Very *mathematical* general physics book

Solution 1:

There are now quite a few excellent ones, but most of these are pitched at fairly sophisticated readers-graduate students or professional mathematicans. The thinking according to such textbooks, of course,is that the readers are very far along in thier mathematical training and are ready to use that mathematics to learn physics at a very high level. Whether or not that's true is debatable. In any event, here's a list to get you started:

Lectures on Quantum Mechanics for Mathematics Students by L. D. Faddeev and O. A. Yakubovskii: Really the only one for undergraduates-a beautiful and justly famous Russian treatment now in English. But even this beginning text requires a good knowledge of differential equations, real and complex analysis, linear algebra and group theory.

Quantum Mechanics for Mathematicians by Leon A. Takhtajan : One of several such texts for second year graduate students, most of which need at least first year graduate courses in analysis,differential geometry and topology. Excellently written,though.

Physics for Mathematicians, Mechanics I by Michael Spivak : Awesome book by the master. Perfect follow up to his 5 volume opus on graduate differential geometry and shows in depth what all that beautiful manifold theory was good for. Let's hope he finishes the projected other 3 volumes.

Quantum Theory for Mathematicians by Brian C. Hall A recent addition to the list I haven't seen. But if it's as good as his Lie groups book, it'll be well worth checking out.

Mathematical Methods in Quantum Mechanics by Gerald Teschl : This book is really more about the methods of quantum theory then the science itself. That being said,it's by a master and presents the material beautifully for second year graduate students.

Quantum Field Theory by Gerald B. Folland: Another terrific textbook by Folland, covering everything about QFT that can be made rigorous at this point,which sadly isn't as much as we'd like. You better have your graduate analysis chops on before tackling this one.

Semi-Riemannian Geometry With Applications to Relativity by Barrett O'Neill : Exactly what the title says it is. A classic,one of the best books on relativity theory for mathematicians. The book to read after Spivak.

Modern Geometric Structures and Fields by S. P. Novikov and I. A. Taimanov: An incredible and completely unorthodox beginning graduate course in differential geometry by one of the premier mathematicians in the world that completely interweaves the careful theory with virtually the entire modern structure of physics-from basic mechanics through relativity theory through the beginnings of string theory and quantum field theory.A good working knowledge of rigorous calculus of one and several variables and linear algebra is all you need to tackle this one. An absolute must have if you're interested in the relationship between mathematics and physics. I DO wish it had more exercises.

UPDATE: I've added a few more in response to some of the other posts here:

Foundations of Mechanics 2nd edition by Ralph Abraham and Jerrold E. Marsden: A remarkable and careful text for both physicists and mathematicians giving one of the first detailed presentations of classical mechanics from the modern differential geometric point of view. Very advanced, really for strong graduate students in mathematics or very strong graduate students in physics. A good follow up to Spivak's mechanics book and a first year graduate mathematics course sequence.

Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems by Jerrold E. Marsden and Tudor S. RatiuL A very mathematical first year graduate course in classical mechanics,again from the moder geometric point of view and emphasizing symmetry. Can act as the preliminary text for the text immediately preceding this one. About the same level as Spivak,but it's not as rigorous and focuses more on advanced physics then geometry. Very good collateral reading to that text.

Symplectic Techniques in Physics by Victor Guillemin and Shlomo Sternberg: I've never seen this book, but I've heard good things about it. It's by 2 masters,so that by itself makes it worth checking out.

Ordinary Differential Equations by V.I. Arnold: A terrific intermediate level course emphasizing the role of linear transformations and manifolds in the study of linear ODEs. A strong course in linear algebra and geometry is needed; this beautiful text is a must for students interested in the geometry of differential equations and physics.

Geometrical Methods in the Theory of Ordinary Differential Equations by V.I. Arnold: Graduate level follow up to the preceding text. An introduction to ordinary differential equations and dynamical systems emphasizing thier role in chaos theory and physics. A very difficult and sometimes mystifying book, but the sheer richness of the ideas and the interplay of math and physics makes it worth the effort.

I'll also mention in passing at the end the wonderful physics textbooks by Walter Griener. Not only are they concise,wide ranging and extremely clear, they have more solved examples then any physics book I've ever seen. They're written by a master. If you can get them in the original German and can read those,that would be even better as some errors creeped in in translation.

Hope that gets you started. Good luck!

Solution 2:

Here is another bunch of texts. Like the ones suggested by Mathemagician1234, they are not general texts. The level of formality is variable.

Classical mechanics

F. Scheck, Mechanics, Springer, 2010. Although not specifically geared toward mathematicians, it makes use of mathematically advanced tools. I consider it the best book on classical mechanics currently available, much superior to Goldstein.

A. Fasano, S. Marmi, Analytical Mechanics, Oxford University Press, 2006. Thorough and complete textbook, strongly mathematically oriented (at undergraduate level).

N. M. J. Woodhouse, Introduction to analytical dynamics, Springer, 2009.

Quantum mechanics:

S. J. Gustafson and Israel, Mathematical concepts of quantum mechanics, Springer, 2011.

F. A. Berezin and M. A. Shubin, The Schroedinger equation, Kluwer, 1991.

F. A. Berezin, The method of second quantization, Academic Press, 1966.

Special relativity

G. L. Naber, The geometry of Minkowski spacetime, Springer, 2010. This is superbly written. There are also two other books from the author on gauge theories.

N. M. J. Woodhouse, Special relativity, Springer, 2003.

Statistical mechanics

F. A. Berezin, Lectures on statistical physics, Online

R. A. Minlos, Introduction to Mathematical Statistical Physics, American Mathematical Society, 2000.

Last but not the least I'd like to suggest two (wonderful) calculus books which, instead, are full of examples taken from physics:

V. A. Zorich, Mathematical Analysis I and II, Springer, 2004.

Solution 3:

Since no one has mentioned it, I must add V.I Arnold's excellent "Mathematical Methods of Classical Mechanics".

As the title suggests, the book focuses on classical mechanics, which is always a good start for a physics education. I'd suggest reading this or other good classical mechanics texts before moving on to various quantum theories, as it's difficult to get a good understanding of the later without insight into Lagrangian and Hamiltonian Mechanics.