Haelfix said:
...Its wonderfully miraculous that the whole business works at all.
"The whole business" fails in QG and in many many other cases.
Haelfix said:
I think a lot of physicists became a lot less worried about Haags theorem after Wilsons work on effective field theory, where it became clear that in general there might need to be a UV completion for most phenomenological field theories of interest.
This direction is completely misleading. There is nothing special at short distances because the UV divergences do not appear as divergences at high transferred four-momentum q. It is the spectral decomposition of perturbation expression that diverges at high values of Fourier parameter. In the coordinate representation the divergence corresponds to the product of delta-functions, roughly speaking. No limiting process can define it because it remains unacceptable. Renormalizations is discarding some divergences just because they are unacceptable. In fact, it is the interaction term that includes the self-action which is unacceptable.
strangerep said:
I see "axiomatize" as merely a synonym for "let's use clear and honest mathematics", which seems like a good thing at any stage. (Of course, choosing particular axioms must be a physically-motivated endeavour).
I am for honest mathematics. I consider discarding corrections as dishonest mathematics.
Fortunately I understand why discarding may sometimes work: it compensates the postulated or "axiomatized" non-physical self-action contribution, contributions that have no physical meaning and make impossible any calculations.
... Haag's original paper basically just pointed out that all known physically-relevant
interactions contained products of creation operators which necessarily lead outside
the free Hilbert space.
So we need new physically-relevant interactions. Self-action is not of that sort, that's the main conclusion to infer from the Haag's theorem.
As I said above, those axioms are merely someone's attempt to construct a
theory using honest mathematics. It might of course be physically incorrect.
But even if we take a phenomenological approach (starting from experimental facts),
one still needs to use honest mathematics in whatever theory one constructs.
Any decent theory is and must be phenomenological. As soon as you start its developing from postulates and axioms, you deal with mathematical study, not with physics. As I said before, only after constructing a working in all respects theory we may "axiomtize" it, not before.
Many of us agree that the actual interaction terms in QFTs are ill-defined. So we need new physical insights in order to advance a physically-relevant interaction. The degrees of freedom should
exchange with the energy-momentum - this is the only purpose of interaction,
not silly "self-action".
Where do our interaction terms come from? In QED it is the Lorentz self-action ansatz jA. Many years after it was "derived" from "gauge" principle. So the actual interaction terms are either "gauge"-derived or products of fields like in φ
4 (why not?). Are they dictated with some experimental data? No, on the contrary: the experimental data are "seen" via prism of silly theoretical constructions. It is very flattering to advance "postulates" and axioms from which everything "follows". So far we see enormous problems (failures) following from such attempts. I see these attempts as irresponsibly counting on luck (renormalizations will hopefully save us) and following bad suites in a lazy way.
