Which are the good introductory books on modern mathematical physics? Which are the good advanced books?
I read Whittaker’s Analytical Dynamics, and I am reading Arnold’s Mathematical Methods of Classical Mechanics. However, I am not very interested in books on classical mechanics, nor books for engineers. The related questions offer mostly books on classical mechanics, or books for engineers or the public. I am aware of Spivak’s and of Landau’s books. I appreciate rigor and I am not interested in popular books. For instance, A Road to Reality is not appropriate but Lectures on Quantum Mechanics for Mathematics Students is.
I am asking in particular about books on quantum theories, gravity and on cosmology. I am also asking about unfalsifiable theories. I mean anything from string theory thorough conformal cyclic cosmology to loop quantum gravity.
To state my background I am a master student of mathematics. I took courses in classical mechanics, continuum mechanics, mathematical models of physics, quantum mechanics, field theory, etc. I audited courses on calculating conformal Feynman amplitudes in $\phi^4$ and in string theory, both of which assumed knowledge in conformal field theories, that I lack.
I have taken or partially audited diverse courses in mathematics. I will make these explicit if need be. I have some knowledge of group theory, representation theory, Lie groups, operator algebras, symplectic geometry, analysis on manifolds, complex manifolds, differential topology, etc.
So, what are good books for a young mathematician who wants to dabble in physics?
From my own experience, I will advise you against every book of mathematical methods written specifically for physicist. From my point of view, is better to learn about mathematics from mathematically written books (it sounds so obvious but is not). For example, many people like Schultz, Geometrical methods of mathematical physics, but I prefer to learn about the common topics in Singer, Thorpe, Lecture notes on elementary topology and geometry. (I don’t say it is not a good textbook, I only say I find difficult learning things on books written in a pretty informal way.)
The most complete work about methods of mathematical physics is probably
that covers functional analysis, Fourier analysis, scattering theory, operator theory.
Since you are interested in cosmology, the best review on Loop Quantum Gravity is that by Thomas Thiemann,
a 900 pages review, equipped with about 300 pages of mathematical methods (mathematical appendices are not a textbook however, but a collection of necessary results, eventually explored in some depth). References therein are very useful also.
Many people like
that joins a good part of your requests. (I haven’t read it, however, I know it since is “famous”.)
A celebrated book on methods of classical mechanics, concerning manifolds too, is
All Arnold’s books are always a great choice. (he wrote about ergodic theory and geometrical methods for differential equations, among the other things.)
There are a lots of more specific books, e.g. dealing with mathematical structure of quantum mechanics, but many of those are more and more specialized and is better to have very clear the general theory before try to get more involved into dangerous subjects such as, to say, quantum field theory. Once one has a strong background, the best opera on the subject of field theory probably is
an enormous amount of things (Zeidler style!) that covers all of the subject. Another excellent text on field theory is that of Haag,
EDIT. I’d like to add some book I’ve discovered more recently and I think fit very well:
Streater, Wightman, “PCT, Spin and all that”,
Teschl, “Mathematical methods in Quantum Mechanics”,
Bogolioubov, Logunov, Todorov, “Axiomatic Quantum Field Theory”,
Lansdman, “Mathematical concepts between classical and quantum mechanics”.
If Lectures on Quantum Mechanics for Mathematical Student work, then you should check Quantum Mechanics for Mathematicians written to provide a somewhat more modern and thorough exposition of Quantum Mechanics for Mathematical Student. (The author is a student of one of the authors of the former book).
In my experience, you can’t go wrong with anything by Arnold. If you are interested in a thorough treatment of mechanics from a mathematical point of view, check out Foundations of Mechanics by Abraham and Marsden. It would say it’s pretty advanced. If I recall correctly, Caltech has made it available for free on the web. Just google about a little and you’ll find the link. It’s not too big on quantum theory, and/or cosmology, however. For the latter, the tome Gravitation my Misner, Thorne and Wheeler is sort of a comprehensive classic.
Hope this helps. Cheerio,
and as always,
Folland’s book Quantum Field Theory: A Tourist Guide for Mathematicians looks good.