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Helicity operator in Dirac equation is given by $$H=\frac{\vec{S}\times \vec{P}}{P^{2}}$$ This operator commutes with dirac hamiltonian.We can also define a helicity(with same form) operator in case of Schroedinger equation.Which will commute with Schroedinger hamiltonian and have eigenvalue $\frac{1}{2}$ or $-\frac{1}{2}$. We know spin also have eigenvalues same as helicity operator.Now spin is the projection of spin angular momentum on $z$ axis but $H$ is the projection of spin angular momentum on momentum axis.My question is how do I relate these two operators in non relativistic case? It looks like helicity operator does all the things spin operator normally do in Non relativistic case also.So isn't more sensible to use helicity operator in non-relativistic limit also instead of spin operator?

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