[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.]