[Note: I'm using QED as a simple example, despite having heard that it is unlikely to exist. I'm happy to confine the question to perturbation theory.]
The quantized Aᵘ and ψ fields are non-unique and unobservable. So, two questions:
A. Can we instead define QED as the Wightman theory of Fᵘᵛ, Jᵘ, Tᵘᵛ, and perhaps a handful of other observable, physically-meaningful fields? The problem here is to insure that the polynomial algebra of our observable fields is, when applied to the vacuum, dense in the zero-charge superselection sector.
B. Is there a way to calculate cross sections that uses only these fields? This might involve something along the lines of Araki-Haag collision theory, but using the observable fields instead of largely-arbitrary "detector" elements. (And since we're in the zero-charge sector, some particles may have to be moved "behind the moon", as Haag has phrased it.)
(Of course, the observable fields are constructed using Aᵘ and ψ. But we aren't obliged to keep Aᵘ and ψ around after the observable fields have been constructed.)
I suspect that the answers are: No one knows and no one cares. That's fine, of course. But if someone does know, I'd like to know too.
[I heartily apologize for repeatedly editting this question in order to narrow its focus.]