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Why is half-integer spin not observed classically?

The other answers are great, and it's very much true that you can totally write down a classical field theory with half-integer-spin fields. That notion still feels kind of odd, though, because all ...
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?

You appear to have two unrelated gaps, the final one being the rotation formula for the vector (triplet) representation, the celebrated Rodrigues rotation formula. $\vec \sigma$ rotates like a vector. ...
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1 vote
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Is the Dirac adjoint in the representation dual to Dirac spinor?

In order for $\overline{\psi} \psi$ to be a Lorentz scalar, we should have $$\lambda^{\dagger}\gamma^0\lambda = \gamma^0$$ or equivalently (assuming $(\gamma^0)^2 = 1$) \lambda^{-1} = \gamma^0\...
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