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Bosons and Fermions in a rigorous QFT 
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#19
Feb1012, 07:45 PM

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#20
Feb1112, 03:24 AM

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Which states do I miss? I can construct nearly arbitrary operators from the creation and annihilation operators; look at the method of coherent states, for example. And please not that up to now we haven't defined the Hilbert space, neither for x and p, nor for the creation and annihilation operators! So if we need to change the entire Hilbert space for some reason we are not forced to do this by introducing creation and annihilation which are nothing else but linear combinations of x and p. There is physics behind it so far. 


#21
Feb1112, 08:36 AM

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In QM, before we define any operator, we set the full Hilbert space as all squareintegrable functions, and we define how x and p (so are a and a+) act on this full Hilbert space, then according to the specific Hamiltonian we have, we can construct the subspace using a and a+, though the construction may not be as simple as harmonic oscillator case. I guess this is what you mean. However in an interacting QFT, the true structure of the full Hilbert space is not clear yet, let alone defining how a and a+ act on the space. The normal treatment is to use the Hilbert space from free theory (a Fock space) and then do things perturbatively. And my point is, if the interaction actually destroys any existence of directproduct structure of the true Hilbert space, then whatever kind of Fock space you use will be just an approximation at best, and if this is the case, I think Fock space does necessarily imply a perturbative approach. 


#22
Feb1112, 01:50 PM

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#23
Feb1112, 03:20 PM

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Googling "Grassmann OsterwalderSchrader" produced these papers which may be relevant.
Jaffe, Constructive Quantum Field Theory "The Euclidean methods also apply to theories with fermions, at least for examples with interactions that are quadratic in the fermions. This is the case for free and for “Yukawa type” interactions, used extensively in physics." Benfatto, Falco, Mastropietro, Functional Integral Construction of the Thirring model "Proposed by Thirring half a century ago, the Thirring model is a Quantum Field Theory of a spinor field in a two dimensional spacetime, with a self interaction ..." 


#24
Feb1112, 07:42 PM

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BTW, the presence of interaction doesn't necessarily "destroy" directproduct structure. It simply means that the Hamiltonian can now mix stuff from different sectors. 


#25
Feb1112, 09:09 PM

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#26
Feb1212, 12:58 PM

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#27
Feb1212, 01:03 PM

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#28
Feb1212, 03:50 PM

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#29
Feb1212, 08:41 PM

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(I should probably refresh my memory of Haag a bit more thoroughly, else I risk talking out of my rear... :) 


#30
Feb1212, 08:56 PM

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The field operators in the exact form of Gauss' law correspond to the physical fields, not the free fields. Some approaches to QFT try to construct physical field operators perturbatively in terms of (increasinglycomplicated) products of the free field operators. Among other things, this procedure must ensure that the new field operators still correspond to suitable Poincare unirreps with the physically correct spin, etc. Do you recall the earlier thread where I talked about dressing the asymptotic electron states using coherent photon operators? After such dressing has been applied, the commutation relations are a bit different. E.g., the commutator between an electron operator and the electric field operator is no longer zero. Instead, it gives the usual Coulomb field of a charged electron. (This is all at low momenta, since the main task there was to deal with IR divergences.) A similar construction (of Dirac) also shows how to banish some parts of the unphysical EM gauge freedom. At least one of the rigorous QFT results that I know proceeds via a related process of dressing transformations applied to the basic operators: J. Glimm, "Boson Fields with the [itex]:Φ^4[/itex]: Interaction in Three Dimensions", http://projecteuclid.org/DPubS?servi...cmp/1103840981 Warning: very few people can safely read that on an empty stomach... 


#31
Feb1312, 03:48 PM

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#32
Feb1312, 04:59 PM

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A Fock space is nothing else but a direct sum of tensor products of copies of a singleparticle Hilbert space H. But it is not necessary that the Hilbert space H is spanned by free fields; any single particle Hilbert space with the correct creation and annihilation operator algebra is sufficient. So if it's possible to construct a suitable transformation from free to 'physical' or 'dressed' fields and a physical Hilbert space then the latter one can be decomposed into physical Fock states. This has been done in QCD in orer to study confinement in the canonical formulation (the problem with the construction of the physical Hilbert space is of course always the same: complete gauge fixing, taming Gribov ambiguities etc.; anyway  these problems do by no means spoil the Fock space approach using physical fields) 


#33
Feb1312, 05:03 PM

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I also think it is interesting to note that classical electromagnetism is also not a rigorous theory, at least when thinking about point charges. Indeed, classical fluid dynamics and general relativity are also not known to be free of singularities. 


#34
Feb1312, 05:11 PM

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DarMM once mentioned something about this, but we never got to hear the full story. 


#35
Feb1312, 05:40 PM

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I remember the 'dressing' and the 'gauge fixing by unitary transformations'; both approaches seem to be similar b/c they partially 'solve' some field equations.
The first approach (dressing) changes the operator algebra; it can be solved explicitly in some 1+1 dim. field theories like the Schwinger model (here one formally solves the Dirac equation by exponentiation using a gauge field string with a path ordered product). The second approach does not change the operator algebra; the Gauss law is solved but b/c a unitary trf. is used, all operator algebras remain unchanged. This may be spoiled by regularization which requires gaugeinvariant point splitting (I can only remember the twodim. case). In the first case the interaction is "hidden" in the dressed fields; they create the physical Coulomb interaction, but the interaction term itself looks trivial algebraically. In the second case the interaction terms are constructed explicitly and in principle they can be expressed using physical Fock space operators. In the second case the (A°=0 & Coulomb gauge) Hamiltonian contains one piece which shows directly the colorelectric Coulomb potential: [tex]H_C = g^2 \int d^3x \int d^3y \,\text{tr}\,J^{1}(x)\,\rho(x)\,(D\partial)^{1}\,(\partial^2)\,(D\partial)^{1}\,J(y)\,\rho(y)[/tex] with D = ∂ + gA, A being the gaugefixed gluon field, J being the FadeevPopov determinant J = det(D∂), ρ = ρ[q] + ρ[A] being the total color charge with quark and gluon contribution (w/o J, D and ρ[A] in H_{C} the usual Coulomb gauge interaction in QED is recovered) 


#36
Feb1312, 05:47 PM

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@strangerep, BTW, another interesting comment in Haag was that since the Hilbert spaces for each representation of the CCRs are different, presumably the selection of the representation depends on dynamics. He then says that the advantage of the Lagrangian approach is that it makes it easy to choose the dynamics based on symmetries, and then construct the appropriate Hilbert space after that. (Again, I don't have the page reference, but it should be in one of the two sections I mentioned above.)
Also, it's really interesting to me that BCS has this "rigourous treatment"  I'd always taken it to be unrigourous since it's modelling a condensed matter phenomenon where one can definitely take a lattice cutoff so that Haag's theorem won't apply. 


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