"Methods of Theoretical Physics for Mathematicians"

Question

I read in the Princeton Companion to Mathematics that even pure mathematicians should know some theoretical physics. However, I see that there are many reference books of mathematical methods for physics, but I cannot find any succint reference book of ideas of physics often applied in mathematics. Could you point out some such reference books [if they exist]?

Answer 1

Have a look at The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose.

Penrose as you probably know a great mathematician (e.g., Penrose tilings) and a mathematical physicist (e.g., major contributions to relativity). This book is a synthesis of his worldview of the physical world. It uses a great deal of mathematics. So if as a mathematician, you want to know what mathematics is used in theoretical physics and what is one coherent acount of thinking about the current state of knowledge, this is a guide.

I especially like Road to Reality because there is a clear narrative / account of the theory. So it's not just a collection of techniques.

Answer 2

Whatever you intend by "condensed", the standard mathematical physics book is Reed/Simon: Methods of modern mathematical physics. It is condensed in the sense that it is written concisely, without a lot of fuss, very elegantly written. There are four volumes, in increasing difficulty. It is written for mathematicians and physicists alike, in any case the treatment is fully rigorous, and you should have taken a course in topology and measure theory (albeit they include a chapter on topology, and a bit on measure theory).

Unlike Penrose, Reed/Simon is in some sense a book for the "working mathematical physicist".

Answer 3

A nice text explaining a lot of modern physics from the viewpoint of a professional mathematician is

Albeverio, Sergio; Høegh-Krohn, Raphael; Fenstad, Jens Erik; Lindstrøm, Tom. Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Academic Press, Inc., Orlando, FL, 1986.