Some Early Criticism of String Theory

From the “First Superstring Revolution” on, there have always been skeptics, even though they often were not very vocal. Perhaps the most well-known piece of such criticism was Paul Ginsparg and Sheldon Glashow’s Desperately Seeking Superstrings, which appeared in the May 1986 issue of Physics Today. I recently became aware of some other similarly critical articles by Noboru Nakanishi, and copies of them have been made available to me. They are:

Comments on the Superstring Syndrome (also from May 1986)

“Superstring Theory” Syndrome (published in the popular magazine “Parity”, September 1986)

Can the superstring theory become physics? (January 1993)

This last paper claims that “the bubble of superstring theory has … bursted”, which, in 1993, was rather premature.

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77 Responses to Some Early Criticism of String Theory

  1. Bert Schroer says:

    What one is missing has more to do with how to get out of the present crisis which already lasts for more than 3 decades. There are indeed suggestions of how to get some conceptual progress on gauge theory in general and the Higgs issue in particular. Some ideas you find in the last section of my Samizdat essay. But the only presently already worked out innovations are in the area of QFT in CST and black hole issues in particular the entropy of the quantum state which causes the entropy associated with the Hawking radiation (also explained there).
    But from the way you phrase your question it seems that you think of particle physics in terms of a sport match with winners and losers (this seems to be your attitude with respect to AQFT) or even worse in terms of a eliminationist anti-conceptual ideology. Perhaps if there would be more than a handful of researchers in algebraic QFT you would not still find yourself in that already 30 years discovered best quasiclassical straitjacket for the physical reality behind the SM (from which all the globalized communities of enormous size have not been able to get away).
    It has been my experience in more than 40 years of professional activity that particle physics needs many different talents to make progress.

  2. Tony Smith says:

    Bert Schroer compared Superstring Theory with Algebraic Quantum Field Theory, saying:
    “… in string theory … people use category theory, gerbs, algebraic geometry, noncommutative geometry…. I think the basic difference [ from AQFT ] is that in ST you do not have to know their deeper mathematical content; the metaphoric physical nature of ST allows you to operate with a basically verbal knowledge and the rest can be done by improvisation and massage …
    AQFT continued von Neumann’s tradition on the mathematical side … and tries to implement the Jordan plea of disconnecting a more fundamental theory (QFT) from its classical bonds … It is the only setting of QFT in which there has been an independent development on par with mathematics on a very deep level of conceptual coalescence … “.

    With respect to AQFT and Standard Model model-builders, Bert Schroer went on to say that he ” … felt … a division of labor.
    There are those who go out and find our truffles … gauge theories ….
    there are others who like to clarify and secure these findings in order to build a reasonable trustworthy basis for starting new discoveries …
    The distance which people maintain from AQFT has little to do with its high demands on mathematics; it is more related to sociology and history …”.

    It seems to me that it would be a good thing to build bridges across the “division of labor” between AQFT approach and the “truffle hunter” model builder approach in which models are often built from gauge theory Lagrangians using path integral quantization.

    In a comment to another thread on this blog, Bert Schroer said “… hyperfinite Type III_1 von
    Neumann algebras … are the heart of local quantum physics …”,
    I don’t see a road-map for exactly how the various parts of the von Neumann algebra correspond to the various parts of a gauge Lagrangian / path integral model.

    Maybe a reason that ST people seem to have more political influence than AQFT people might be that the ST people (even if by “improvisation and massage”) take pains to describe purported connections between their ST models and gauge Lagrangian / path integral models.

    Without a road map to connect gauge Lagrangian / path integral models with the relevant von Neumann algebras, it is understandable why the model builders would feel distant from the AQFT people,
    if the main voice that the model builders hear from AQFT people is harsh criticism of basic model builder tools such as path integrals, as opposed to discussion of what kind of limiting processes etc might connect the two approaches.

    Tony Smith

  3. woit says:

    I kind of agree with Garbage that I wish people would post more substantive comments, especially ones relevant to the more technical postings. I recognize that this may be too much to expect; the number of people out there really knowledgeable about some of these subjects is vanishingly small and undoubtedly they have better things to do than contribute to this blog.

    But, please avoid posting repetitive comments that just add to the noise level. It’s true that many of the postings here deal with the same issue, the current status and problems of string theory. This is largely because this is an active story and new material keeps coming in, but the same general comments about what’s wrong with string theory are not enlightening. I’m already deleting quite a few of these, and will delete more in the future.

  4. Ari Heikkinen says:


    It’s interesting that you encourage people to post more technical comments instead of purely philosophical, yet the very first of the links to the papers you posted (Glashow et al.) about early criticism of string theory is purely philosophical without a single equation or proof. Yet they manage, in addition to the criticism of string theory in that very paper, call Einstein, probably the greatest thinker since Newton, incompetent (Quote: “He had to fail, simply because he didn’t know enough physics.”).

    I’m sorry to say, but it seems more and more to me as if different standards are applied when evaluating the work of string theorists compared to evaluating work of non-string theorists.

  5. Peter Woit says:


    My comment had absolutely nothing to do with string theory vs. non-string theory, other than noting that I’m already deleting a lot of repetitive comments people write in here criticizing string theory, and encouraging people to stop doing it. As far as comments here go, yes I do have a double standard: I virtually never delete anything people write in that is positive about string theory, and/or critical of my point of view on it, but often delete things that are negative about string theory or just endorsing my criticisms.

    Also, when I link to papers, it is definitely not because I agree with everything in them. The ones mentioned here were mentioned for historical interest, just because I only very recently became aware of the Nakanishi ones.

  6. Bert Schroer says:

    Particle Physicist
    I think that there has been a misunderstanding, my statement should have been:
    …..who consider it as a superfluous distraction from their grand new designs”….”The word new was missing. If there was some hubris to their achievements to discover the SM this is human, understandable and did not cause any harm. I meant the hubris that one could get much more without any additional conceptual physical investment. I think it is clear by now that those ideas which were involved in the discovery of the SM and which people have used for 30 years to get beyond are sapped.
    Tony Smith
    All these things you mention have been discussed in previous blogs of mine. They have appeared in a specific context and taking them out of this context does not move me to explain things for the nth time again just because weblogs do not seem to have a memory for more than one week. I have got to know your point of view (basically use the old ideas perhaps with slight updating) and I am not trying to proselyte. There is a high entrance fee for AQFT and I am not offering a cheaper rate to anybody (and answering all your questions and statements for the nth time does not help anybody).
    It is strange that you remind me of the only thing which I have in common with ST, namely the conviction that more of the QFT as it is in the books is not taking us out of our present misery.

  7. TheGraduate says:


    It’s unfair of you to blame other people when you’re the one who hasn’t been clear in a prominent place about what you want on your blog.

    You said “But, please avoid posting repetitive comments that just add to the noise level.”

    What is ‘noise’ supposed to mean? If you have not defined it in an easy to find place on your blog, how are people supposed to know? And on the issue of repetitiveness … so agreeing with people is disallowed?

    “As far as comments here go, yes I do have a double standard: I virtually never delete anything people write in that is positive about string theory, and/or critical of my point of view on it, but often delete things that are negative about string theory or just endorsing my criticisms.”

    This approach is very, very curious indeed. I am not how sensible it is to make insulting you the easiest way of making sure a comment is not deleted but to each his own.

  8. Peter Woit says:


    I probably should post somewhere some kind of “policy”, but the problem is that it isn’t so easy to precisely formulate such a thing. In practice the main problems have always been people who don’t know what they’re talking about, people who want to change the topic of discussion to their pet interests, and people who think it is a good idea to post comments that repeat others or their own, adding nothing new. The best way I know of characterizing what I’m trying to achieve is keeping the signal/noise ratio as high as possible. Sure, what I consider “noise” someone else may consider signal and vice-versa. But if you’re writing a comment that is adding nothing new to the discussion that people are likely to find interesting, you shouldn’t do it, it’s that simple.

  9. TheGraduate says:


    This may not seem plausible to you but asking others to live up to a standard which you refuse to articulate truly is the definition of ‘unreasonable’.

  10. Arun says:

    Ari writes:

    “Yet they manage, in addition to the criticism of string theory in that very paper, call Einstein, probably the greatest thinker since Newton, incompetent (Quote: “He had to fail, simply because he didn’t know enough physics.”).”

    Ignorance is not the same as incompetence. Unless you mean to say that the discoveries since 1955 are irrelevant to producing a theory of everything (e.g, anomaly cancellation), Einstein certainly did not know enough physics. And it is very likely we still do not know enough physics to produce a theory of everything, despite all the ST claims to the otherwise.

    Think about it – how would you express the fact that Einstein did not know about quarks and QCD, among a great many things? “He did not know enough physics” is certainly a legitimate expression of Einstein’s ignorance.

  11. Peter Woit says:


    I just articulated it as best I can: don’t post comments unless they’re adding something new to the discussion that will be interesting to other readers (who you should assume are smart and well-informed…).

    And the only reason I’m not deleting your last comment arguing with me about this is that you’re insulting me by describing me as unreasonable…

  12. TheGraduate says:


    Well I certainly did not get into posting on your blog to insult you or any one else. I only aim to accurately describe. I apologize. (You may delete this post after you have read it, as it will have served its purpose.) I will try my best in the future not to add to the noise. You may also delete the ‘unreasonble’ comment if you wish. Although, it may not seem like it, most of my comments are aimed at promoting better conversation.


  13. Tony Smith says:

    In my comment 52, I asked Bert Schroer “… exactly how the various parts of the von Neumann algebra correspond to the various parts of a gauge Lagrangian / path integral model. …”.

    Bert Schroer, in reply said “… All these things you mention have been discussed in previous blogs of mine. … I have got to know your point of view (basically use the old ideas perhaps with slight updating) …”.

    I agree that my point of view is to stay close to “the old ideas” of a gauge Lagrangian / path integral model,
    I do not see where Bert Schroer has in “previous blogs” explained “exactly how the various parts of the von Neumann algebra correspond to the various parts of a gauge Lagrangian / path integral model.”.

    I HAVE seen a dialogue in the thread in Peter’s blog “Schoer’s Samizdat” in which:

    Arun in comment 106 said something substantially equivalent to my present question:
    “… It seems to me [ Arun ] that AQFT practitioners find the Lagrangian path integral method of QFT to be unreasonably successful.
    Well, if AQFT is indeed the way to do QFT,
    then the success (and limitations) of the Lagrangian path integral should be explainable within AQFT.
    Presumably the successful Lagrangian formulations are taking care automatically somehow the setting up of the unique hyperfinite type III_1 factor algebra, the modular positioning of a finite number of abstract monades, etc., etc. …”


    Bert Schoer replied in comment 108:
    “… “… arun,
    this is all deja-vue, we (maybe without you) discussed this in this blog in April/May. … I [ Bert Schroer ‘ will ignore this and answer it to the best of my [ Bert Schroer’s ] knowledge
    1) Funct5ional integrals are extremely limited …
    2) you cannot solve any 2-dim. massive system, even in case it admits a Lagrangian presentation …
    3) Most free fields cannot be characterized as being of Euler-Lagrange type (above spin 2 I [ Bert Schroer ] do not know any) but their use is perfectly legitimate in causal perturbation theory (where you only use interaction polynomials in free fields) although the most important ones can

    4) The Lagrangian approach where it works in your sense is an artistic device:
    you write something on paper in good faith, develop it in perturbation theory,
    see that it does not make sense (the integrals diverge) and
    use your hindsight and good physical sense to repair it (renormalization) and
    at the end you realize that you have a perfectly reasonable result which however (if you try to check whether it its into your original functional representation) fails to satisfies your original functional representation. …
    The artistic device of functional integrals (in QM it is mathematical) works because the whole idea of locality is so strongly incorporated that you cannot fail to extract the right result (if you have a good covariant formalism which was not available before 1949). …”.

    In my view, Bert Schroer’s reply is only a negative attack on “the old ideas”, and is NOT responsive to the issue raised then by Arun and now by me:

    If the von Neumann algebras of AQFT describe physics as well as “the old ideas” of the Lagrangian / path integral Standard Model,
    what are the exact correspondences between von Neumann algebra and the structures of “the old ideas” ?

    does AQFT in its present state suffer from an ailment similar to that of the numerous vacua of superstring theory, that is:
    are there a lot of von Neumann algebras and nobody in AQFT can identify a single one that corresponds to the Standard Model ?

    From reading Bert Schroer’s hep-th/0610225, it seems to me that the latter is the case, as he says there:
    “… the physical reality of relativistic local quantum matter in Minkowski spacetime originates from the (modular) positioning of a finite number of copies of one abstract monade …
    The rich content of a quantum field theoretical model including its physical interpretation (particles, spacetime and inner symmetries, scattering theory….) is solely encoded in the relative positions of a finite number of its monades …
    The requirement is that the positioning should be natural within the logic of modular operator theory (using the concept of modular inclusions and modular intersections). For higher than 3 spacetime dimensions the presently known descriptions still look somewhat
    concocted …
    there are some still rather vague ideas of how the fact that the S-matrix has the interpretation
    of a relative modular invariant … may be used in model constructions.
    They are based on … the hope that its use (i.g. in a perturbative bootstrap-formfactor program) could have a chance to improve this situation. I [ Bert Schroer] expect
    that by combining some ideas of the old S-matrix approach with this recent
    framework of modular wedge localization one will obtain new insights into the
    construction of QFT.
    It is my [ Bert Schroer’s ] intense impression that … it is not possible to make progress in
    particle physics without a new problematization of the bootstrap-formfactor idea. …”.

    Bert Schoer’s appeal to bootstrap ideas to try to construct models in more than 3 spacetime dimensions indicates to me that AQFT suffers from a problem similar to that of the many solutions of Superstring Theory:
    a large and poorly described Landscape / Swamp of solutions,
    with no known correspondence of any of them with the Standard Model.

    In the absence of a concrete answer to the question raised by Arun in comment 106 of the “Schroer’s Samizdat” thread and my question in comment 52 of this thread,
    I will plod along using “the old ideas perhaps with slight updating”.

    Tony Smith

  14. A.J. says:


    I think what Tony was suggesting is that it’s time for someone to write up an “Introduction to Algebraic QFT for Particle Theorists”, showing how the essential ideas of effective field theory (regularization, renormalization, anomalies, gauge symmetries, and perturbative expansions) are interpreted in the “local quantum physics” framework, in a language suitable for particle physicists. I don’t mean that one shouldn’t use Tomita-Takesaki theory, of course, but that one should treat it as something within the grasp of most particle physicists, rather than claiming that “there is a high entrance fee” to AQFT. The essentials of factors can be explained to anyone with a basic grasp of Hilbert spaces.

  15. steve says:

    Chris W: I like your formulation of the crackpot-policing issue in terms of time allocation. From an econ background, it makes a lot of sense.

    If passive ignoring were the only behavior directed at well-credentialed heretics, though, it wouldn’t be such a striking phenomenon. The aggression or venom involved in ridiculing, cutting off the funding of, and otherwise ostracizing the people who keep picking at anomalies almost seems to take as much time as would be required to address their concerns. I guess the number of active persecutors relative to passive ignorers is pretty low, though, so your theory still has a lot of power.

  16. Chris W. says:

    I, as many physicists do, expect the LHC to clear the way for science soon. (from Garbage)

    To the extent that one wants the in question science to be about quantum gravity and unification, I wonder how much it will help. If the current theoretical situation allows for millions of potentially applicable models, and fundamental assumptions remain up in the air as one or the other of the models is refuted by the data, then we’ll still be groping for a way to narrow the field on other than purely empirical grounds. (I’m assuming here that the individual models will be of limited interest by themselves.)

    On the other hand, if one is mainly interested in taking the next small steps beyond Standard Model, with some interesting anomalies to chew on, then a return to doing science seems more likely…although that depends on what the LHC turns up.

  17. Bert Schroer says:

    Dear contributers and readers
    When I actively participated in the weblog for the first time in April this year, the problem of alternatives to ST was an issue of considerable common interest. It was in that context that I started to mention the algebraic approach to QFT which most participants have only met (if at all) in the context of chiral theories or factorizing models (since in both cases there is no Lagrangian Lagrangian construction of such models) and whose contributions to 4-dim QFT are meanwhile very impressive and everytime mor constructive and concrete.
    I tried to convey a bit about its content but as time went on it got a bit out of hand and I now realize that a weblog is not the right place. What caused me to write these lengthy contributions was the lack of a good reference; the book of Haag which was written in the late 80s as a nice motivating text (and it still continues to be) which however does not intend to connect readers to the technical level which would be necessary to reach a working knowledge (not to mention the impressive progress during the last 2 decades).
    I think the lack of a textbook which includes in addition to a presentation of the concepts also sufficient technical-mathematical details makes any discussion on a weblog very difficult. Even though there are only very few people working with these methods, one would hope that this will change in the not so distant future. Since weblogs are not the right place I will stop right here admitting: mission not accomplished.

  18. Sailor Moon says:

    Ha! It looks like Ginsparg will let you make trackbacks to ~his~ papers!

  19. Arun says:

    While waiting for a up-to-date AQFT textbook, I’d just point out that the nature of successful scientific theories is that they explain the apparent success and limitations of previous scientific theories. Thus,in this sense, Special Relativity contains Newtonian mechanics, General Relativity contains Newtonian gravitation, and so on.

    Anything that claims to supplant the Lagrangian/ path-integral/ perturbation-theory Standard Model of electro-weak & strong interactions, which is our most successful theory so far, needs to show how, in the appropriate limit, the Standard Model emerges.

  20. Tony Smith says:

    Arun said that if AQFT (or anything else) “… claims to supplant the Lagrangian/ path-integral/ perturbation-theory Standard Model of electro-weak & strong interactions, which is our most successful theory so far,
    [ then it ] needs to show how, in the appropriate limit, the Standard Model emerges. …”.

    Alain Connes is certainly very knowledgeable about von Neumann algebras etc
    he does take pains to connect his Non-Commutative Geometry (NCG) models to the Standard Model (although I don’t think that he seems to be opposed to using Lagrangians etc in his model-building).

    In his paper with Chameseddine and Maroclli at hep-th/0610241, Connes says
    “… The input of the model is extremely simple. It consists of the choice of a finite dimensional algebra … a direct sum C + H + H + M3(C)
    … geometric considerations on the form of a Dirac operator … lead to the identification of a subalgebra … of the form C + H + M3(C) …”
    (where C = complex numbers, H = quaternions, and M3(C) = 3×3 matrix algebra).

    Is there a way to show how that particular choice of algebra, in the context of that NCG model,
    would correspond to
    a particular “(modular) positioning of a finite number of copies of one abstract monade” from which, according to Bert Schroer’s hep-th/0610225, “the physical reality of relativistic local quantum matter in Minkowski spacetime originates”

    If there is, then it would help me to understand AQFT.

    If not, then it might still be helpful to know why not,
    and what are the difficulties in making such a correspondence.

    If Bert can shed any light on that, I would appreciate him making at least one more appearance on this blog thread to do so.
    In any event, I thank Bert for a lot of interesting comments, which I have enjoyed reading, although I regret that my understanding (although expanded from what it was before I read) is limited.

    Tony Smith

  21. Marty Tysanner says:


    You commented:
    I think the lack of a textbook which includes in addition to a presentation of the concepts also sufficient technical-mathematical details makes any discussion on a weblog very difficult.

    Given your wealth of knowledge and experience, have you considered writing something like a review article of the “state of the art” of AQFT, complete with sufficient mathematical content to offer a path for those who might consider it a serious subject for study? The advantage of an in-depth summary of this kind is that you would have complete editorial control over the content and style. Publishing costs are free, and so is the cost to the student…

    This would be a significant undertaking, but it would be a potentially lasting contribution, and one which would have far greater potential to interest the wider community in the possibilities of AQFT than anything a weblog could offer. Plus, even if you didn’t have time to finish the article you could post whatever you have, and that would still offer something of substance to a potential student of AQFT. Anyway, it’s just a thought…

  22. Bert Schroer says:

    The situation in the absence of a good pedagogical text is not that desperate as it appears in my last blog. Different from ST talks and articles often start with simple historically well-known results and problems. I have never seen a paper or participated in a talk on AQFT which started with hyperfinite type III_1 factor algebras, but I already found myself lost in talks (when I missed an early chance to get away) which started with type II_1 superstrings (with most of the audience nodding their heads approvingly). So it is not surprising that there are some very good reviews of some central concepts in AQFT (modular localization, charge confinement/liberation, local covariance of QFT in CST,…), but reading them outside the reach of experts may be frustrating because there is nobody whom you can ask if you get stuck. Certainly a normal weblog like this is not the right place and it may be worthwhile to think about a better solution.
    Concerning the question of writing a book about algebraic methods, I once started to write some notes but then concluded that a time of rapid changes is perhaps not the right one. I do intend to return to this project next year (after having cleared up some pending problems as well as my desk).
    What has been a bit frustrating to me is that that people always try to construct an antagonism between the functional integral approach and the perturbation theory resulting from AQFT ideas. In my earlier contributions (April-June) I took great pains to clarify this issue of AQFT being inclusive; renormalized physical correlation functions just don’t care about the (even mathematically illegitimate) way you obtain them because you can check their correctness independent of the sins you committed on the way to obtain them. The number of renormalizable Lagrangian field theories is quite limited, there is a finite number of abstract actions and the rest is obtained in terms of families by playing around with varying numbers of field components (the last classified family was the gauge theory family which was the only one which possesses the asymptotic freedom property). But the number of non-Lagrangian families is infinitely larger as the studied 2-dim. factorizing setting shows. In that case you classify your theories by its simplest algebraic structure (as in case of chiral theories). I don’t see any sense in craving for Lagrangian in 4-dim. while living quite comfortably with algebraic classifications and constructions in lower dimensions.
    Concerning contributions to particle physics of people with great mathematical rigor, insight and talents like Connes and Witten I prefer to be silent because anything critical I say, as objective as it may be, will be misconstrued as regicide (whereas in reality I have an immense admiration for their mathematical knowledge and skills). The furthest I went was in a weblog discussion in (I think) May at this weblog with an anonymous participant named “the great inquisitor”. He understood precisely what I wanted to convey and did not misuse it, neither did anybody else (perhaps I have been just lucky).

  23. N. Nakanishi says:

    Looking at the discussions between Bert Schroer and the outsiders of Algebraic QFT, I would like to comment that there is a standpoint in between theirs. I emphasize that any of non-superstring, non-path integral formalism, and non-perturbative approach does not necessarily implies Algebraic QFT, that is, there is the most traditioinal approach, canonical operator formalism of Lagrangian QFT. Recently, Abe and myself have developed a method for finding the solution to the canonical operator formalism of QFT in the Heisenberg picture. I believe that this method is quite natural, and several models have been explicitly solved by this new method. For a review, see N. N., Prog. Theor. Phys. 111 (2004), 301-337.

  24. Chris Oakley says:


    I am not aware of your work, but it sounds interesting and will look it up when I am in the library next.
    I agree with the approach you describe above, although I would like to see the Lagrangian formalism replaced by something that emanates entirely from quantum considerations. I am (sort of) working on it…

  25. Bert Schroer says:

    AQFT is a different method of implementing QFT it is not a different theory which modifies any of the underlying principles of QFT (as ST or LQP). There is no antagonism and I have tried (without much success) to get this message across. Fortunately QFT is a totally intrinsic theory (it even contains the concepts of its own interpretation) and the correctness of a result can be checked without using the same methods which were used to derive it, it is definitely not just a bunch of recipes as ST.
    There are however problems for which the standard canonical quantization methods fail. Take e.g. chiral theories, factorizing models, QFT in CST or the question of computation of localization entropy (which I addressed in two of my publications). If you know how to deal with those problems in a canonical setting go ahead and do it; people on this weblog who are reluctant to learn trans-canonical methods will be thankful.
    I think we both are conservative. You are conservative about the formalism whereas I am extremely conservative about the principles but care less about what formalism you use to implement them. I have no problems to agree with your critique of ST , supersymmetry and the so-called noncommutative theory.
    all the best

  26. N. Nakanishi says:


    I do not understand the contents of what you mean by “problems for which the standard canonical quantization method fails”. In the method of Abe and myself, the representation of the operator algebra is given by constructing the set of all Wightman functions for the fundamental fields. If your problems can be discussd in the Wightman framework, I guess that they can also be dealt with my method.
    The important characteristic common to both canonical operator formalism and Algebraic QFT is the existence of the operator level analysis. Izumi Ojima, my colleague, who previously established canonical operator formalism of the Yang-Mills field and is presently working in Algebraic QFT, has recently emphasized the fundamental importance of avoiding the theories in which the representation level is directly considered (such as path-integral formalism). I expect that he makes some comments in this debate.

    (By the way, my first name is Noboru.)

  27. Bert Schroer says:

    For the three problems I mentioned previously I do not know how to compute without using the setting of algebraic QFT.
    The Wightman framework may be a nice way to present the structure of QFT, but I am not aware of any model computation which has been carried out in terms of them. The nonlinear positivity requirements are just too hard for any explicit construction and all successful calculations I know start with a Hilbert space and operators therein.

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