salparadise
Dec3-10, 11:57 AM
I'm taking a course on group theory in physics, but the teacher is really bad at making the bridge between the maths and the physics.
As homework I have to do the exercise below. I think I know how to do it but I'm also posting it to see if someone could please recommend a good reference book where this kind of questions are treated. A reference that clearly explains Young diagrams (not just Young tableau) is also something I can't find. I've consulted the following books: Georgi.H and Cornwell.
1. The problem statement, all variables and given/known data
Symmetry group S3. Taking into account the direct product of the 2D irreps as follows:
\psi_{i}\otimes\psi^{}_{j} = \Psi_{1}+\Psi_{1}+\Psi_{2}
where:
\Psi_{1} = \psi_1\psi^{'}_1 + \psi_2\psi^{'}_2\\
\Psi_{1'} = \psi_1\psi^{'}_2 + \psi_2\psi^{'}_1\\
\Psi_{2} = (\psi_1\psi^{'}_2 + \psi_2\psi^{'}_1 , \psi_1\psi^{'}_1 - \psi_2\psi^{'}_2 )^T\\
Write the most general scalar potential up to power four, made exclusivly with
two S3 doublets, namely \psi and \chi.
3. The attempt at a solution
Knowing that the product of the 2D irreps of S3 is 2⊗2=1+1'+2, and knowing that a scalar invariant potential can only be formed by spaces of trivial representation. We only need to form all possible products of \psi and \chi up to power 4 and at the end only take the resulting 1 irrep (trivial one) terms.
Thanks in advance
PS - If this should be in another forum section, please let me know.
As homework I have to do the exercise below. I think I know how to do it but I'm also posting it to see if someone could please recommend a good reference book where this kind of questions are treated. A reference that clearly explains Young diagrams (not just Young tableau) is also something I can't find. I've consulted the following books: Georgi.H and Cornwell.
1. The problem statement, all variables and given/known data
Symmetry group S3. Taking into account the direct product of the 2D irreps as follows:
\psi_{i}\otimes\psi^{}_{j} = \Psi_{1}+\Psi_{1}+\Psi_{2}
where:
\Psi_{1} = \psi_1\psi^{'}_1 + \psi_2\psi^{'}_2\\
\Psi_{1'} = \psi_1\psi^{'}_2 + \psi_2\psi^{'}_1\\
\Psi_{2} = (\psi_1\psi^{'}_2 + \psi_2\psi^{'}_1 , \psi_1\psi^{'}_1 - \psi_2\psi^{'}_2 )^T\\
Write the most general scalar potential up to power four, made exclusivly with
two S3 doublets, namely \psi and \chi.
3. The attempt at a solution
Knowing that the product of the 2D irreps of S3 is 2⊗2=1+1'+2, and knowing that a scalar invariant potential can only be formed by spaces of trivial representation. We only need to form all possible products of \psi and \chi up to power 4 and at the end only take the resulting 1 irrep (trivial one) terms.
Thanks in advance
PS - If this should be in another forum section, please let me know.