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9

For quantum mechanics, the original is still the best: Dirac's "The Principles of Quantum Mechanics". It's clear, it's terse, and it's comprehensive. All other books take most of their material from this source. For a basic short introduction to quantum mechanics, you can't beat: Feynman Lectures on Physics Vol III This is very good and intuitive, ...

9

As it is, differential forms don't tell you the whole story--strictly speaking, differential forms only deals with covectors and wedge products of covectors and then uses the hammer of the Hodge star to be able to clumsily do inner products. To me, it is too far removed from the vector calculus you may already know. Instead, I strongly urge you to look ...

7

I would really recommend the book by Frankel, The Geometry of Physics. He deals with all the fundamental concepts of topology and differential geometry, but gives clear and detailed applications to classical mechanics, electromagnetism, GR and QM. He is not too formal, but develops really a lot of useful tools using differential forms. Another book, which ...

6

I always loved Callen's Thermodynamics. In 200-250 pages you get the whole structure of thermodynamics more clearly than anywhere else. And it is wonderfully written, a fun and easy read. When I was in college I essentially studied these pages in a long weekend. Of course back in those days I had the time to fully concentrate on a single thing.

6

:-) The best gentle introduction to basic twistor theory that I know of is the book by Huggett and Tod If you don't have access to that book and some other answers don't surface in the meantime I'm happy to write a few bits and pieces here, but will have to wait until the weekend. (I may be biased, but I think it's well worth learning, as the MHV ...

5

The canonical textbook is the two-volume set by Polchinski. David Tong has very nice notes up following this text. You should be able to find various review articles on the arXiv as well, for instance: http://arxiv.org/abs/hep-th/0207249 http://arxiv.org/abs/hep-th/0207142 Hope that helps...

5

The true fathers of quantum mechanics – Heisenberg, Born, Jordan, and later Bohr – started with matrix mechanics; it was the picture in which the classical equations of motion were easier to be understood as a limit of the new theory. That was in 1925. That was how quantum mechanics was born for the first time. Within a year, wave mechanics was born and ...

5

The relation is very deep and has a rich mathematical structure, so (unfortunately) most stuff will be written in a more formal, mathematical way. I can't say anything about Donaldson theory or Floer homology, but I'll mention some resources for Chern-Simons theory and its relation to the Jones Polynomial. There is first of all the original article by ...

5

The classical reference is Landau & Lifshitz, The Classical Theory of Fields, from the Course of Theoretical Physics. As all Landau & Lifshitz books, masterpieces [in my opinion] full of content but sometimes a little difficult to grasp for beginners.

4

Have a look at my paper ''Phenomenological thermodynamics'' http://www.mat.univie.ac.at/~neum/ms/phenTherm.pdf , which summarizes the core of thermodynamics in 18 pages, essentially starting from scratch. (It is also available as Chapter 7 of my online book http://lanl.arxiv.org/abs/0810.1019 , and can be read independent of the remainder of the book.) ...

4

Another interesting application is that Chern-Simons Theory in 3d is equivalent to General Relativity in 3 space-time dimensions. GR in 3 dimensions is quantisable and following a nice playground for quantum gravity. http://ncatlab.org/nlab/show/Chern-Simons+gravity has a nice reading list about that topic at the References. Maybe a good start is "Edward ...

4

There isn't any mathematically precise definition. These are physical objects, and they acquire their definition in a given model which allows for calculations. The same physical object can appear in different models in different roles, so the strings have different mathematical definition in different limits of the full M-theory. The closest thing to a ...

4

As I think you are serious, you should start studying mechanics at a higher level, and classical electricity and magnetism. As I am of an older generation, the books I found good as a preparation for later understanding quantum mechanics and quantum field theory are Classical Mechanics by Goldstein, and Classical Electricity and Magnetism by Panofski and ...

4

Set your goals on something concrete that you really would like to know but do not yet understand at all, such as ''Why does water freeze?'' or ''What is an elementary particle?''. (And if that is settled, go for something more advanced but again concrete, etc.) Given the goal, search for the answer, starting with Wikipedia (and later Google Scholar), and ...

4

I do not know much about books on Anderson localization but there was a conference some time ago celebrating the $50$ years on the proposal by Anderson in which you can find many useful references. The list is here. I can recommend you the course by van Tiggelen. He's quite an expert in this subject and Les Houches lectures tend to be very pedagogical. ...

3

You need to know the rudiments of the application of algebraic topology to the classification of bundles on manifolds. If you're self teaching using the internet, it would be useful to look up "characteristic classes", and work backwards from there, filling in the gaps that you need. Nakahara is a good introduction to this material, as is Eguchi, Gilkey ...

3

If you want to learn topology wholesale, I would recommend Munkres' book, "Topology", which goes quite far in terms of introductory material. However, in terms of what might be useful for physics I would recommend either: Nakahara's "Geometry, Topology and Physics" Naber's "Topology, Geometry and Gauge Fields: Foundations" Personally, I haven't read ...

3

Mathematical rigor is not the most important thing when first learning strings, there are many things that are not possible to formulate rigorously, because the best language for doing this isn't known. In addition to Polchinsky (which is excellent), I recommend reading Green Schwarz and Witten, and also the original papers, since these have points of view ...

3

Here is my answer from a condensed matter physics point of view: Quantum field theory is a theory that describes the critical point and the neighbor of the critical point of a lattice model. (Lattice models do have a rigorous definition). So to rigorously define/classify quantum field theories is to classify all the possible critical points of lattice ...

3

Boundary layer theory come in as a method to "simplify" the mathematics in fluid mechanics, so that it is solvable analytically. It separates the fluid as two regions: where the viscosity effect is important, i.e. boundary layer where viscosity is not important This approach has been proven useful for large range of applications, that's why aerodynamics ...

3

I would like to recommend to you the following lecture notes by V.P. Nair. These lecture notes contain a very concise chapter about twistors, their relation to massless wave equations and their use in the construction of Yang-Mills amplitudes. The importance of this work to me is that, here, Nair connects these two applications to another (may be less ...

3

Olaf has already given most of the references I would recommend. But in the case of Chern-Simons theories and knot theory, there are two (plus one) other very nice references. These are all written by physicist to physicists, so no modular functors, Cobordisms and so on. 1) Marcos Marino - Chern-Simons Theory and Topological Strings (arXiv:hep-th/0406005v4) ...

3

Goldstein's and Jackson's are the examples of widely used graduate level textbooks, however it should be used already in your MSc course. Griffith's in the other hand is widely used in physics undergraduate EM course. I don't know the level of math and physics that you have, but probably it would be good to start studying Landau-Lifshitz's Course on ...

3

A thorough overview of theoretical physics, from a uniform point of view and with lots of explanations is given in the 10 volumes on Theoretical Physics by Landau and Lifshitz. Though not covering the newest things they start from scratch (though assuming some background that you surely have with a MSc) and are a nearly ideal foundation on which to build ...

3

This may not be exactly what you are looking for, but I am going to recommend two specific texts. Misner, Thorne, and Wheeler, Gravitation, Chapters 4, 9, and end of 14 Solidly in the realm of physics but they have a lot of tidbits of interpretation in there. Choquet-Bruhat and DeWitt-Morette, Analysis, Manifolds, and Physics, Chapter IV.C I mean, this ...

3

Take a look at the following Fluctuations And Localization In Mesoscopic Electron Systems Introduction to Wave Scattering, Localization and Mesoscopic Phenomena

3

In addition to what has already been said I would add: The original paper is at this point historical; you are right to look for other sources. You should also read about weak localization and coherent backscattering which are closely related. A good reference is Chapter 5 of the 1st edition Electronic Transport in Mesoscopic Systems by S. Datta.

2

Have a look first at several chapters in Stone and Goldbart, "Mathematics for Physics" (the free preprint is here) before entering into more specific books. I think you may want to see chapters 1, and parts of 2 and 9. You may find some parts of what you want in classic books of the "Comprehensive Mathematical Methods for Physics" type, but they don't ...

2

I don't know at what level you want the problems. Some excellent sources for olympiad level problems are : Irodov Krotov 200 puzzling physics problems The last one is especially good. For undergrad level physics problems, it is best you refer to individual textbooks.

2

Scott Aaronson has just published a new book about quantum computing. According to the nice introductary comments the author himself has written to his book here (scroll down to the second half of the article if you only want to learn about the book), it should explain and introduce both the physical and mathematical concepts quantum computing is based on. ...

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