The Bargmann-Wigner method in Galilean relativity
The equations of motion of a spin one particle as derived from Levy-Leblond's Galilean formulation of the Bargmann-Wigner equations are examined. Although such an approach is possible for the case of free particles, inconsistencies which closely parallel those encountered in the Bargmann-Wigner equations of special relaticity are shown to occur upon the introduction of minimal electromagnetic coupling. If, however, one considers the vector meson within the Lagrangian formalism of totally symmetric multispinors, it is found that the ten components which describe the vector meson in Minkowski space reduce to seven for the Galilean group and that in this formulation no difficulty occurs for minimal electromagnetic coupling.
More generally it is demonstrated that one can replace Levy-Leblond's version of the Bargmann-Wigner equations by an alternative set which leads to the correct number of variables for the vector meson. A final extension consists in the proof that for all values of the spin the (Lagrangian) multispinor formalism implies the Bargmann-Wigner equations. Thus the problem of special relativity of seeking a Lagrangian formulation of the Bargmann-Wigner set is found to have only a somewhat trivial counterpart in the Galilean case.