On the Galilean-invariant equations for particles with arbitrary spin

Abstract

In our preceding paper [1] the equations of motion which are invariant under the Galilei group G have been obtained starting from the assumption that the Hamiltonian of a nonrelativistic particle has positive eigenvalues and negative ones. These nonrelativistic equations as well as the relativistic Dirac equation lead to the spin-orbit and to the Darwin interactions by the standard replacement pμ → πμ = pμ − eAμ. Previously it was generally accepted the hypothesis that the spin-orbit and the Darwin interactions are truly relativistic effects [2]. In [1] only the equations for the particles with the lowest spins s = 12 , 1, 3 2 have been obtained. What puts the equations [1] in a class by themselves is that the transformation properties of a wave function are rather complicated (nonlocal) and it is difficult to establish their invariance under the Galilei transformations after the replacement pμ → πμ. In the present note equations for arbitrary-spin particles are obtained which possess as good physical properties as the equations [1]. Moreover the wave function has simple transformation properties in the case of the equation describing the interaction with an external field as well as in the case of the absence of interaction. We shall start from the assumption that under the Galilei transformation x → x′ = Rx+ V t+ a, t → t′ = t+ b, (1)