Weinberg introduces the idea of Lorentz group representation describing how vectors in the Hilbert space of definite momentum states should change due to a L.T. It is understandable that to preserve probability amplitudes this transformation must be unitary. But then why isn't the operator $S=e^{\frac{1}{2}\alpha_{ij}\sigma_{ij}}$ , encoding L.T. in the space of spin 1/2 particles, unitary as well (where $ \sigma_{ij}=\frac{1}{4}[\gamma_i,\gamma_j]$)? Don't we want to preserve amplitudes also in this space?
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2$\begingroup$ Does this answer your question? Dirac spinor's non-unitary representation of the Poincaré group leads me to conclude that Dirac spinors are not "quantum states" in the usual sense $\endgroup$– Connor BehanMar 8 at 18:47
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$\begingroup$ Possible duplicates: physics.stackexchange.com/q/669780/2451 and links therein. $\endgroup$– Qmechanic ♦Mar 8 at 18:49
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