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Rigorous Quantum Field Theory. 
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#235
Apr610, 07:24 PM

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PF Gold
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#236
Apr610, 09:34 PM

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I think in order to move forward we need to understand exactly what we are doing in QFT. It would be nice to revisit (Wightman's) axioms to see what is their physical meaning (if any). For example, one axiom postulates how quantum fields transform with respect to inertial frame changes (see my post #227). DarMM has agreed with me that quantum fields are not directly observable objects/properties. This means that the mentioned transformation law cannot be verified in experiments even in principle. So, this transformation law is simply an unjustified assumption. There is a good chance that this assumption is just wrong. Then no matter how "rigorous" our math is, we'll not get anything useful from a wrong axiom. Eugene. 


#237
Apr710, 01:13 AM

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PF Gold
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People were probably saying the same thing about the Dirac delta in 1930. Do you also think that the "inconsistencies" of the delta "function" made it pointless to develop distribution theory? 


#238
Apr710, 02:21 AM

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since there's still quartic products of creation operators therein. What am I missing? 


#239
Apr710, 03:10 AM

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1. If you integrate [itex]:\phi_{0}^{4}:[/itex] against a test function it always results in a denselydefined operator. This is in constrast to [itex]\phi_{0}^{4}[/itex], which after smearing does not give a denselydefined operator 2. From a Feynman graph point of view, any graphs associated with [itex]:\phi_{0}^{4}:[/itex] do not contain tadpole loops. Tadpole loops are the only ultraviolet divergent loops in 2D, so it is ultraviolet finite. However when I say a well defined OVD, I mean (1.). (2.) is just for perturbative intuition. 


#240
Apr710, 03:20 AM

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Wightman's axioms go beyond that and postulate that the same transformation law should be valid for interacting fields as well. As far as I know, there is no justification for this requirement. Moreover, Haag's theorem (in the formulation given by Greenberg) says that if interacting fields transform like that (plus some other conditions, which I find reasonable and therefore omit) then the theory must be equivalent to the noninteracting one. I have a strong feeling that if one succeeds in constructing the interacting field operators in QED (which is an absurdly accurate theory, as you say) one would find that the "Lorentz" transformation law does not apply to them. Unfortunately, as far as I know, nobody was able to construct interacting fields in QED in any reasonable approximation and study their inertial transformations. However, this kind of study has been performed in a simple model example. H. Kita, "A nontrivial example of a relativistic quantum theory of particles without divergence difficulties", Progr. Theor. Phys., 35 (1966), 934. Eugene. 


#241
Apr710, 03:44 AM

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1. All pure scalar theories in 2D 2. All pure scalar theories in 3D 3. All Yukawa theories in 2D 4. All Yukawa theories in 3D 5. YangMills in 2D 6. The Abelian HiggsModel in 2D and 3D 7. The GrossNeveu model in 2D and 3D 8. The Thirring model and finally 9. All scalar theories in 4D. The caveat on (9.) is that the only purely scalar theory which exists in 4D is probably the trivial one. However any field theory which exists has been proven to have this transformation property. This list is basically every single theory we have constructed and understood nonperturbatively. So for every theory we have nonperturbative knowledge of, the transformation law holds. The list of theories which exist nonperturbatively and don't obey the transformation law is an empty list. Hence I would say the assumption is justified, or at least far more justified than its negation. Haag's theorem says that if the theory lives in the same Hilbert space as the free theory and obeys relativistic transformations and is translationally invariant, then it is free. That is it says: (Same Hilbert space) + (Normal transformation law) + (Translationally invariance) => Noninteracting It does not say: (Normal transformation law) => Noninteracting. 


#242
Apr710, 03:49 AM

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Eugene. 


#243
Apr710, 07:27 PM

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(Or are they easy to derive but I'm still missing something?)  [Edit: I sense a note of frustration in your post #241, so I just like to say two things: a) THANK YOU for going to the effort in those earlier posts, and THANK YOU in advance for (hopefully) future episodes of the climbingtheladder saga. b) I do want to understand these things rigorously, including how one goes about proving convergence since (among other things) acquiring such functionalanalytic skill is clearly valuable in any other nonWightman approach that one might wish to investigate.  


#244
Apr810, 04:17 AM

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I should also say the theorem is much harder to prove in the Hamiltonian approach that I'm discussing. In the FunctionalIntegral (PathIntegral) approach it's just a matter of evaluating a single Feynman diagram. See Glimm and Jaffe's book Section 8.5, Proposition 8.5.1. [1] Jaffe, A. : Wick polynomials at a fixed time. J. Math. Phys. 7, 1250 — 1255 [2] Segal, I. "Notes toward the construction of nonlinear relativistic quantum fields, I. The Hamiltonian in two spacetime dimensions as the generator of a C*automorphism group." Proc. Natl. Acad. Sci. U. S. 57, p.1178—1183 


#245
Apr810, 05:35 AM

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P: 303

Let's say the representation of the algebra which gives you the finite volume theory is [tex]\rho_{\Lambda}[/tex]. All [tex]\rho_{\Lambda}[/tex] are unitarily equivalent, only the infinite volume theory [tex]\rho_{\infty}[/tex] is unitarily inequivalent. Also, something which allows making estimates and bound easier, the [tex]\rho_{\Lambda}[/tex] are all unitarily equivalent to the Fock/Free rep [tex]\pi[/tex]. So the entire construction of the theory is "merely" a matter of passing from one rep to another. In higher dimensions though things are not so easy. As I will explain in detail, in three dimensions due to ultraviolet divergences one must renormalize. In the algebraic approach this shows up in the fact that the ultraviolet cutoff theory and the theory with no UV cutoff have the same C*algebra, but different reps. Put another way, even though the algebra is again unchanged, the finite volume reps [tex]\rho_{\Lambda}[/tex] are not unitarily equivalent to the Fock/Free rep [tex]\pi[/tex] Also unlike the 2D case [tex]\rho_{\Lambda}[/tex] and [tex]\rho_{\Lambda'}[/tex] for [tex]\Lambda \neq \Lambda'[/tex] are unitarily inequivalent. Let me sum up. In the Algebraic approach, ultraviolet divergences associated with mass and vacuum renormalization show up as changes in representations as you take some limit. In the 2D case there is only ever one change in rep. If you take the UV limit, the rep stays the same. When you then take the infinite volume limit the rep change only shows up in the limit. In the 3D case there is a change of rep in the UV limit. Then there is a change of rep for every single value of [tex]\Lambda[/tex] in the ultraviolet limit. In the 4D case things become incredibly difficult, unlike all previous cases the algebra itself changes as you take the UV limit. It's not just a rep change. It's difficult enough to control the reps, but controlling the algebra is something truely difficult. The change in the algebra itself is associated with coupling constant renormalization. (If anybody is curious, Field Strength renormalization is associated with something you can't really see in the Algebraic approach. I'll explain it when I do my post on the 4D field.) 


#246
Apr1010, 09:07 AM

P: 399

from the book of Zeidler http://www.flipkart.com/book/quantum...dge/3540853766 i heard that all the 'divergent' quantities were encoded in the linear combination of dirac delta funciton
[tex] \sum_{n\ge 0}c_{n}\delta ^{n} (x) [/tex] so when taken x=0 the expression was divergent. As far as i know EpsteinGlasser method allowed you to recover the Scattering Smatrix perturbatively plus a distributional contribution involivng dirac derivatives, also the fact that '2 distributions can not be multiplied' avoided us from getting finite result could anyone give a lazyman intro to EpsteinGlasser theory ?? 


#247
Apr2310, 09:51 AM

P: 90

DarMM, or anyone else for that matter, I'm trying to figure out the rigorous construction of the [tex]\varphi_2^4[/tex] and I'm reading Glimm and Jaffe, "Quantum field theory and Statistical mechanics  Expositions". Problem I find it a rather hard nut to crack. Tons of technicalities and I'm also failing to grasp the big picture, i.e. how are all the technicalities supposed to fit together. So question is, do yo know of any "pedagogical" account on the rigorous construction of [tex]\varphi_2^4[/tex] in a Minkowski setting? That is, no HaagKastler nor OsterwalderSchrader. Would really appreciate any refrences. DarMM, you mentioned you found your notes... I'm guessing they're not in electronic format, are they?
zetafunction > I did not use the EpsteinGlaser approach, so this is just the idea of how it works. Essentially, if you know the time ordered product of one Wick monomial, then by causality you know the TOP of 2 Wick monomials. And if you know the TOP of 2 WM, then you know the TOP of 3 WM. And so on. Here, when I say you know I mean "you can construct". For instance, ion the case of the usual [tex]\varphi^4[/tex] theory, causality will allow you to construct the following chain of TOP [tex]T[:\varphi^4:] \longrightarrow T[:\varphi^4::\varphi^4:] \longrightarrow T[:\varphi^4::\varphi^4::\varphi^4:] \longrightarrow \dots [/tex]. Double dots denote normal ordering and the fields are free fields. Now, the problem with the above chain is that you have products of distributions which are generally ill defined for coinciding points. The extension of the TOP of 2 or more WM to the diagonal, i.e. to coinciding points, then amounts to renormalization. And the extension is also not unique, which corresponds to the usual renormalization ambiguities. Note that there are no divergencies here, everything's finite. 


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