# Why Lorentz transf. representation in spin 1/2 particles Hilbert space is not a unitary operator? [duplicate]

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?