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I'm studying Electroweak theory and I've come across the following problem.

I want to look at the scalar in electroweak theory in a general $n$-dimensional irreducible representation of $SU(2)$. For this I think I need a general expression for the angular momentum matrices $J_x$, $J_y$ and $J_z$, but I don't see how to find such a generalization. I know how to obtain them in 2 and 3 dimensions and I suppose I could do it for a fixed dimension, but not for any general $n$.

Could someone point me in the right direction?

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    $\begingroup$ what’s wrong with $J_\pm\vert jm\rangle=\sqrt{(j\mp m)(j\pm m+1)}\vert j,m\pm 1\rangle$? It’s about as general as it gets… $\endgroup$
    – ZeroTheHero
    Commented Mar 12, 2023 at 14:48

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You must surely know that for a $n=2j+1$ $$ J_z|j,m\rangle = m|j,m\rangle, \quad -j\le m\le j,\\ J_{\pm} |j,m\rangle= \sqrt{j(j+1)- m(m\pm 1)}|j,m\pm 1\rangle, $$ so there are your $n$-by-$n$ matrices.

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