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I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\frac{1}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\quad K_{0}=\frac{1}{4}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}).$$ In particular, I would like to know their 3x3 matrix representation. Mufti, Schmitt and Sargent III give a 2x2 matrix representation (and also a 4x4 matrix representation) that satisfies the commutation relations $$[K_{+},K_{-}]=-2K_0,\quad[K_0,K_{\pm}]=\pm K_{\pm}.$$ However, they do not provide details on how they derived these matrices.

I would be grateful if somebody could provide me a derivation of these matrices or simply state the result for the 3x3 case.

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  • $\begingroup$ You get that directly from the commutation relations. Try first getting the $3\times 3$ irrep of $\mathfrak{su}(2)$ using commutation relations alone, then just do the same for $\mathfrak{su}(1,1)$. $\endgroup$
    – ZeroTheHero
    Commented Sep 15, 2020 at 16:00
  • $\begingroup$ @ZeroTheHero I am a student with zero experience in group/representation theory (will learn that in the future) and I am not familiar with the terminology you are using. I do know QM well, so I'd really appreciate a detailed derivation. $\endgroup$
    – eemg
    Commented Sep 15, 2020 at 16:17
  • $\begingroup$ Also see WP. $\endgroup$
    – Cosmas Zachos
    Commented Sep 15, 2020 at 16:22
  • $\begingroup$ I appreciate your conundrum but it's not right to simply supply the answer. Look up "adjoint representation" and try to do this for angular momentum first, then do the same for $\mathfrak{su}(1,1)$ $\endgroup$
    – ZeroTheHero
    Commented Sep 15, 2020 at 16:52

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They told you $K_-=-J_-$. Did you fiddle with $$ K_+= \sqrt{2} \begin{bmatrix} 0&1&0\\0&0&1\\0&0&0\end{bmatrix}; \\ K_-= -\sqrt{2} \begin{bmatrix} 0&0&0\\1&0&0\\0&1&0\end{bmatrix}; \\ K_0= \begin{bmatrix} 1&0&0\\0&0&0\\0&0&-1\end{bmatrix} ? \\ $$

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  • $\begingroup$ these ones do satisfy the commutation relations. May I ask how you found them? It would be enough for me to know the derivation of one matrix. Once I know this I can do the other two. $\endgroup$
    – eemg
    Commented Sep 15, 2020 at 17:34
  • $\begingroup$ Standard undergrad QM spin matrices, given your authors' mapping tip. $\endgroup$
    – Cosmas Zachos
    Commented Sep 15, 2020 at 17:35
  • $\begingroup$ Well, I still do not quite understand how you derived them, but I will mark it as accepted answer because I was mainly looking for these matrices. I mean, the link to the spin matrices was not very helpful. I do know how to derive the spin matrices because I know how the angular momentum operators $L_z$ $L_{+}$ and $L_{-}$ act on the $\lvert \ell, m\rangle$ states. However, for the $K$ matrices above I do not know how they act on states. $\endgroup$
    – eemg
    Commented Sep 16, 2020 at 15:20
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    $\begingroup$ Read the paper. If the spin matrices obey the angular momentum algebra, flipping the sign of $J_-$, as advised, yields for you the algebra of Ks. So any matrix representation of the J s will then produce a matrix representation of K s, this way. You are asked about an abstract matrix representation of a given algebra! $\endgroup$
    – Cosmas Zachos
    Commented Sep 16, 2020 at 15:28

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