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Which are the good introductory books on modern mathematical physics? Which are the more advanced ones?

I already read Whittaker's Analytical Dynamics, and I am reading Arnold's Mathematical Methods of Classical Mechanics. However, I am not so much interested in books about classical mechanics, neither am I interested in books for engineers. I have seen the related questions but their answers either offer books on classical mechanics, books for engineers, books for the public, or books on introductory mathematics, neither of which is any of what I need. I am already aware of Spivak's and of Landau’s books, so no need to include those. 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 and on cosmology. I did not put it in the title, because I want to include unfalsifiable theories. I mean anything from string theory thorough conformal cyclic cosmology to loop quantum gravity.

To state my background I am in the senior year of an undergraduate mathematics program. I took courses in classical mechanics, continuum mechanics, and mathematical models of physics. The last one concluded with an application of quantum mechanics in soliton theory. I audited a course in quantum field theory and a course in the calculation of conformal Feynman amplitudes. This course assumed knowledge in conformal field theories, which I lack.

I have taken or at least partially audited most of the offered courses in pure mathematics. If someone asks in a comment, I will make those explicit. I have at best poor knowledge of group theory, representation theory, Lie groups, operator algebras, symplectic geometry, analysis on manifolds, complex manifolds, differential topology, or algebraic number theory. I plan to cover some of those in part during the summer semester and in a master program.

So what are good books for a young mathematician who wants to dabble in physics?

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Would you think Advanced Mathematical Methods for Scientists and Engineers is too basic? –  Daniel R Jan 11 at 22:13
1  
This book is not too basic, but I am not interested in books written for engineers, @Daniel. I am looking for books introducing modern theories in mathematical physics and for books, I should consider once I have read the introductory books. –  Student Jan 11 at 22:25
    
Do you any background in Hilbert spaces? –  Mike Miller Jan 11 at 23:22
    
Yes, I do. @Mike –  Student Jan 11 at 23:28
2  
This is a book about mathematics and its applications to physics, but I'm currently enjoying Bleecker and Booss-Bavnek's Index Theory with applications to Mathematics and Physics which has no prerequisites other than knowing Hilbert spaces and the willingness to work. –  Mike Miller Jan 11 at 23:49

4 Answers 4

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).

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As a mathematical physicist, 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

  • Reed, Simon, Methods of modern mathematical physics,

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,

  • Thomas Thiemann, Modern and canonical quantum general relativity,

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

  • Deligne et al., Quantum fields and strings: a course for mathematicians,

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

  • Abraham, Mardsen, Foundations of mechanics.

Another is

  • Choquet, Bruhat, Analysis, manifolds and physics.

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

  • Zeidler, Quantum field theory,

an enormous amount of things (Zeidler style!) that covers all of the subject. Another excellent text on field theory is that of Haag,

  • Haag, Local quantum physics.
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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,

Fiat Lux!!!

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