The quantized A? and ? fields are non-unique and unobservable. So, two questions

Question

The quantized A? and ? fields are non-unique and unobservable. So, two questions:

A. Can we instead define QED as the Wightman theory of F??, J?, T??, 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 A? and ?. But we aren't obliged to keep A? and ? 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.

Explanation / Answer

Suppose that QED exists in the strongest feasible sense. This means that appropriately smeared fields in A? and ? with compact support are self-adjoint operators on some Hilbert space with a common dense nuclear domain, such that the operators (anti)commute for spacelike separated smeared fields, and formal expansion of the time-ordered correlation functions reproduces the standard perturbation expansion.

In this case, the gauge invariant even polynomial expressions of degree at most two are Wightman fields defining the vacuum sector of QED, and they generate a C?-algebra satisfying the Haag-Kastler axioms. This is the observable subalgebra of the field algebra.

As photons are massless, the standard Haag-Ruelle collision theory is not applicable, and as charged fields are missing, the scattering theory is not asymptotically complete. To get an asymptotic completion one would have to proceed in a DHR-like fashion and reconstruct intertwiners between the (uncountably many) superselection sectors of the theory. But DHR assumes a mass gap, hence the theory is not applicable. Nevertheless, if QED exists, the intertwiners exist, too, and are in fact heuristically known. However, the asymptotic charges states (electrons) are only infraparticles, as they (unlike bare electrons) carry their own elecromagnetic field. An asymptotic scattering theory of relativistic infraparticles (which should involve coherent state superselection sectors) has not been worked out so far.

But work by Derezinski treats the nonrelativistic case rigorously, and work by Kulish and Faddeev indicates nonrigorously that nothing should go wrong in the relativistic case.

Thus a lot is known about how things should look like, but in the relativistic case there are neither constructions nor proofs. The best that has been done rigorously (by Salmhofer, I believe) is to construct QED as a field theory whose fields are formal power series in the coupling constant, but this is far from what is needed.

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