Monday, December 23, 2019

quantum mechanics - Fock Subspaces and Weight Vectors


This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons.


I've got the following question to work out;



Let F2 denote the Fock subspace of all 2-boson states, spanned by homogeneous polynomials of degree 2 in the boson creation operators b1, b2, b3 acting on the physical vacuum state |0. From lectures, we know that F2 is a six dimensional gl(3)-module. Explicitly give an orthonomal basis of F2 that is symmetry adapted to the subalgebra chain o(2)o(3)gl(3), and express the highest weight vector of F2 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 F2 must be (2,0,0)

  • F2=c2

  • 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.




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