Gauge invariance lost using Maxwell's equations I read a paper on superconductivity, and the calculations use Newton's second law and the Maxwell-Faraday law to write the "free electron current" as
$$
\textbf{J}_e=-\frac{ne^2}{m_e}\textbf{A}
$$
where $\textbf{A}$ is the vector potential ($\textbf{B}=\textbf{rot A}$). After that, it said that the Coulomb gauge is chosen ($\text{div }\textbf{A}=0$), and it's verified since $\text{div }\textbf{J}_e\propto \partial_t \rho =0$. But it also says that we have lost gauge invariance. Why? 
Moreover, I don't think I understand properly what implies gauge invariance. Can someone help me? 
 A: You have lost gauge invariance because a specific gauge has been chosen, the Coulomb gauge $\nabla \cdot {\bf A} = 0$.  
Gauge invariance of a theory implies some redundancy in how certain fields in the theory are defined, in that the physics is unchanged by different definitions of these fields.  In classical electromagnetism, one has $\nabla \times {\bf A} = {\bf B}$.  Notice that we can add the gradient of an arbitrary scalar function, $f$, to the vector potential, ${\bf A}' = {\bf A} + \nabla f$, without changing the magnetic field, since the curl of the gradient of a scalar function vanishes.  
Via Maxwell's equation, $\nabla \times {\bf E} = -\partial {\bf B}/\partial t$, and the fact that ${\bf E} = -\nabla V - \partial {\bf A}/\partial t$, you can see a similar thing happen with the scalar potential, $\phi$: $\phi' = \phi - \partial f/\partial t$. This redundancy in the scalar potential is perhaps most intuitive, since only changes in the electric potential are ever physically measurable, and changes are unaffected by the addition of a constant.  
Often when doing calculations it's possible to simplify a problem by fixing the gauge, that is, by choosing a specific function $f$.  In the Coulumb gauge, $f$ is chosen so that $\nabla^2 f = 0$, giving $\nabla \cdot {\bf A} = 0$.  Once we've selected a gauge, though, the theory is no longer gauge invariant.  This makes it sometimes difficult to know whether the result obtained is specific to the gauge choice, or a result true of the gauge invariant theory.
