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In many books on particle physics, it is first shown that the Lie algebra of the Lorentz group is isomorphic to

$$ su(2) \oplus su(2) , $$

then by this fact, it is implicitly assumed that for each representation of $su(2)\oplus su(2)$, one can construct a representation of the Lorentz group.

But it is well known that the group $SO(3)$ has a Lie algebra isomorphic to $su(2)$, but half of the latter's representations do not lead to a representation of the former.

The question is then, does every representation of $su(2)\oplus su(2)$ lead to a representation of the Lorentz group? even though it cannot be unitary?

In particular, does the representation labeled as $(1/2, 0)$ lead to a representation of the group?

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  • $\begingroup$ I don't think you always want exact representations of the Lorentz group. Rather, you want all projective representations of the Lorentz group which arise as exact representations of its universal cover. If we were only interested in representations of the Lorentz group then we would wouldn't have the spinor representations. $\endgroup$
    – SigmaAlpha
    Commented Oct 1, 2017 at 9:39
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/357183/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Oct 1, 2017 at 9:52
  • $\begingroup$ This Q&A of mine explains why it suffices to look at the algebra in quantum-mechanical contexts. $\endgroup$
    – ACuriousMind ♦
    Commented Oct 1, 2017 at 10:54

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