I finally posted the paper I've been working on. The claim will probably be controversial, so I tried to check my results carefully. But it's still possible that I've missed something.
Physical consequences of the QED theta angle
We describe a simple gedanken experiment which illustrates the physical effects of the QED theta angle, a fundamental parameter of Nature that has yet to be measured. The effects are manifest in quantum phases analogous to those in the Aharonov-Bohm effect, although they are not intrinsically topological. We also derive the quantum phases using a functional Schrodinger approach, and generalize the results to non-Abelian gauge theories.
The conventional thinking is that because FF-dual in QED is a total derivative, doesn't affect the classical equations of motion, and isn't related to any topological vacuum structure, it can't have physical consequences. However, I seem to be able to construct gedanken experiments in which there is a quantum mechanical effect: a relative phase that can lead to interference between photons. This is analogous to the Aharonov-Bohm effect, in which the electron phase is affected by the vector potential, even if there is no magnetic force on the electron.
I derive the effect first using the path integral, but also using the functional Schrodinger equation, which makes the A-B analogy more transparent. The presence of a theta term shifts the canonical momentum in QED, which shifts the momentum operator in the Schrodinger equation, and leads to a phase just as in A-B.
I'm giving a talk on this later today at the ITP/KITP Beijing. The slides are here.
My host told me I should have given it a sexier title like "New quantum phases in Abelian gauge theory" ... Kids these days ... :-)