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).
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.
1) Mathematical Methods of Classical Mechanics, by Arnold
2) A Mathematical Introduction to Fluid Mechanics, by Chorin-Marsden
1) Mathematical Foundations of Quantum Mechanics, by Mackey
2) The Theory of Groups and Quantum Mechanics, by Weyl
1) General Relativity for Mathematicians, by Sachs-Wu
2) The Large Scale Structure of Space-Time, by Hawking-Ellis
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
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
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
The answer to this question depends sensitively on how much physics you want to learn.
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.
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.