Reddit reviews Group Theory in Physics

J.F. Cornwell, Group theory in physics: an introduction (link)

W. Ludwig, Symmetries in physics: group theory applied to physical problems(link)

M. Tinkham, Group theory and quantum mechanics (link)

W.-K. Tung, Group theory in physics (link)

E.P. Wigner, Group theory and its applications to the quantum mechanics of atomic spectra (link1, link2)

N. Jeevanjee, An Introduction to Tensors and Group Theory for Physicists (link)

G. Costa, Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries (link)

B. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (link)

R. McWeeny, Symmetry: An Introduction to Group Theory and Its Applications (Dover Books on Physics)(link)

> It's fair to say that isn't what people are talking about here, but I guess that is what my original question was meant to refer to. I've read that when adding SSB to a theory with massless Goldstone bosons, that certain properties of the massless bosons carry over into the massive bosons (for example the "longitudinal polarization" of the massive boson being given by the massless one, though I don't know any of the details of how).

> If that's the right way of characterizing it, then on the surface it seems that there is a conversion of some kind going on (i.e. massless to massive) rather than being two otherwise completely unrelated types of particle. I guess that's what I'm trying to understand; is it a conversion, and if so, in what ways are they related or not related.

It's actually the locality of gauge invariance that causes the "conversion" most directly. What I mean by that is that if you have a quantum field theory with the right kind of potential, if you impose global gauge invariance, then you wind up with, say, one massive boson and one massless Goldstone boson. But if you impose local gauge invariance on the same theory, you instead wind up with two massive bosons. One of them will be the same massive boson from the globally invariant case, in the sense that it's the same field involved. But the other one comes from the gauge field that enters into the covariant derivative. The term in the Lagrangian that represented the Goldstone boson in the globally invariant case gets canceled out by the term resulting from the gauge transformation of the gauge field. (This is kind of hard to explain without math, so I'd suggest you take a look at chapter 14 of Halzen and Martin, in particular sections 14.7-14.8.)

Anyway, the main point is that in one theory, you have 5 degrees of freedom from particle polarizations and one more from a gauge transformation, and in the other theory you have 6 degrees of freedom from polarizations alone. If you want to call that a conversion, then you can. But personally I prefer not to, since to many people "conversion" implies some sort of physical process that you go through to turn one thing into another, and of course that's not what's happening here. (Basically I want to avoid prompting people to ask "why don't we convert photons into massive particles to slow them down?")

> Yeah, that's called a "quasiparticle" or perhaps a "collective excitation" right? I hesitated to ask about them because I understand these are not really fundamental but are more like an approximation or a different way of describing the system -- like describing conduction in terms of electron holes rather than electrons, or coordinates in polar form rather than linear.

I'm not really sure, actually - I don't know that much about the theory of superconductivity, at least not in enough detail to figure out what happens to photons inside a superconductor.

> One thing I may be confused on, and it might just be bad wording so I'm going to ask. You said there are "really four symmetries -- or more precisely, four generators" of SU(2)xU(1). Is it right to say that there is a symmetry for each generator? And what is the difference then between a symmetry and a generator? I was under the impression that SU(2)xU(1) was itself a single symmetry/group, and the generators were related to the charges rather than the quanta. Is that wrong?

Yeah, that was bad wording. Forget the thing I said about four symmetries; that was a relic from something I'd started typing up before it occurred to me that I could probably just talk about generators without confusing you too much. All I meant was that the symmetry has four degrees of freedom, in a sense; they correspond to the four generators of the SU(2)xU(1) group.

Now, you could also construct a theory that had a U(1)xU(1)xU(1)xU(1) symmetry, and that would also have four generators and thus four degrees of freedom to the symmetry. But the commutation relations of the generators would be different. In matrix notation (in the fundamental representation), the generators of SU(2)xU(1) are the three Pauli matrices and the number 1 (or the 2x2 identity matrix, if you prefer), but the generators of U(1)^4 are just the number 1, four times. Generators play a role in group theory similar (in certain respects) to the role that basis vectors play in linear algebra, so having different commutation relations among the generators is kind of like having two spaces with the same number of dimensions but with different metrics.

As for learning more about group theory, I'm no expert on it myself, but I've heard good things about this book by Wu-Ki Tung.