Spin operators and representations of the Poincar\'e group

We present the rigorous derivation of covariant spin operators from a general linear combination of the components of the Pauli-Lubanski vector. It is shown that only two spin operators satisfy the spin algebra and transform properly under the Lorentz transformation, which admit the two inequivalent finite-dimensional representations for the Lorentz generators through the complexification of the $SU(2)$ group. In case that the Poincar\'e group is extended by parity operation, the spin operator in the direct sum representation of the two inequivalent representations, called the new spin distinguished from the Dirac spin, is shown to be equivalent to axial and Hermitian spin operators for particle and antiparticle. We have shown that for spin $1/2$, the Noether conserved current for a rotation can be divided into separately conserved orbital and spin part for the new spin, unlike for the Dirac spin. This implies that the new spin not the Dirac spin provides good quantum observables.