  Unparticles as suggested recently by Georgi have surprising phenomenological
implications, distinctive from any other new physics that we know of. But they
must interact very feebly with ordinary matter to have avoided detection thus
far. We determine how feebly they can interact with the electron, using the
precisely measured quantities in QED: the electron g-2 and the bounds on
invisible and exotic positronium decays. The most stringent bound comes from
invisible ortho-positronium decays: the effective energy scale entering the
vector unparticle-electron interaction must exceed 4 x 10^5 TeV for a scaling
dimension 3/2 of the vector unparticle. The lower bounds on scales for other
unparticles range from a few tens to a few hundreds TeV. This makes the
detection of unparticles challenging in low energy electron systems.
