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In physics, the generators of the fundamental representation of $SU(2)$ are taken to be $\sigma_i/2$ instead of $\sigma_i$ where $\sigma_i$ denotes the $i$th Pauli matrix. This is because, as I understand, if we take $$T_i=\sigma_i/2,$$ the Lie algebra of $SU(2)$ looks identical to that of $SO(3)$. But why would we like to match the $SU(2)$ Lie algebra with that of $SO(3)$ in the first place?

Or is it the other way around? Is it because the $SU(2)$ Lie algebra is identical to the $SO(3)$ Lie algebra, we make the choice that $T_i=\sigma_i/2$? But in that case, I don't know how to derive the $SU(2)$ Lie algebra without starting from any matrix representation of $T_i$'s.

Qmechanic
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    So the question is about conventions and the choice of basis? The specific convention is quite practical, cf. e.g. https://physics.stackexchange.com/q/394152/2451 – Qmechanic Jul 10 '23 at 06:58

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