Hello, just had a quick silly question about the coulomb gauge here, though I guess it applies for gauge transformations in general. The problem is, I'm concerned about my gauge choice not being consistent with the equations of motion. For example, suppose I'm working with a nonrelativistic theory describing a single complex scalar coupled to a magnetic field. The part referencing A_i is upto constants and signs [tex] \mathcal{L} = B_i B_i + A_i ( h^*\partial_i h - \partial_i h^* h ) + A_i A_i h^* h [/tex] (this isn't gauge invariant because I haven't included the other relevant terms, but these are the only ones that contribute to the EOM) If I impose the coulomb gauge \partial_i A_i = 0, the gauge fixed lagrangian is [tex] \mathcal{L} = - A_i \nabla^2 A_i + A_i h^* \partial_i h + A_i A_i h^* h [/tex] giving equation of motion [tex] \nabla^2 A_i = h^* \partial_i h + 2 A_i h^* h [/tex] If I took the divergence of this I would get 0 on the left but not on the right. Is this something one has to worry about then? Fixing gauges that are inconsistent with equations of motion? I think I'm rather just making a silly mistake, since given any old A, there's no reason I can't take it to coulomb gauge. Thanks for the help!
If you work with a gauge dependent Lagrangian, how can you arbitrarily fix a gauge? Either the Lagrangian is valid only in a specific gauge (then you have to impose the gauge condition used to derive the Lagrangian) or the Lagrangian is simply not describing electromagnetic fields correctly and you should not use it.