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I am a Masters student of math interested in physics. When I was an undergraduate, I took the introductory course of physics, but it is just slightly harder than high school physics course. To be precise, it just taught us how to use calculus in physics, without involving the higher knowledge of math such as manifold, PDE, abstract algebra and etc. By the way, the knowledge in that course is "discrete", the connections between fields of physics is omited.

My question is, what I should do to learn "real physics" by myself? What books or materials should I read?

LZB
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    This has some good answers: https://mathoverflow.net/q/2917/167834. It only covers QM, though. This is more general: https://mathoverflow.net/q/51395/167834. – Alessandro Della Corte Sep 08 '22 at 06:19
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    Be aware that there's a substantial difference between "real physics" as done by physicists (lots of experimental aspects which guide the reasoning, and non-rigorous calculations) and "physics for mathematicians" which takes care to mathematically justify each step but also usually has to simplify things, or can only consider special cases, to make some progress, and lags real physics by years or decades depending on the area. – Thomas Sauvaget Sep 08 '22 at 06:49
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    This webpage has lots of good pointers. I'd particularly recommend John Baez's advice and the Physics StackExchange list. – Timothy Chow Sep 08 '22 at 15:28
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    Start with the book Mathematical Physics: Classical Mechanics – rfloc Sep 08 '22 at 19:59
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    Some of the suggestions here and in the links are more of an answer to "How does one learn the mathematics of physics devoid of physical intuition?" – Tom Copeland Sep 08 '22 at 21:44
  • I think some things you can do quite good, other things are more difficult. It similar I think some computer science undergrads who go on to pursue math phds. You might do better at physics as like an engineer? But may not do as well theoretically. You can look at David Hilbert vs Albert Einstein. As an example. – marshal craft Sep 09 '22 at 12:15
  • Contrast Robert Geroch's "General Relativity from A to B" with Bernard Schutz's "A First Course in General Relativity". I like both, but as Schutz says, "For treatments that take a more thoughtful look at the fundamentals of the theory, consult . . . Geroch . . . ." Schutz's "Geometrical Methods of Mathematical Physics" is short and sweet, but even more rarefied w.r.t. the physics. – Tom Copeland Sep 09 '22 at 17:58
  • Maybe you are looking for something along the lines of "The Geometry of Physics" by Peter Frankel: The basic ideas at the foundations of point and continuum mechanics, electromagnetism, thermodynamics, special and general relativity, and gauge theories are geometrical, and, I believe, should be approached, by both mathematics and physics students, from this point of view. – Tom Copeland Sep 09 '22 at 20:05
  • Physics for Mathematicians, Mechanics I: Michael Spivak ? – BCLC Sep 10 '22 at 09:26
  • If you want to learn ''physics'' and not ''mathematics for physics'', you should read books which are targeted towards physics students and avoid books which have titles like ''quantum mechanics for mathematicians'' as they are devoid of physical intuition and also include axiomatic stuff which is generally irrelevant for experiments. – Hollis Williams Sep 10 '22 at 11:06
  • You might like to read "Geometry and physics" by Atiyah, Dijkgraaf, and Hitchin to get a feeling for the synergy between mathematics and physics beyond the basics in recent decades (https://royalsocietypublishing.org/doi/10.1098/rsta.2009.0227). – Tom Copeland Sep 16 '22 at 18:43

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I can recommend Leonard Susskind's Theoretical Minimum:

A number of years ago I became aware of the large number of physics enthusiasts out there who have no venue to learn modern physics and cosmology. Fat advanced textbooks are not suitable to people who have no teacher to ask questions of, and the popular literature does not go deeply enough to satisfy these curious people. So I started a series of courses on modern physics at Stanford University where I am a professor of physics. The courses are specifically aimed at people who know, or once knew, a bit of algebra and calculus, but are more or less beginners.

The name "theoretical minimum" is a reference to the notoriously rigorous exam a student needed to pass in order to study with Lev Landau. See also this discussion.

Carlo Beenakker
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    Landau's "Theoretical minimum" was based on his own books, multi-volume "Course of theoretical physics" which is available in English translation. – Alexandre Eremenko Sep 09 '22 at 01:27
  • @CarloBeenakker, I looked at the beginning of Susskind's Lect. 1 and perused Lect. 3 on Classical Mechanics. To begin with a dynamical systems, 'set theoretic' approach using the two sides of a coin as states of a system is just backasswards Zenoistic to me. As one of my favorite profs used to say in his Baltimore accent, "Ehhhh, forget the mathematics, let's look at the physics." Then he would argue what the governing equation should be from, e.g., looking at the physics at the extremes of parameters. Feynman approaches least action with physical intuition--Susskind, not so much. – Tom Copeland Sep 09 '22 at 16:53
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The Feynman lectures on physics would also be a suitable entry point.

oliversm
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    (https://www.feynmanlectures.caltech.edu/) Hard to find a set of books that combine physical intuition and mathematics in introductory classical mechanics / dynamics, stat mech, electromagnetism, and quantum mechanics to the degree that Feynman did which is appropriate for someone familiar with basic calculus. – Tom Copeland Sep 08 '22 at 22:17
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    Feynman's lectures can be somewhat irritating for a mathematician: he assumes absolutely no knowledge of mathematics, see for example his long and clumsy explanation of the exponential function:-) These are great lectures, but perhaps not "for mathematicians". – Alexandre Eremenko Sep 09 '22 at 01:31
  • It seems Feynman's books are toooooo long to read. – LZB Sep 09 '22 at 06:26
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    @AlexandreEremenko I agree that Feynman's lectures are not "for mathematicians," but then again, I don't interpret the original question as necessarily asking for books "for mathematicians." I read it as primarily asking for how to study physics on one's own. A master's student in mathematics may not yet be locked into a mathematician's mindset. – Timothy Chow Sep 09 '22 at 13:52
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    @Timoty Chow: I can only rely on my own experience. For the first time I read the first few volumes of Feynman lectures when I was a high school student. And even at that time I noticed that he tries to avoid using any mathematics. This feature (of these and other similar books) certainly did not please me. – Alexandre Eremenko Sep 09 '22 at 17:02
  • @AlexandreEremenko, volume and page numbers for the exp? I'd like to compare F's presentation with the historical development of the exp after the introduction of the log. – Tom Copeland Sep 09 '22 at 17:05
  • @LZB, can't learn physics through soundbites. – Tom Copeland Sep 09 '22 at 19:51
  • I've just started reading "The Maxwellians' by Hunt. Moral of the story: too much reliance on one or the other--physical intuition or pure math--introduces blinders. It's in the blending that constructive insights are made in physics. Theoretical physics is life on the razor's edge. – Tom Copeland Sep 10 '22 at 00:33
  • @AlexandreEremenko, Feynman used the exp in introducing the gaussian distribution and decay functions assuming the students were already familiar with it. Later in Vol. I in Section 22 Algebra, he introduces the natural base e as an outgrowth of the invention of the log following its historical and natural development as a milestone in algebra--Napier introduced the log in 1614 with logs of the sine and cosine; Briggs, the log tables in 1620; and Cotes, $i\theta = \ln[\cos(\theta)+\sin(\theta)]$ in 1714. Note Feynman's algorithm in https://en.wikipedia.org/wiki/Logarithm. – Tom Copeland Sep 10 '22 at 19:25
  • That is, Cotes derived $i\theta = \ln[\cos(\theta) + i \sin(\theta)]$ (refs on the history of the logarithm and the anti-logarithm in https://mathoverflow.net/questions/382121/what-are-some-of-the-earliest-examples-of-analytic-continuation/382156#382156). – Tom Copeland Sep 11 '22 at 21:55
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I think it's not difficult to find physics books explicitly written for mathematicians. For instance, Takhtajan's "Quantum Mechanics for Mathematicians" or "General relativity for mathematicians" by Sachs and Wu.

But I personally think you should start by familiarizing with experimental physics. The classical "Atomic Physics" by Max Born can be a good read before anything more formal, for instance. Also, if you are interested in classical electrodynamics, I would recommend "Classical electricity and magnetism" (another old one, 1960s...), by Panofsky and Phillips, for similar reasons. The energy associated with the EM field, for instance, is treated with an experimental attitude which is useful, I believe, to have a better grasp on the physical meaning of energy conservation involving EM phenomena. Special relativity is also introduced with attention to its experimental basis.

In this way you may add depth, in a dimension otherwise hardly accessible, to your mathematical understanding.

Alessandro Della Corte
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    A more basic book in quantum mechanics is "Lectures on QM for mathematicians" by Faddeev and Yakubovskii. Takhrajan's book is a kind of continuation, or "next level", as he says himself. – Alexandre Eremenko Sep 09 '22 at 01:29
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Some excellent books for mathematicians who begin to study quantum mechanics are

Alexander Givental, Introduction to Quantum Mechanics (available from the author for a very modest price).

L. D. Faddeev and O. A. Yakubovskii, Lectures on quantum mechanics for mathematics students. Translated from the 1980 Russian original by Harold McFaden. With an appendix by Leon Takhtajan. Student Mathematical Library, 47. American Mathematical Society, Providence, RI, 2009.

On classical mechanics, I recommend L. Landau and E. Lifshitz, Course of theoretical physics, vol. I, and, of course, V. Arnold, Mathematical methods of classical mechanics.

Alexandre Eremenko
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For more advanced stuff written by mathematicians, Lectures on Quantum Field Theory by Borcherds, Quantum Fields and Strings: a course for mathematicians (well, easier parts of it, probably - this is a two-volume set!), and, more broad and accessible, written by a rare universalist (that is, physicist but also mathematician), The Road to Reality: A Complete Guide to the Laws of the Universe by Penrose.

მამუკა ჯიბლაძე
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I would suggest Mathematical Methods of Classical Mechanics by V.I Arnold, it does not go over any quantum mechanics discussions, but keep in mind that our intuition to do things like quantization arises from classical concepts.

caverac
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I recommend strongly against Leonard Susskinds theoretical minimum. It's written for people who have very rudimentary knowledge not one who has already a bsc in math.

Let me give you some pretext. Pretty much everything in Physics is somehow related to the ideas in Differential Geometry. So having good knowledge of this will help.

Now, for the actual book, Roger Penroses Road to Reality

This book introduces a lot of the Differential Geometry pre requisites and does classical , quantum and relativistic physics in its context.

I really liked this book and I firmly believe that one can get quite far if they are to simply understand all the ideas in the book.

The issue with this book it doesn't teach at all to calculate stuff. You can check out Kai S Lams classical mechanic book for this

tryst with freedom
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  • I do have a bsc in math. – LZB Sep 09 '22 at 06:20
  • Take that book and skip to like 8 or 9th chapter. I almost 90% sure it'll appeal more to you than susskund @LZB – tryst with freedom Sep 09 '22 at 06:59
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    my understanding was that the OP wished to learn "real physics" (their words), which I interpret as acquiring insight into physical phenomena based on high level concepts and ideas, and in particular the unifying nature of these concepts; one needs little math for that, and I find Susskind's course quite helpful in that respect; it is not the place to go if one seeks mathematical rigor, but that is a very different question (a question which Tom Copeland describes as asking "How does one learn the mathematics of physics devoid of physical intuition?"). – Carlo Beenakker Sep 09 '22 at 08:31
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What kind of physics are you interested in? If you are interested in rigorous introduction to statistical physics, there are classic references such as Grimmett's random cluster model textbook, Yvan Velenik's statistical mechanics textbook, or Duminil-Copin's lecture notes on the Potts model.

If you want a more non-rigorous (i.e. more real-physics flavor) approach to modern statistical physics, John Cardy's RG textbook is good, but I have to warn you the treatment of disordered systems in Chapter 8 is highly non-rigorous and even incorrect in certain cases.

PeaBrane
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In case you are interested in general relativity, I suggest taking a look at a book I wrote, A Mathematical Introduction to General Relativity.

It presents general relativity to undergraduate mathematics students in a mathematically rigorous fashion in a `definition-theorem-proof' format. The only physics needed to read it is a basic college course in Physics. A pdf file of sample chapters can be found on the publisher's website above, and a google preview is linked here . The book is intended for self-study, and the solutions to all the (over 200) exercises are included in the book.

Amol Sasane
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