Haag's Theorem loophole?
My understanding of Haag's theorem (see link below) is that there is a mismatch between the Hilbert spaces of free and interacting particles. The argument seems to be that we require both the free and interacting vacuum states to be invariant under Poincare transformations. Now since the entire Hilbert space of the free particle is built by acting on the vacuum with creation operators which are not invariant under Poincare transformations, then our only choice for the interacting vacuum is to have it be proportional to the free vacuum. Here we get a contradiction, since the free vacuum cannot be an eigenstate of both the free Hamiltonian and the full Hamiltonian.
http://philsciarchive.pitt.edu/arch...rfinalrevd.pdf) All that's good and well, except for one thing. Why do we require the free vacuum to be invariant under Poincare transformations? The free fields aren't the physical ones, the interacting ones are. I see no reason why the nonphysical free vacuum can't transform nontrivially under a translation or a rotation. If, instead, we require only that the interacting vacuum be invariant under Poincare transformations, the mismatch between the Hilbert spaces disappears. 
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If you abandon the principle that the (free) vacuum is not annihilated by the (free) Poincare generators, then... how do you construct a Fock space? What Lie algebra do you start from?? And how do you construct the interacting Hilbert space? 
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Before, we have [tex]U(\Lambda)0\rangle = 0\rangle [/tex] Now, we would some nontrivial transformation [tex]U(\Lambda)0\rangle = \sum c_i i\rangle[/tex] where the coefficients of this transformation are chosen such that the interacting vacuum is invariant. 
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One approach to deal with this problem is to take this difference between bare and dressed states for granted, consider different Hilbert spaces for noninteracting and interacting theories, etc, etc. Personally, I find this approach cumbersome and not appealing. There is, however, another line of thought. We can ask ourselves, are we sure that the interaction adopted in QFT (take QED as an example) is the correct one? After all, this interaction (the "minimal" coupling between the photon field and electron current; I think this is what Bob_for_short had in mind when writing about the "wrong interaction term" in QFT.) was "derived" by using rather shaky analogies with classical Maxwell's electrodynamics. The only experimental data supporting this choice of the QED interaction is related to scattering. But it is wellknown that there are many (scatteringequivalent) Hamiltonians, which produce exactly the same Smatrix. So, perhaps, we can choose QED interaction in another form, such that 1) the selfinteraction is not present anymore; 2) there is no difference between free (bare) and interacting (dressed) states, just as in ordinary quantum mechanics; 3) the Smatrix remains the same (i.e., agreeing with experiment) as in renormalized QED; 4) ultraviolet divergences are not present. This strategy (known as the "dressed particle" approach) was first formulated in a beautiful paper O. W. Greenberg and S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim. 8 (1958), 378. It appears rather promising, as you can check by following multiple references to this work. Within this approach, Haag's theorem does not present any difficulty. See, for example M.I. Shirokov, "Dressing" and Haag's theorem. http://www.arxiv.org/abs/mathph/0703021 
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Understanding the "free" particles as elementary excitation modes of one compound system makes it unnecessary to add the selfaction term. One can write just an interaction term. For example, scattering of charges can be constructed as a potential scattering of compound systems (just like atoms) with inevitable exciting their internal degrees of freedom (photon oscillators). 
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etc, satisfying the usual CCRs, and these operators constitute an irreducible set, (meaning that any operator on the Fock space can be expressed as a polynomial or function of the a/c ops). That means you still have the usual (free) representation of the Poincare generators, so there are still operators [itex]U(\Lambda)[/itex] such that [itex]U(\Lambda)0\rangle =0\rangle[/itex]. Then.... Quote:
representation of the Poincare group". Let's call these operators [itex]W(\Lambda)[/itex] and let's call the interacting vacuum [itex]\Omega\rangle[/itex], where [itex]W(\Lambda) \Omega\rangle = 0[/itex], but [itex]W(\Lambda) 0\rangle \ne 0[/itex]. But the original (free) a/c ops constitute an irreducible set, so [itex]W[/itex] must be expressible as function of them (if indeed the free and interacting Hilbert spaces coincide). One would also expect to find "interacting" a/c operators, i.e., [itex]\alpha_k^*[/itex], etc, corresponding to singleparticle states in the interacting theory (which are eigenstates of the interacting Hamiltonian). These must satisfy [itex]\alpha_k \Omega\rangle = 0[/itex]. The task is then to express the [itex]\alpha_k[/itex] ops in terms of the [itex]a_k^[/itex] ops. That's pretty much what the "dressed particle" approach involves (also known under the phrase "unitary dressing transformation"). As well as the references mentioned by meopemuk, there's also this review article: Shebeko, Shirokov: "Unitary Transformations in Quantum Field Theory and Bound States" Available as nuclth/0102037 Unfortunately, one generally encounters infinite quantities when perturbatively constructing the transformation, which must be "renormalized" in a manner reminiscent of standard renormalization. One thus encounters a different variation of Haag's theorem in that the interacting representation cannot live in the Fock space constructed from the free a/c ops. 
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First, one can define the usual "Fock space constructed from the free a/c ops". Then one automatically gets the noninteracting representation of the Poincare group there. The next step is to construct the interaction part of the Hamiltonian (the Poincare generator of time translations) as a normallyordered polynomial in "a/c ops". Of course, in order to be physically admissible, this polynomial must satisfy a few conditions: 1. Each term in the polynomial must have at least two annihilation operators and at least two creation operators. This is necessary to avoid selfinteractions in the vacuum and 1particle states, i.e., to make sure that these states are lowenergy eigenvectors of the interacting Hamiltonian. 2. The momentumdependent coefficient functions in each term must vanish sufficiently rapidly away from the "energy shell". This is necessary to ensure that all loop integrals encountered in Smatrix calculations are finite, so that there are no ultraviolet divergences. 3. In parallel with the above construction of the interacting energy one needs to build the "interacting boost" operator having similar properties, and making sure that commutation relations of the Poincare Lie algebra remain intact. This would guarantee the relativistic invariance of the theory. 4. Finally, one needs to make sure that the Smatrix calculated with the above Hamiltonian is exactly the same as the Smatrix of the renormalized QFT. This can be done by properly adjusting coefficient functions in each perturbation order. Then the new theory is guaranteed to agree with all existing experiments. I don't have a full mathematical proof that this construction is possible, but there are many indirect indications that there are no fundamental obstacles on this path. 
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According to my experience, any interaction smears quantum mechanically the charge. Probably your interacting (dressed) electron is still pointlike because you have not completed the dressing, isn't it?

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The pointlike character of particles and the spread of their wave functions are two separate issues. In QM, the stationary wave function of a free electron is a plane wave, i.e., it is completely delocalized. However, this does not prevent us from talking about the pointlike electron. 
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