Haag's Theorem loophole?

strangerep

Which other axiom does it break?

strangerep

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 believe the point is that an interacting theory cannot share a Hilbert space with
a free theory.

IIRC, 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.

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 phenomenlogical approach (starting from experimental facts),
one still needs to use honest mathematics in whatever theory one constructs.

Perhaps you should re-phrase that statement. It sounds like you're
against honest mathematics, which I'm sure is not what you mean.

meopemuk

The same can be said about our most accurate theory - QED. I am pretty sure that Shirokov's condition (1') does not apply in QED. Does it make QED invalid in your opinion?

meopemuk

Apparently Haag didn't know about the class of "dressed particle" theories in which interactions having only products of creation operators are forbidden. In these theories, each normally-ordered interaction term must have at least two creation operators and at least two annihilation operators. This guarantees that the interaction does not act on the vacuum and one-particle states, so all problems associated with self-interaction, vacuum polarization, etc. are solved trivially.

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.

Bob_for_short

"The whole business" fails in QG and in many many other cases.
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.
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.
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.
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 φ⁴ (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.

meopemuk

I agree with you that these interactions are physically unacceptable. Expressed in terms of creation-annihilation operators they contain terms composed of (for example) creation operators only. This means that the vacuum of the interacting theory is different from the vacuum of the free theory. Also "physical" particles of the interacting theory are different from "bare" particles of the free theory. This difference creates all sorts of confusions and inconsistencies, one of which is Haag's theorem.

I also agree that we need to look for other interactions which do not lead to the "vacuum polarization" and "self-action". 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.

Bob_for_short

I can only add that the standard theory but without counter-terms is good for nothing - it predicts nothing physical. For example, the elastic charge scattering that never happens. Worse, it does not exist at all, so the references to the interacting theory vacuum are meaningless.
In general I agree with the "dressing" approach but I see it differently - we have to "dress" charges non-perturbatively. Then we can arrive at a construction with very nice and promising features: a theory of potential interaction of compound ("dressed") systems.

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.

strangerep

Oh... by "perfectly valid" you meant "agreeing extremely well with experiment" ?
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.

One must still construct such theories perturbatively, and divergent integrals arise in that process
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). :-)

meopemuk

I meant that generic "dressed particle" theory provides a trivial solution to the problem of "physical" vacuum and 1-particle states. By construction, these states simply coincide with corresponding states of the non-interacting theory.

Divergent loop integrals are not a problem as well: if interaction potentials are chosen correctly (momentum-dependent 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.