Starting from the Lagrangian density: [itex] L= -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{m^{2}}{2} B_{\mu}B^{\mu}[/itex] we can derive the E.o.M. for the field [itex]B[/itex] which read: [itex] ( \partial^{2} + m^{2}) B^{\mu} - \partial^{\mu} (\partial B) = 0 [/itex] In the case of a massive field, I am not sure how I can kill out the partial of B through the field equations.... [itex] \partial B=0 [/itex] Does this come as a constraint/boundary condition of minimizing the action? or is there something I cannot see? In most cases they state it's a Lorentz Gauge, however I am not sure how this can be indeed shown...
http://www.theory.nipne.ro/~poenaru/PROCA/proca_rila06.pdf P.180 , eq 19,20 and the 1st paragraph of p.181 gave the answer... One has to take the derivative of the Equations of Motion [itex]\partial_{\mu}[/itex] and [itex]\partial B[/itex] comes out zero... (unfortunately for me I came out with the wrong EoM missing a minus sign and I couldn't even think of doing it)