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I have added to perturbation theory and to AQFT a list of literature on perturbative constructions of local nets of observables.
This is in reply to a question Todd was asking: while the rigorous construction of non-perturbative interacting QFTs in dimension $\gt 2$ is still open, there has at least been considerable progress in grasping the perturbation theory and renormalization theory known from standard QFT textbooks in the precise context of AQFT.
This is a noteworthy step: for decades AQFT had been suffering from the lack of examples and lack of connection to the standard (albeit non-rigorous) literature.
This progress is at the level of deformation quantization and combinatorics of formal series, and no essential new convergence result at all in last several decades. So one may have formal expansions, but not things working at the level of operators.
This progress is at the level of deformation quantization and combinatorics of formal series, and no essential new convergence result at all in last several decades.
I think that’s precisely what I said: perturbation theory, but not non-perturbative theory.
the best place to start reading about perturbation theory in the AQFT framework is the following, which I just added to the relevant entries:
The observation that in perturbation theory the Stückelberg-Bogoliubov-Epstein-Glaser local S-matrices yield a local net of observables was first made in
- V. Il’in, D. Slavnov, Observable algebras in the S-matrix approach Theor. Math. Phys. 36 , 32 (1978)
which was however mostly ignored and forgotten. It is taken up again in
- Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds Commun.Math.Phys.208:623-661 (2000) (arXiv)
(a quick survey is in section 8, details are in section 2).
I think that’s precisely what I said: perturbation theory, but not non-perturbative theory.
We disagree in definitions. Non-perturbative is usually used for the methods which do not involve perturbation expansions. Perturbative means using perturbative expansions. Still for perturbative expansions there is a question of convergence of perturbative expansion which is BY DEFINITION a question on perturbative theory.
Most perturbative expansions in past where in the sense of asymptotic series, with Borel summation arguments. The newest phase is dominated by much more formal and in that analytic sense less achieving formal expansions of deformation quantization, what is from the analytic point of view, the most trivial case of perturbation theory.
We disagree in definitions.
Do we? What is it actually that you are disagreeing with, I don’t understand. Something on the $n$Lab? Something I said here?
To me it seems that what I said here is that standard perturbation theory in QFT fits into the AQFT axioms: concretely: the interacting fields form a local net in perturbation theory. This is just an observation on the standard lore.
From there, you can then investigate convergence and/or resummation if desired.
Your statement explicitly in 3 is that the convergence of perturbative series is a question of non-perturbative theory (??). I disagreed with that in 5.
Before that I remarked that there has not been much recent progress in true perturbative theory of asymptotic series (which is truly relevant to AQFT) but only of a formal series (which is not a valid input for AQFT as it does not define a true operator but a series in formal variable which does not belong to Hilbert space/operator algebra formalism over complex numbers; AQFT is not over the field $\mathbf{C}[[h]]$, though the latter is relevant for some QFT questions in mirror symmetry), what you subsumed into non-perturbative distinction. Read above. Alain Connes has argued a couple of years ago in a paper very strongly about the distinction between deformation quantization and true quantization.
only of a formal series (which is not a valid input for AQFT as it does not define a true operator but a series in formal variable which does not belong to Hilbert space/operator algebra formalism over complex numbers;
In AQFT no Hilbert space appears explicitly in the formulation of the net. It is a net of algebras. Here, yes, of algebras of formal power series. But still of algebras.
Did you have a look at the articles I am talking about?
Here, yes, of algebras of formal power series. But still of algebras.
These are not even remotely $C^\ast$-algebras, Urs. Formal power series has a completely different ultrafine topology.
Hilbert space or operator algebra, it is not a big difference. On the Hilbert space side one can push this into the change of the ground field (possibly non-archimedan). In any case it does NOT fit into AQFT.
Did you have a look at the articles I am talking about?
Not much, is it relevant for our discussion ? The first article is at the physics level of rigor and the second is working with difficult analytic conditions id est true topologies and not related to the advance in deformation quantization (the setup which does not ensure the input to those analytic conditions).
You see for decades there are lots of papers working hard with analyticity and not getting to any major advance. For example, even there is no proof that the usual QED as defined perturbatively gives meaningfuol results from the point of view of AQFT. ON the other hand, more recently there has been huge progress in deformation quantization, including formal perturbation theory. That line on the other hand does not connect to the analytical conditions needed to to have the meaning for AQFT. This is all I am saying.
Section 1.4
Deformation quantization [16] is another matter. Unlike quantization, deformation quantization is a systematic procedure.
…
However, deformation quantization is not quantization. Generically, it leads to a deformation over a ring of formal power series (in the formal variable $h$), not a deformation with a complex parameter. It does not lead to a natural Hilbert space $H$ on which the deformed algebra acts.
Not much, is it relevant for our discussion ?
I would think that in a discussion on “latest changes” of an $n$Lab entry the content of that entry would be relevant.
This is probably why I don’t understand you: I say: “hey, I have added this to the nLab”. In reply you say something, as if disagreeing. I don’t understand what you are disagreeing with.
It seems to me much rather you feel like adding some information to the topic, and instead of trying to make it look as if in disagreement with me, you might just add your knowledge to the Lab.
These are not even remotely $C *$-algebras,
No, and they need not. The point is that the interacting fields in perturbation theory do form a local net. This is – by the very point of deformation theory – an intermediate step in between the classical theory and its full quantization. To turn it into a net of $C^*$-algebras one will have to apply any of the usual steps that go from formal quantization to the full theory. If only these would work for interesting theories.
Have a look at the introduction of the Brunetti-Fredenhagen article that this thread here is about: here.
This is interesting remark, you consider the local nets not in full flegded AQFT but some sort of local nets over formal power series ring ? I have not seen such a definition of local net. This can be interesting as a toy of deformation theory, but as I say, the true quantization is not deformation theory as Witten in the above citation points out. The articles which you pointed out say that the TRUE perturbation theory with analytic conditions, not the one over a formal variable forms a local net. Now you say more, this may be interesting, but what are then the definitions and teh references for that ?
You said in 1:
there has at least been considerable progress in grasping the perturbation theory and renormalization theory known from standard QFT textbooks in the precise context of AQFT
I do not care that you said that in the Latest Change discussion, as long as it is interesting to discuss I discuss. It seems you say in 13 that I should care about the classification of the nForum entry thread in order to qualify for discussion (???).
I should also point out that the Fredenhagen articles are mainly about not so popular Epstein-Glaser renormalization procedure which is rather different and was always appealing to mathematical treatment. Vladimir Glaser is originally from Zagreb, the best “Croatian” mathematical physicist in history.
I would think that in a discussion on “latest changes” of an nLab entry the content of that entry would be relevant.
If somebody starts a thread with some claims those claims are also relevant. When reading nForum thread I do not necessarily read the entry, unless the thread has convinced me that I can help there or that there is something interesting to read there. Similarly, when I read the abstract of an arxiv paper I do not necessarily download the paper.
I have not seen such a definition of local net.
You could have a look at the references that I am trying to discuss here. The locality axioms on a net do not involve the $C^*$-condition.
This can be interesting as a toy of deformation theory,
That’s the point: the algebras of observables in ordinary perturbation theory do form a local net.
The articles which you pointed out say that the TRUE perturbation theory with analytic conditions, not the one over a formal variable forms a local net.
The algebra of interacting fields considered there is an algebra generated from certain elements in a formal power series algebra, subject to some relations. It’s all nicely described in Brunetti-Fredenhagen. Maybe later I find the time to summarize some of it on the Lab.
It seems you say in 13 that I should care about the classification of the nForum entry thread
No. I am thinking that it is weird that in a thread about perturbation theory in AQFT you feel it is irrelevant to take notice of the content of this topic and the references on perturbation theory in AQFT.
Fredenhagen’s paper 9903028 you quote has indeed formal $\lambda$ just at the beginning (say 3.1) when defining general expressions, like perturbed Lagrangean. All the essential estimates (say those in section 6) take $\lambda\in\mathbf{R}^+$ and do not make sense for formal $\lambda$. See for example section 5.1 on where it is stated explicitly (otherwise things like formulas (33),(34) do not make sense). He imposes special conditions of analytic nature; usually such are hard to prove for concrete theories. (It is a bit easier here as it is only a scalar theory as in this paper. there are results of similar nature in some other special cases by Glimm, Jaffe and others. But say, nobody can yet prove consistency/existence of perturbative QED.)
In 3.2. he just proves the properties of quantities in each order, not of the sum of the series.
take notice of the content of this topic and the references on perturbation theory in AQFT
Those as a rule for any substantial result use hard and very special estimates for asymptotic series, not formal series. Hence the big recent progress on formal deformation quantization does not really help, yet. Maybe it will be input into something tomorrow, but so far, I indeed do not find the arguments of full relevance for true Hilbert space/operator algebra level theory there.
Don’t misunderstand me. The papers you suggest are very nice (thanks for suggesting them), but they are still in line with and advance difficult and special results in true perturbation theory and do not prove that the huge combinatorial and cohomological results in formal deformation theory can be transferred for the use in AQFT. One can not get far there with formal deformation arguments which I have seen so far.
That’s the point: the algebras of observables in ordinary perturbation theory do form a local net.
Maybe it is not even that difficult to see statement like that, provided one can prove first (or make hard conditions ensuring it) that particular kind of perturbation theory for particular class of Lagrangeans is a consistent theory. Once one has an existence result, then it fits to many alternative axiomatics and formalisms.
Maybe it is not even that difficult to see statement like that,
It’s a direct consequence of a basic property of the S-matrix: one needs that for interactions $A$, $B$, $C$ with the support of $A$ and $C$ seperated by a Cauchy surface, one has
$S(A + B + C) = S(A + B) S(B)^{-1} S(B + C)$Thanks, so the only thing needed is that for a real $\lambda$ the total perturbation series for S-matrix converges.
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