On physical origin of Berry connection of free relativistic electrons

No consensus has been reached on the appropriate operators for the spin and orbital angular momentum of free relativistic electrons. But there is a tendency to believe the existence of a spin-orbit coupling through the Berry connection. Here we show that this is not the case. Because the positive-energy solution of the Dirac equation is associated with the state of the electron, it is reasonable to postulate that only when projected onto the positive-energy subspace can observable operators on the Hilbert space of the Dirac equation represent physical quantities of the electron. On this basis, we find, after mapping the projected operators onto an irreducible representation, that the Berry connection describes the position of a quantum reference point with respect to the laboratory reference frame, relative to which the position of the electron is canonical. The orbital angular momentum of the quantum reference point about the origin of the laboratory reference frame is expressed in terms of the Berry connection. It combines with the canonical orbital angular momentum of the electron about the quantum reference point to give the orbital angular momentum of the electron about the origin of the laboratory reference frame.