I've got an assignment due in a few hours, and I'm at a complete loss as to how to even start it, really. I haven't encountered any Dirac notation before, so I'm having a lot of trouble attempting the following;
Let $F_2$ denote the Fock subspace of all 2-boson states, spanned by homogeneous polynomials of degree 2 in the boson creation operators $b^{\dagger}_1$, $b^{\dagger}_{2}$, $b^{\dagger}_{3}$ acting on the physical vacuum state $\left|0\right\rangle$. From lectures, we know that $F_2$ is a six dimensional $gl(3)$-module. Explicitly give an orthonomal basis of $F_2$ that is symmetry adapted to the subalgebra chain $o(2) \subset o(3) \subset gl(3)$, and express the highest weight vector of $F_2$ as a linear combination of these basis vectors.
Now, just going through lecture notes, I tried to work out a few things step by step.
- The $gl(3)$ highest weight vector of $F_2$ must be $(2, 0, 0)$
- $F_2 = c_2 \dotplus \Delta^{\dagger}F_0$
- dim($c_2$) = 5
- $c_2 = \{\Psi \in F_2 | \Delta\Psi = 0\}$
- Define $\Psi_2 = k(b^{\dagger}_1 + i b^{\dagger}_2)^2 \left|0\right\rangle$
Now, I'm not even sure where to go from there. I've been told to normalise that $\Psi_2$ term, but I'm not even sure how to go about that. I've also been told to act $L_3$ on the $\Psi$ terms, where $$L_3 = -i(b^{\dagger}_{1}b_{2} - b^{\dagger}_{2}b_{1})$$
But I'm not even sure how to even start going about this. Of course I'd love some help on this question, but, links to easy-ish resources that might point me in the right direction, or even give some better background into this would also be much appreciated.