I only have an undergraduate level math but am quite fascinated with how lie groups are applied in elementary particle physics.  Any advice on how I can learn lie groups theory (with an application to physics in mind)?  What pre-requsite do I need, and how much group theory I need to master before I can start lie groups, and how much lie group theory do I need to understand elementary particle theory?
120471
Ok so I read the following article the other day:  and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to do with groups" So I picked up a copy of Dummit and Foote's book on abstract algebra and skimmed through it. It didn't say anything about Lie groups however. $E_8$ is coming to be rather famous so maybe other people are interested in this question too. Lets suppose I wanted to learn about lie groups. What books should I read to be ready to learn about Lie groups and what is a good book that talks about Lie groups. I'm guessing its a combination of group theory (representation theory in specific) and also differential geometry. Is this correct? Thank you very much for your time.
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I know enough amount of set theory and logic to study grad-level math. However, I don't know more advanced set theory and logic, such as the ones on Kunen's or Shoenfield's texts. Although it's good to learn such advanced topics, I may not have enough time to do so, since I will probably put my emphasis on algebra, analysis or geometry as well as mathematical physics.   Although how much set theory and logic I should learn is totally up to what sort of analysis, algebra or geometry I will study, I want to know how much of them typical competitive algebraists/analysts/geometers know. If you are one of them, how much do you know?
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I'm interested in learning how to use geometry and topology in physics. Could anyone recommend a book that covers these topics, preferably with some proofs, physical applications, and emphasis on geometrical intuition? I've taken an introductory course in real analysis but no other higher math.
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The question I guess is pretty clear. I am a physics undergrad wishing to pursue research in quantum gravity(string theory?). What are the subjects I should learn other than the usual compulsory undergrad courses such as quantum mechanics, and GR, etc? Things that come to my mind are QFT, lie algebras, differential geometry, algebraic geometry etc? Other than this, could you suggest any topic that are not entirely essential but interesting in mathematical physics, gauge theory etc (e.g. solitons, monopoles etc). Actually I am looking for a mini project of sorts which would give me an idea of what these fields really are. I have done some lie algebras, classical field theory and am currently studying QM.
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I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard to put words on my question and I hope that you will be able to help me ask it clearly.  I'm trying to link what I know from mathematics to what we are writing in physics for Noether's theorem. If I understand correctly we are looking at the symmetries of the action i.e. under which symmetry groups is it invariant. Noether's theorem allows us to calculate a conserved current in the case of a continuous symmetry (Lie groups), by means of so-called infinitesimal symmetries which I believe to be elements of the Lie algebra (so the tangent space to the neutral element) of the Lie group.  I'm guessing that if the action is invariant under the symmetry then it's "variation" should be 0 when we vary the system (space-time coordinate, or field) using this symmetry; it is exactly this step that I would like to understand better, how can I formalize this step correctly mathematically? How should I understand this variation, and how does it's calculation give rise to the elements of the Lie algebra?
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I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$.  My friend who told me this thought that it involved de Rham cohomology, but I don't really know anything about the cohomology of Lie groups so this doesn't help me much.  Presumably there are some pretty strict conditions we can get from talking about invariant differential forms -- if you can tell me anything about this it will be a much-appreciated bonus :)  (A necessary condition for a manifold to be a Lie group is that is must be parallelizable, since any Lie group is parallelized (?) by the left-invariant vector fields generated by a basis of the Lie algebra.  Which happens to mean, by some pretty fancy tricks, that the only spheres that even have a chance are the ones listed above plus $S^7$.  The usual parallelization of this last one comes from viewing it as the set of unit octonions, which don't form a group since their multiplication isn't associative; of course this doesn't immediately preclude $S^7$ from admitting the structure of a Lie group.  Whatever.  I'd like to avoid having to appeal to this whole parallelizability business, if possible.)
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How do you effectively study physics? How does one read a physics book instead or just staring at it for hours?   (Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)
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I'm a mathematician. I have good knowledge of superior analysis, distribution theory, Hilbert spaces, Sobolev spaces, and applications to PDE theory. I also have good knowledge of differential geometry. I would like to study the Semiclassic Analysis, but perhaps I must first study the foundations of quantum mechanics. So I would like to know what book you recommend me to begin studying quantum mechanics. I'm primarily interested in a mathematical point of view. I have seen some books of this type, but I would like to have some other opinion.  Thanks for every reply
123556
I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?  The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.  I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.
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I am a mathematics major without any undergraduate level courses in physics. I am lately interested in physics. Is there any chance for me to go into good theoretical physics graduate program without any background in physics?   I am not sure if this kind of question is acceptable here, though.  ADDED My concern is do Universities even accept students with math majors into physics department without physics courses?
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The professor in my class of mathematical physics introduces the definition of groups and said that group theory is the mathematics of symmetry.   He gave also some examples of groups such as the set of all real numbers under addition.   My question is what symmetry are we talking about here when we consider the set of all real numbers under addition?
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I am looking for a preferably introductory text on Yang-Mills theory with mathematical basis and language. (Differential forms) I have a background in QCD and QED, however, I want to go more in details in geometrical interpretations of Yang-Mills theory. I am sorry if this question has asked before. Thanks in advance.
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What textbook would you recommend for a bachelor in theoretical physics to study ? I don't want to focus too much on chaos, just having a broad view of every interesting characteristic is enough. Physical meaning behinds equations should be explained.  Some related resources:   , but obviously those are math-y.  While there are some overlaps, the evolution of a simple oscillator can evoke interesting dynamics characteristics. , but I'm not sure if studying only fluid dynamics is too narrow for me.
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Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I think this is true for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{C})$, or at least that's what physics books lead me to believe!   Is it true in general for semisimple Lie algebras?
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I'm interested in cutting edge string theory studied by research physicist. I'm wonder what  mathematics is needed and how far am I in terms of mathematics background needed and how much more mathematics do I need in order to study it properly and reach my goal. My mathematics level is at calculus/Linear algebra.
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I am looking for a good mathematical rigorous introduction to solid state physics. The style and level for this solid state physics book should be comparable to Abraham Marsdens Foundations of mechanics or Arnols mechanics book for classical mechanics or to Thirrings Physics course for quantum mechanics.   Any recommendations?  Edit: As a reaction to Peter Shor's comment, I try to narrow the scope of the question a bit and give some more specific subareas of solid state physics I am in particular interested in:   semiconductors and applications the quantum hall effect superconductivity
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I am a second year undergraduate and studying quantum mechanics from sakurai's 'Modern Quantum Mechanics'. Is it a good idea to solve problems from sakurai, which are mostly mathematical in nature? I need a textbook that has physically relevant problems, maybe going even into condensed matter, or field theory in its exercises. This would probably help me to appreciate and understand qm better. Sorry if this question is too localised but I just had to post it.