Taken from Quantum Field Theory in a Nutshell by Zee, problem II.3.1:
Show by explicit computation that $(\frac{1}{2},\frac{1}{2})$ is indeed the Lorentz vector.
This has been asked here:
but I can't really digest the formality of this answer with only a little knowledge of groups and representations.
By playing around with the Lorentz group generators it is possible to find the basis $J_{\pm i}$ that separately have the Lie algebra of $SU(2)$, and thus can be separately given spin representations.
My approach has been to write $$J_{+i}=\frac{1}{2}(J_{i}+iK_{i})=\frac{1}{2}\sigma_{i}$$ $$J_{-i}=\frac{1}{2}(J_{i}-iK_{i})=\frac{1}{2}\sigma_{i}$$ which implies that $$J_{i}=\sigma_{i}$$ $$K_{i}=0$$ However I don't really get where to go next.
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