Analytical Study of the Spin Projection Operator

Abstract: We present an analytical study of the spin projection operator of the spin state and the helicity state. It is pointed out emphatically that the former is Lorentz covariant, while the latter is a nonrelativistic two-component operator. In the special case of $\bm p=p_z$, the helicity state and the spin state are formally identical. However, even so their spin projection operators are still different and so the helicity state is not a special case of the spin state. This makes the spinor is the degenerate state of the two different spin projection operators. The calculation on the lifetime of polarized muons shows that this difference will inevitably lead to the left-right polarization-dependent lifetime asymmetry.