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Gauge Symmetry breaking in superconductivity
The London brothers realized that the first London equation is not gauge invariant. They were further concerned by the implication of a description of the electron by means of a quantum wave function. As a matter of fact the considerations by Fritz London on this topic are the basis of the modern interpretation of quantum mechanics (QM). Let us briefly see how these topics are connected, a topic that was later elucidated by John Bardeen.
If the current density, an observable in the quantum mechanical sense, has to be proportional to the vector potential, which is arbitrary because of electromagnetic gauge choice, the latter becomes observable too. By the way a similar paradox appears in the Aharonov-Bohm effect. The paradox is partially solved, since the choice is not unique, as it was shown later by John Bardeen in Phys Rev 81 469. We shall now discuss this point (see also Branson UCSD 2003).
According to QM each electron is described by a complex wave function {$\psi$} (for the time being let's assume it is just an independent electron picture, the argument can be extended to many body wave functions). The wave function may be written as a real function, times a phase factor, {$\psi(\mathbf r)e^{i\theta}$}. The complex phase is also not observable according to the standard interpretation of QM. It may be changed for instance by a constant without altering the state and the predicted value of observables. The phase of the wave function may also by changed by a (space and time) function {$\Theta(\mathbf r)$}, as long as it satisfies the same Hamiltonian with the same energy eigenvalue {$E$}. This becomes effectively a local gauge choice, like that of the EM field, and the two are linked. Let's see how, by imposing the minimal substitution
{$$ \frac 1 {2m} \left(\frac {\hbar} {i}\boldsymbol \nabla - q \boldsymbol A\right)^2 \psi = \left(i\frac {\hbar\partial} {\partial t} -qV\right) \psi = (E-qV)\psi$$}
Indeed we may rewrite London equation in another gauge, where {$\mathbf A^\prime = \mathbf A + \delta \mathbf A$} and {$V^\prime = V + \delta V$}, allowing at the same time a local change of phase {$\psi^\prime=\psi e^{i\Theta}$}. Notice that the phase may depend on {$\mathbf r, t$}, but the description must be invariant, i.e. we must impose that {$\psi^\prime$} obeys the Schrödinger equation for the same eigenvalue {$E$} in the new gauge
{$$\frac 1 {2m} \left(\frac {\hbar} {i}\boldsymbol \nabla + {\hbar\boldsymbol\nabla\Theta} -q \boldsymbol A - q\delta \boldsymbol A\right)^2 \psi = \left(E - \frac{\hbar\partial \Theta} {\partial t}-qV-q\delta V\right)\psi$$}
This means that the EM gauge and the local phase of the wave function must change simultaneously as
{$$\boldsymbol \nabla \Theta = \frac {q} \hbar \delta \boldsymbol A,\qquad \frac {\partial \Theta}{\partial t} = - \frac q \hbar \delta V$$}
This is true for any coherent state in an electromagnetic field.
As Bardeen pointed out this relaxes the requirement of describing fields in the London gauge and the first London equation becomes
{$$ \boldsymbol J = -\frac {ne^2}m \left(\boldsymbol A - \frac \hbar q \boldsymbol \nabla \Theta\right)$$}
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