1. ### I Calculating group representation matrices from basis vector/function

Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail): 1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...
2. ### I Quark Model Families and Masses

Consider the pseudoscalar and vector meson family, as well as the baryon J = 1/2 family and baryon J = 3/2 family. Within each multiplet, for each particle state write down its complete set of quantum numbers, its mass, and its quark state content. Furthermore, for each multiplet draw the (Y...
3. ### A SU(2) Invariant Lagrangian

Consider two arbitrary scalar multiplets ##\Phi## and ##\Psi## invariant under ##SU(2)\times U(1)##. When writing the potential for this model, in addition to the usual terms like ##\Phi^\dagger \Phi + (\Phi^\dagger \Phi)^2##, I often see in the literature, less usual terms like: \Phi^\dagger...
4. ### I Adjoint Representation Confusion

I'm having a bit of an issue wrapping my head around the adjoint representation in group theory. I thought I understood the principle but I've got a practice problem which I can't even really begin to attempt. The question is this: My understanding of this question is that, given a...
5. ### A How is the invariant speed of light enocded in SL(2,C)?

In quantum field theory, we use the universal cover of the Lorentz group SL(2,C) instead of SO(3,1). (The reason for this is, of course, that representations of SO(3,1) aren't able to describe spin 1/2 particles.) How is the invariant speed of light enocded in SL(2,C)? This curious fact of...
6. ### Algebra Good book on representation theory of groups

Hi I am a physics graduate student. Recently I am learning representation theory of groups. I understand the basic concepts. But I need a good book with lots of examples in it and also exercise problems on representation theory so that I can brush up my knowledge.The text we follow is "Lie...
7. ### How can I construct the 4D real representation of SU(2)?

An element of SU(2), such as for example the rotation around the x-axis generated by the first Pauli matrice can be written as U(x) = e^{ixT_1} = \left( \begin{array}{cc} \cos\frac{x}{2} & i\sin\frac{x}{2} \\ i\sin\frac{x}{2} & \cos\frac{x}{2} \\ \end{array} \right) = \left(...
8. ### Meaning of representations of groups in different dimensions

Problem This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions. Relevant Example Take SO(3) for example; it's the...
9. ### Scalar as one dimensional representation of SO(3)

Hi to all the readers of the forum. I cannot figure out the following thing. I know that a representation of a group G on a vector spaceV s a homomorphism from G to GL(V). I know that a scalar (in Galileian Physics) is something that is invariant under rotation. How can I reconcile this...