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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]?

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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.

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    $\begingroup$ What is it? Could you add some remarks on the book? $\endgroup$ – Dal Dec 24 '14 at 13:48
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    $\begingroup$ Penrose as you probably know a great mathematician (e.g., Penrose tilings) and a mathematical physicist (among other things he has made 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, this is a good resource. $\endgroup$ – Simon S Dec 24 '14 at 13:52
  • $\begingroup$ $+1$ Thank you very much for pointing it out then :). $\endgroup$ – Dal Dec 24 '14 at 13:55
  • $\begingroup$ Do you have any other (also more technical) recommendations? $\endgroup$ – Dal Dec 24 '14 at 13:55
  • $\begingroup$ I recommend you look at this book at your library/bookshop; I think you'll find it technical. It also has references if you want to dig deeper. Beyond this, I can't think of anything particularly on point. $\endgroup$ – Simon S Dec 24 '14 at 13:56
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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".

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  • $\begingroup$ What I meant by condensed is that I'm not searching for many different books on different topics in theoretical physics from the point of view of a mathematician. I'm looking for something similar to the books titled "Mathematical methods for physicists", which normally include a lot of topics from different areas of mathematics. Anyway, thank you for the suggestion. $\endgroup$ – Dal Dec 24 '14 at 17:37
  • $\begingroup$ @Dal: in that case it is the perfect book. it deals with basically anyting. $\endgroup$ – Mister Benjamin Dover Dec 24 '14 at 20:14
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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.

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