Can an Electromagnetic Gauge Transformation be Imaginary? The Hamiltonian of a non-relativistic charged particle in a magnetic field is 
$$\hat{H}~=~\frac{1}{2m} \left[\frac{\hbar}{i}\vec\nabla - \frac{q}{c}\vec A\right]^2$$.
Under a gauge transformation of the magnetic potential:
$$\vec A ~\rightarrow~ \vec A + \vec\nabla \chi,$$
the wavefunction of the particle transforms as
$$\Psi~\rightarrow~ \Psi\exp(\frac{iq\chi}{\hbar c}).$$
When $\chi$ is real, the wavefunction simply gains an extra phase factor.
However, when $\chi$ is imaginary, there is a measurable change to the wavefunction. This seems to contradict the fact that the magnetic field is invariant under the gauge transformation. How do I resolve this?
 A: $\chi$ can be any reasonable function, real-valued, imaginary-valued, whatever. No variable change can change physics although new wave function and its new equation may be different from the old ones ;-)
A: $\chi$ is a real-valued function. This is part of the definition of the gauge transformation, since $U(1)$ is a one (real) dimensional group. In general, when talking about gauge transformations in particle physics, group parameters are restricted to be real by convention.
In principle, I suppose you could perform a transformation on the wavefunction that looks just like a $U(1)$ gauge transformation except that the parameter can be complex. But the resulting group of transformations would not be $U(1)$, it would be some two-dimensional group, because a complex number parametrizes two dimensions.
A: You can use electromagnetic gauge transforms with complex, rather than real $\chi$. However, I don't think they would be as useful as transforms with real $\chi$, because, if $\chi$ is not real, the equations of motion change, so there is no gauge invariance (see, e.g., Eqs. 20,21 of my article in the European Physical Journal C (free access, http://download.springer.com/static/pdf/480/art%253A10.1140%252Fepjc%252Fs10052-013-2371-4.pdf?auth66=1381456528_6b6a376576161b4f3d18182317776008&ext=.pdf ), where the equations of motion after a gauge transform with a complex $\chi$ (which is equal to $\alpha$ of my article, up to a constant factor) are written out for the Dirac field interacting with electromagnetic field. To avoid confusion, please see the note between Eqs. 16 and 17).
Let me also note that magnetic field does not change under a gauge transform with a complex $\chi$.
A: I think because the wavefunctions are required to be normalized so that $\psi^{*}\psi$ represents the probability or probability density of finding the particle, so their amplitude are not allowed to scale arbitrarily. That's why the gauge field can only be real.
