proof of gauge invariance for quantum 1D ring

This is a question on gauge invariance in quantum mechanics. I do some simple math on a 1D wave-function with periodic boundary conditions, and get that gauge invariance is violated. What am I doing wrong?

Consider one coordinate dimension configured as a ring. The gauge dependent momentum operator can be written:

$p_{op}=-i \frac{\partial}{\partial x} - k$

Units have been chosen so that $\hbar = 1$, $k$ is an arbitrary real constant different for each gauge and $x$ represents the coordinate.

The gauge dependent eigenfunction can be written

$\psi(x)= Ae^{i(n+k)x}$

where A is a constant determined by normalization. As is well known in quantum mechanics, an operator applied to one of its eigenfunctions should yield a real constant eigenvalue multiplying the same eigenfunction: Thus

$[-i \frac{\partial}{\partial x} - k] Ae^{i(n+k)x}= nAe^{i(n+k)x}$

so that the real number n is the eigenvalue, which must be determined by the boundary conditions.

The boundary condition for this periodic system must be that the wave function should join onto itself smoothly everywhere. Thus, if the coordinate is chosen such that x extends from –$\pi$ around the ring to $\pi$ then the eigenfunction in equation 3 must have (n + k) = m, where m is an integer.

Under these conditions, the eigenvalue n in equation 3 will be n = m – k. This eigenvalue depends explicitly on k, and so is not gauge invariant.

I'm assuming this simple situation should be gauge invariant, but I don't see where I goofed. I'd appreciate any help.

Answer

I would like to try an answer for my own question.

The straightforward interpretation of the 3 equations in the original post is that they do present a legitimate conflict. This conflict disappears when the spatial domain is extended to infinity, and the role of the boundary conditions goes away. So, the conflict, the violation of gauge invariance for the finite ring, suggests that either the ring is simply not covered by quantum mechanics, or there is something about the boundary conditions implied by the ring geometry that doesn't work.

So are there other boundary conditions that could be tried? The obvious ones are to demand periodicity not of the wave function, but of the probability density and probability current density. These are both real (not complex) quantities, would let the phase of the wave function have a discontinuity at the boundary, so long as the gradient of the phase was smooth.

Such a choice for the boundaries would allow the continuous eigenvalue spectrum of the infinite line to apply as well to the ring, which would restore gauge invariance. But this comes at the expense of inhomogeneous and nonlinear boundary conditions.

The nonlinear boundary conditions could be accepted if they are consistent with the standard type of Hilbert space of Hermitean operators, and the principle of superposition.

In a paper I recently posted on arxiv.org I go into some detail about how the nonlinear boundaries are indeed consistent with both gauge invariance and the standard structure of Hilbert space. The bottom line is that the nonlinearity creates a continuous spectrum of eigenvalues, not all of which are superposable in Hilbert space. The nonlinearity in the boundary allows a subset to be superposable. The combination of continuous eigenvalues and superposable discreet eigenvalues makes a band structure for the Hilbert space, rather than simple discreet levels.

PLEASE CLICK HERE for the larger explanation.

So I think the correct solution of this problem is quite significant for quantum systems coupled to their environments, such as Josephson junctions used as qubits.