Continuity equation for Schrodinger equation with minimal coupling

[nembokid]

The Schrodinger equation with the minimal coupling to the Electromagnetic field, in the Coulomb gauge $$\nabla \cdot A$$, has a continuity equation $$\partial_t \rho = \nabla \cdot j$$ where $$j \propto Re[p^* D p]$$ (D is the covariant gradient $$D= \nabla + iA$$.

My question is: is there any continuity equation which generalized the preceding one, without having to fix the Coulomb gauge? I think that, being the Schrodinger equation nonrelativistic, a choice of a noncovariant gauge is necessary, but maybe some ugly-to-see equation still exists.

thank you

 

[Bob_for_short]

It seems to me that this continuity equation is gauge-invariant. It means the probability conservation.

 

[nembokid]

It's gauge invariant as long as you perform gauge transformations compatible with the Coulomb gauge constraint (which does not fix completely the gauge, as is well known). But what about an equation which does not require this constraint?

 

[Bob_for_short]

It is the same. Start from the Schroedinger equation with an arbitrary A and φ and find the equation for ρ.

 

[nembokid]

But this equation for $$\rho$$ has no longer the form of a temporal derivative plus the gradient of a current (at least, I can't see how to put it in this form).

 

[Bob_for_short]

It should not be a gradient but divergence of current: divj.