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In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to theoretical physics for working mathematicians? I have heard of Armin Wachter and Henning Hoeber's, but I don't know if it is rigorous enough (i.e., for example, there are enough proofs of the theorem given).

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    $\begingroup$ There are many areas of physics. Surely some are more pertinent/relevant to mathematicians than others. Hopefully someone answering keeps this in mind in their recommendations. $\endgroup$ – Stan Shunpike Dec 28 '14 at 2:35
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    $\begingroup$ The idea that one book could be a comprehensive companion to any well-established academic field strikes me as a bit far-fetched. $\endgroup$ – Pete L. Clark Dec 28 '14 at 5:38
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    $\begingroup$ And yet there exists a Princeton Companion to Mathematics. $\endgroup$ – Allen Knutson Dec 28 '14 at 9:53
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    $\begingroup$ Gerard 't Hooft has a web page aimed at giving guidance to those teaching themselves physics. It has some lists of good books. staff.science.uu.nl/~Gadda001/goodtheorist/index.html $\endgroup$ – Dan Fox Dec 28 '14 at 11:59
  • $\begingroup$ @Allen: Yes, but no Princeton Comprehensive Companion to Mathematics. (Perhaps I am taking that one word too seriously...) $\endgroup$ – Pete L. Clark Jan 5 '15 at 1:49
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If you allow such a comprehensive reference to re-introduce basic mathematics, then either as a layman or a working mathematician your prayers are answered by the following (he even prefaces by saying that his intended layman-audience must have some mathematical sophistication):

The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose

Now let's try to break down the subjects.

Classical Mechanics:
1) Mathematical Methods of Classical Mechanics, by Arnold
2) A Mathematical Introduction to Fluid Mechanics, by Chorin-Marsden

Quantum Mechanics:
1) Mathematical Foundations of Quantum Mechanics, by Mackey
2) The Theory of Groups and Quantum Mechanics, by Weyl

General Relativity:
1) General Relativity for Mathematicians, by Sachs-Wu
2) The Large Scale Structure of Space-Time, by Hawking-Ellis

Electrodynamics:
1) Electromagnetic Theory and Computation: A Topological Approach, by Gross-Kotiuga
2) On the Mathematical Foundations of Electrical Circuit Theory, by Smale
3) This is a plug for Gauge theory:
3a) On Some Recent Developments in Yang-Mills Theory, by Bott
3b) On Some Recent Interactions Between Mathematics and Physics, by Bott
3c) Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields, by Wu-Yang
3d) From Superconductors and Four-Manifolds to Weak Interactions, by Witten

Standard Model:
The Algebra of Grand Unified Theories, by Baez-Huerta

Quantum Field Theory and String Theory:
1) Quantum Physics: A Functional Integral Point of View, by Jaffe-Glimm
2) Geometry and Quantum Field Theory, 1994 IAS lectures
3) Quantum Fields and Strings: A Course for Mathematicians, 1996 IAS lectures

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    $\begingroup$ Can you give some references to Bott's papers? $\endgroup$ – Paracosmiste Dec 28 '14 at 10:58
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    $\begingroup$ +1 for recommending Roger Penrose. Road to Reality is a great start to understanding physics from a mathematical perspective. Penrose starts from the foundation of numbers and by the end goes to twistor theory. $\endgroup$ – Jeel Shah Dec 29 '14 at 16:25
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If you are not interested too much in details, the following book can play the role of a comprehensive companion: http://www.amazon.com/Unified-Grand-Theoretical-Physics-Edition/dp/1439884463 (A Unified Grand Tour of Theoretical Physics, by Ian D. Lawrie).

Truly comprehensive systematic introduction to theoretical physics can be found in the (many-volume) well known "Course of Theoretical Physics" by Landau and Lifshitz: http://en.wikipedia.org/wiki/Course_of_Theoretical_Physics as well as in the more contemporary German counterpart by Walter Greiner: http://onphysicsbooks.blogspot.ru/2009/01/walter-greiner.html

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The answer to this question depends sensitively on how much physics you want to learn.

For a brief overview of all of physics, two good choices are The Six Core Theories of Modern Physics by Charles Stevens and The Theoretical Minimum by Leonard Susskind.

If you want to delve more deeply then I think it is best to go for a book that treats just one subfield of physics, such as classical mechanics or quantum field theory. Some good suggestions are listed in the related Physics StackExchange question.

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Take a look at Physics and Partial Differential Equations, by Tatsien Li and Tiehu Qin, published by SIAM.

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Try Eberhard Zeidler's multi-volume Quantum Field Theory. This is extremely comprehensive.

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To give a partial answer: this is a nice companion to quantum physics for mathematicians (especially those that are into category theory and/or operator algebras): Deep Beauty: Understanding the Quantum World through Mathematical Innovation, ed Hans Halvorson.

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