sound smart when you know the answer. =) –  Michael Hoffman Nov 5 '09 at 1:58

There is a problem with this kind of question, namely for many mathematicians the most interesting mathematical physics is a new vast area on the interface of quantum field theory and geometry/topology emerging from about late 1960s till now. You will find no word on this new mathematical physics in the classical books like Reed-Simon, Morse-Feshbach (Methods of mathematical physics, 1953 and later ed.), Vladimirov (Equations of mathematical physics) and even older Courant-Hilbert which focus on the integral and differential equations of mathematical physics, special functions, generalized functions (distributions), representations of classical groups and functional analysis. For your classical hydrodynamics indeed the classical textbooks and reference books suffice, but for people interested in a bit more modern mathematical physics we could add (in various level of exposition and specialization)

  • Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, Analysis, manifolds and physics, 1982 and 2001

  • Albert Schwartz, Quantum field theory and topology, Grundlehren der Math. Wissen. 307, Springer 1993. (translated from Russian original)

  • Bernard F. Schutz, Geometrical methods of mathematical physics (elementary intro)

  • Eberhard Zeidler, Quantum field theory. A bridge between mathematicians and physicists. I: Basics in mathematics and physics. , II: Quantum electrodynamics

  • Charles Nash, Differential topology and quantum field theory, Acad. Press 1991.

  • P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, eds. Quantum fields and strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

  • Gregory L. Naber, Topology, geometry, and gauge fields: interactions

  • Mikio Nakahara, Geometry, topology and physics

  • Peter Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, UK, 1995.

  • James Glimm, Arthur Jaffe, Quantum physics: a functional integral point of view, Springer

  • Sternberg, Shlomo (1994), Group theory and physics, Cambridge University Press.

  • V. I. Arnold, Mathematical methods of classical mechanics, Springer (1989).

  • V. Guillemin, S. Sternberg, Symplectic techniques in physics, Cambridge University Press (1990)

  • Leon A. Takhtajan, Quantum mechanics for mathematicians, Graduate Studies in Mathematics 95, Amer. Math. Soc. 2008.

  • Marian Fecko, Differential geometry and Lie groups for physicists

  • V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, AMS and Courant Institute, 2004.

  • R. E. Borcherds, A. Barnard, Lectures on QFT, arxiv:math-ph/0204014

  • Paul Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, Dirichlet branes and mirror symmetry, Amer. Math. Soc. Clay Math. Institute 2009.

  • R. S. Ward, R. O. Wells, Twistor geometry and field theory (CUP, 1990)

  • N. N. Bogoliubov, A. A. Logunov, I. T. Todorov, Introduction to axiomatic quantum field theory, 1975

  • O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge Univ. Press 2003.

  • Martin Schottenloher, A mathematical introduction to conformal field theory

  • Philippe Di Francesco,Pierre Mathieu,David Sénéchal, Conformal field theory, Springer 1997

  • T. Miwa, M. Jimbo, E. Date, Solitons: Differential equations, symmetries and infinite dimensional algebras, Cambridge Tracts in Mathematics 135, translated from Japanese by Miles Reid

  • V. Kac, Vertex algebras for beginners, Amer. Math. Soc.

  • Ludwig D. Faddeev, Leon Takhtajan, Hamiltonian methods in the theory of solitons, Springer

  • V.E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Univ. Press 1997.

  • N. P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics 1998. xx+529 pp.

  • Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf

  • A. Cannas da Silva, A. Weinstein, Geometric models for noncommutative algebras, 1999, pdf

I have placed these references in new nlab entry books and reviews in mathematical physics which will be updated at times wuth more specialized references.

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The above mentioned nlab entry is now already much improved (subsections, more references, links) in comparison to the above raw list. –  Zoran Skoda May 18 '10 at 16:36

For a mathematician who wants to learn some (classical) physics, the first book I'd recommend is Arnold's "Mathematical Methods of Classical Mechanics".

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Courant and Hilbert is great. However, it predates many significant developments in mathematics and physics in much of the 20th century. A more recent such work is Reed and Simon's Methods of Modern Mathematical Physics.

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Large parts of Courant and Hilbert is orthogonal to the content of Reed and Simon. I wouldn't suggest one as the replacement of the other. –  Willie Wong Jun 18 '10 at 22:23

While not directly geared at the interest you mentioned, if a mathematician were trying to learn quantum gravity(perhaps to work in this field from the math side), I would recommend they consult the duo of books by Rovelli and Thiemann.

The first book focuses on the physics side, and builds your background in the physics of quantizing gravity from the loop quantum approach. The second book focuses on the mathematical methods used and the specific mathematical models currently in use. Additionally, both books provided me with much needed exposition in the field, feeding my desire as a mathematician to understand why many of these math structures apply to these physical phenomena.

The first book is Rovelli: Quantum Gravity, the second is Thiemann: Modern Canonical Quantum General Relativity.

I warn you they are large books.

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It is a bit misleading to say that "a mathematician can learn quantum gravity" from these books. Where they cover classical material about classical gravity this may be the case, but beyond that these books discuss a mathematical construct whose relation to quantum gravity a bit uncertain. –  Urs Schreiber May 18 '10 at 16:15
That is a good point Urs, I didn't mean to suggest these were in some way comprehensive, just some useful references to get some perspective on the field. I am far from an expert so this advice should be taken with a spoon of salt. –  B. Bischof May 19 '10 at 3:04

Another book is Robert Geroch's Mathematical physics, although this may perhaps be more properly characterized as a book on modern mathematics for physicists.

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As no one has given any references that aim to improve physical intuition, I would recommend Einstein's classic "The Meaning of Relativity". This short book gives good physical insights into the theories of special and general relativity, and the mathematics should be easy enough for someone with a knowledge of differential geometry. It contains a good portion of text that aims to explain Einsteins reasoning as the basis for the theories, as well as some more philosophical ponderings.

For someone less knowledgable in differential geometry, I think that Einstein's book, coupled with J.G. Simmonds' "A Brief on Tensor Analysis" would be a good combination. Although it is a math book, Simmonds goes a long way in offering physical intuition as a means of motivating differential geometry and tensor analysis for physics and engineering students. Also, this book is short, and at an undergraduate level, so the first two chapters should read extremely fast for a seasoned mathematician.

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Thanks! Intuition is always key! –  Michael Hoffman Jun 18 '10 at 14:09

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