Invariant definition of relativistic spin states in classical and quantum theories with an external field

The invariant spin operators $\hat M=H^{\mu\nu}\hat\Pi_{\mu\nu}/2$ and $\hat M^{'}=E^{\mu\nu}\hat S_{\mu}\hat P_{\nu}/m_0c$, where $\hat\Pi_{\mu\nu}$ and $\hat S_{\mu}$ are spin operators, $H^{\mu\nu}$ is the tensor of an external electromagnetic field, and $E^{\mu\nu}$ is the tensor that is the dual of $H^{\mu\nu}$, are considered. The spin invariants $\hat M$ and $\hat M^{'}$ in the rest frame of the particle determine the spin projection onto the direction of the magnetic field. It is shown that in the Bargmann–Miehel–Telegdi approximation both spin invariants $\hat M=\hat M^{'}$ are integrals of the motion in both the classical and the quantum theory.