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It is a well-known phenomenon that mathematicians and physicists working on closely-related topics (say in gauge theories, or in general relativity etc.) generally approach the problems from very different angles and often have trouble understanding each other.

In this light, which book(s) do you wish theoretical physicists working on topics intersecting your own research area had read so they could more readily appreciate your work and/or more easily explain their own work to you in terms accessible to you?

To avoid this being opinion-based, please explain what essential concepts or techniques the theoretical physicist would learn from this book(s).

gmvh
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    I read this Book in 2012-2013 Alexei Kushner, et al. Contact Geometry and Nonlinear Differential Equations https://www.amazon.fr/Contact-Geometry-Nonlinear-Differential-Equations/dp/0521824761 –  Sep 16 '20 at 09:46
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    The best Book about Geometry of Quantization I read in 2013-2014 was N. M. J. Woodhouse , Geometric Quantization, https://www.amazon.fr/Geometric-Quantization-N-M-Woodhouse/dp/0198502702 –  Sep 16 '20 at 09:55
  • Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry which are very useful for Geometry of 3-forms in 6-dimensions (I read them in 2012) https://arxiv.org/abs/1008.0973 –  Sep 16 '20 at 09:58
  • See the World of Daniel Friedan about Cosmology and Ricci flow https://www.physics.rutgers.edu/pages/friedan/ –  Sep 16 '20 at 10:04
  • Positivity theory(Positivity of semi flat metric) and Hitchin fibration is open reasarch domain which is used in advanced area of Mathematical Physics such as C.Vafa. or Hausel et al. https://www.physicsoverflow.org/39137/any-progress-on-strominger-yau-and-zaslow-conjecture –  Sep 16 '20 at 10:11
  • Gauged Linear Sigma Model https://www.google.com/url?sa=t&source=web&rct=j&url=http://gokovagt.org/proceedings/2016/02ggt16-tianxu.pdf&ved=2ahUKEwi4583MvO3rAhUppIsKHbEJBg0QFjAJegQICBAB&usg=AOvVaw2b4z_RKTQhX1ONL2HpF0JE –  Sep 16 '20 at 10:33
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    I recall once hearing a physicist say that all functional analysts should read Dirac's Principles of Quantum Mechanics; he seemed sure that they had not all done so. – Ben McKay Sep 16 '20 at 15:51
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    I somewhat dislike the implicit premise of this question, but maybe I am reading too much into it. My very inexpert impression is that theoretical physicists are actually ahead of most of us in using modern techniques. – Mark Wildon Sep 17 '20 at 10:05
  • I think it goes both ways - there's physics maths that doesn't yet exist as maths maths, and there's existing maths that physicists rediscover (sometimes over and over again). In any case, I explicitly also ask for the case where the reading would enable the physicist to better explain their own research to mathematicians. – gmvh Sep 17 '20 at 10:16
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    @MarkWildon : I interpret the question to be analogous to a non-native speaker of English asking native English speakers for recommendations for a manual for idiomatic English that is geared toward speakers of English as a second language. – Timothy Chow Sep 17 '20 at 16:50
  • @HassanJolany Why don't you consolidate your comments into an answer? – gmvh Sep 19 '20 at 12:46
  • @TimothyChow That's a nice analogy! Indeed the question is intended to be mostly about things that aid communication between mathematicians, mathematical physicists, and theoretical physicists. – gmvh Sep 19 '20 at 12:48
  • I find it interesting that nobody so far has suggested any book dealing with categories, sheaves or stacks. These provide mathematical languages that (with the [occasional] exception of categories) aren't used in physics, but are quite relevant to a fair amount of mathematics that does have a non-empty intersection with the mathematics of theoretical physics. I would have thought that knowing those languages would be rather useful for conversations between physicists and mathematicians. – gmvh Sep 21 '20 at 06:31

4 Answers4

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"Les tenseurs" by Laurent Schwartz. This is the best book on the subject and I often feel that physics books do not really make a good job at presenting tensors. Too bad it has no English translation.

YCor
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coudy
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Quantum theory is an application of generalized probability theory (i.e., quantum logic). The most detailed introduction to this approach is: "Geometry of Quantum Theory", by V. S. Varadarajan https://smile.amazon.com/Geometry-Quantum-Theory-V-Varadarajan/dp/0387961240/ref=sr_1_3?crid=7P7GHVPJLJ95&dchild=1&keywords=varadarajan+quantum&qid=1600337321&sprefix=Varadarjan%2Caps%2C194&sr=8-3

Prof. David Edwards
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  • Second edition: https://www.amazon.com/Geometry-Quantum-Theory-V-S-Varadarajan-dp-0387493859/dp/0387493859/ref=dp_ob_title_bk – gmvh Sep 28 '20 at 09:23
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Nakahara's Geometry, Topology and Physics.

It was chapter $10$ that allowed me to fully engage with connections on fibre bundles and the relationship to Gauge theories. The earlier chapters on Holonomy and de Rham Cohomology groups was instrumental in introducing these concepts from a Physics perspective for me in the early part of my PhD.

asymptotic
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I always thought spinors were treated in a quite obscure way in physics. In particular it's difficult to make sense of the Dirac adjoint $\bar \psi= \psi^\dagger\gamma_0$ of a spinor (take a look at the wikipedia entry if you don't believe me) until one reads an introductory book on Clifford algebras, particularly one which emphasizes the non-Euclidean case and the natural metric on spinors, like for instance Garling's.

Fabien Besnard
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