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Haag's Theorem loophole? 
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#55
Oct1009, 03:57 AM

P: 1,746

In my opinion it is very unfortunate that historical development of QFT has ignored this simple and important condition. This omission made QFT more complicated and mysterious than it should be. 


#56
Oct1009, 02:31 PM

P: 1,160

Fortunately I understand why discarding may sometimes work: it compensates the postulated or "axiomatized" nonphysical selfaction contribution, contributions that have no physical meaning and make impossible any calculations. Many of us agree that the actual interaction terms in QFTs are illdefined. So we need new physical insights in order to advance a physicallyrelevant interaction. The degrees of freedom should exchange with the energymomentum  this is the only purpose of interaction, not silly "selfaction". Where do our interaction terms come from? In QED it is the Lorentz selfaction 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. 


#57
Oct1009, 04:22 PM

P: 1,746

I also agree that we need to look for other interactions which do not lead to the "vacuum polarization" and "selfaction". The question is how can we change the interaction Hamiltonian without destroying all the good physical results produced by QED in the last 60 years? In my opinion, the "dressed particle" approach is the way to go. 


#58
Oct1009, 04:45 PM

P: 1,160

In my opinion, a perturbative dressing cannot be achieved. It is like obtaining an atomic wave function (bound state) from the perturbation theory  it is technically impossible. So we have to start from dressed charges from the very beginning. 


#59
Oct1009, 08:25 PM

Sci Advisor
P: 1,908

OK. The adjective I'd use for such successful QFTs is "perplexing"  that the mathematics of the limiting process in general renormalization is still not well understood in the most important QFTs. which must be dealt with similarly to standard renormalization. The limiting process and convergence are not understood in general (afaik), so I don't think all those problems are yet solved trivially (at least, not in the mathematical sense). :) 


#60
Oct1009, 10:27 PM

P: 1,746

Divergent loop integrals are not a problem as well: if interaction potentials are chosen correctly (momentumdependent coefficients multiplying products of creation and annihilation operators vanish rapidly away from the "energy shell") then all integrals converge. The problems that you mentioned appear only if we want to construct a "dressed particle" theory that is equivalent (i.e., yields the same scattering matrix) to a "normal" QFT theory, such as QED. This construction can be done by means of the messy "unitary dressing transformation". Yes, this transformation can be done only perturbatively, its convergence is not established, and it must deal with all the nasty legacies of QED, such as divergencies. 


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