Interacting Particles

Abstract

In the interacting case the IR problem constitutes an essential complication. We therefore directly consider QED. We define particle probes like in the free case by (14.37), (14.46), and (14.51), though the photon case will be shown to need some elaboration. The fields in these definitions are now the interacting ones. In the fermion probes D± we use the physical fields Ψ, Ψ̄, because we are only interested in physical states. There is a problem here: even though formed with the physical fields, the probe

$$ {D^ - }(a) = \overline \Psi ({f_a})*\overline \Psi ({f_a})$$

is not a gauge invariant operator, hence not a true observable. The same holds for D⁺.¹ But we do not claim the probes to represent realistic detectors. We use them only as a convenient means of motivating and describing the concept of a particle as a localized object. Therefore the use of these probes is justified despite their lack of gauge invariance. In any case, in the asymptotic region in which we are interested, the interacting fields behave sensibly as free fields. And in the free theory there are no gauge transformations of the second kind, hence the problem does not occur. A more satisfying way of handling the problem would be to work with genuine observables, for example, with D(a) = ∫dxf(x-a)j _µ(x), where f is a real test function which is essentially concentrated around the origin both in x-and p-space.