Schroer Notes and Other Links

Bert Schroer has sent me some notes comparing the Lagrangian path integral and algebraic approaches to quantum field theory, which others may also find interesting. I have a very different perspective than he does, but have gone through the experience of at one time believing that basically all there is to QFT is to choose an action functional and then apply straightforward techniques to evaluate the path integrals you get. Non-perturbatively, this works beautifully for Yang-Mills theory, but runs into serious problems in many other cases, and it becomes clear that the path integral method, for all its virtues, does hide some very real problems.

Update: Some more notes from Schroer on AQFT.

Some other unrelated links:

There’s an extremely well-known story about the young Gauss, and it turns out that, as almost always with such stories, the truth of what actually happened is rather elusive. American Scientist has a wonderful article about this by Brian Hayes entitled Gauss’s Day of Reckoning.

The AMS has announced a Leonard Eisenbud Prize for Mathematics and Physics. It will be awarded every three years for a work published in the preceding six years that brings the two fields closer together. The first award will be made in January 2008. The prize was established by David Eisenbud (currently director of MSRI) and his wife in honor of Eisenbud’s father, who was a mathematical physicist.

While I was away the Museum of Natural History here in New York sponsored a debate on the Multiverse, entitled “Universe: One or Many?”. For some reports on the debate, see here, here and here. The last of these is from local blogger “mighty dasmoo”, who really, really, doesn’t like Michio Kaku.

The EPP 2010 Panel will release its final report to the public at a press conference in Washington on April 26.

Update: New Scientist also has some quotes from the “Universe: One or Many?” debate:

Kaku, of the City University of New York, spoke at one point of the possibility of tunnelling into other universes through space-time foam, harnessing the power of negative energy. “Genesis happens all the time,” he said. “Continuous genesis in an ocean of Nirvana, and the ocean is an 11-dimensional hyperspace.”

As Kaku spoke, Krauss, of Case Western Reserve University in Cleveland, Ohio, looked as if he was about to have an aneurysm. He turned to Kaku. “If there are an infinite number of universes,” he declared, “I can’t imagine one in which I agree with what you just said.”

Update: Another report on the debate is here.

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36 Responses to Schroer Notes and Other Links

  1. Thomas Love says:

    What, the Gauss story isn’t true? I suppose next you’ll tell me that George Washington didn’t cut down the cherry tree!

    The third blogger’s take on Kaku agrees with mine. I’ve read a few of his books and won’t read another.

  2. JC says:


    Interesting read.

    What other quantum systems differ greatly between operator and functional integral treatments, besides the spinning top?

  3. Lubos Motl says:

    I’ve read Prof. Schroer’s text against the path integrals and it makes no sense to me. It is probably not even wrong.

  4. MathPhys says:

    Kaku is phenomenal. I’m always surprised by how he gets invited to speak at events next to respectable people.

  5. Bert Schroer says:

    Answer to JC:
    The functional integral representation in its strict measure-theoretic setting is limited to standard quantum mechanical Hamiltonians (kinetic energy + V(q)).
    The formal Duistermaat-Heckmann aspect (which leads to the “Schulman paradox”) certainly shows up for any movement of a particle on a group manifold. The model must be integrable.
    By the way, I am not against the functional integral approach, I only indicate its limitations. In fact even in QFT where the approach becomes “artistic” there is nothing wrong with using it. My only plea is that one should not forget that it is artistic in order not to loose an enigmatic piece of future progress.

  6. From what I’ve read about Gauss, I think it’s entirely possible that he was given either a dozen or so numbers, or a sequence of that length, and he simply added them up in his head. No formulae required.

    The part about his summation “trick” may have simply been tacked on later, or rather, the story was tacked onto the trick to pique the interest of otherwise disinterested students!

    Certainly, there are people alive today who could mentally add up quite large lists or sequences of numbers, quickly and accurately, with no roughwork. It’s well within the bounds of possibility that Gauss was such a person.

    Tales often grow in the telling, but it’s important to stick to the facts as well.

  7. jb says:

    I don’t find the Gauss story dubious at all. I found the trick for calculating 1+…+10 when I was six or seven (of course, I failed to rediscover it when I was asked 1+…+100 several years later), and if some fellow mathematician told me he got 1+…+100 when he was eight, I wouldn’t be surprised. Even more so, I have no reason to doubt the Gauss story — he did many much more impressive things and was also a whiz with arithmetic. And of course it’s ridiculous to think that if it really happened, there would be primary documents proving it.

  8. Joe Zhou says:

    I’ve no doubt that Gauss just simply saw the summation trick when he looked at the problem. I consider myself rather dumb, always being one of the slowest in class to do an integral or to come up with proofs. But even I was visited by occasional flashes of insight in childhood. An uncle told me when I was 11 or so about how Zu Chongzhi from the 5th centry found the volume of the sphere by subtracting a cone from the cynlinder. That night, I applied Zu’s principle to found the volume for any figure of rotation, and used that generalization to find the center of mass for the half circle, without any inkling of calculus or trignometry. And I am not even smart, almost failing the entrance exams for middle school that same year. Such flashes must be fairly common to intelligent people working on challenging problems.

  9. nayagam says:

    Brian Hayes has written some wonderful articles in American Scientist .The article on Gauss is of course good, but I liked his article The Spectrum of Riemannium more.

  10. robert says:

    A feat of infant prodigeousness that rivals that of Gauss’ summation of a finite number of terms of an arithmetic series is Dyson’s summation of an infinite number of terms of a geometric series, which he did to alleviate the boredom when put down in is crib for a nap

    And we have this tale from an unimpeachable source.

  11. Chris Oakley says:

    A feat of infant prodigeousness that rivals that of Gauss’ summation of a finite number of terms of an arithmetic series is Dyson’s summation of an infinite number of terms of a geometric series, which he did to alleviate the boredom when put down in is crib for a nap

    Yes, but the Nobel prize was awarded for taking an infinite series, most of whose terms are infinite, and getting a finite result. Maybe Dyson is just as glad not to be one of them.

  12. Karthik says:

    I once listened to Michio Kaku’s interview on the BBC. It was like a big fantasy novel.

    Science is increasingly becoming like religion where the scientists (priests) interpret the Universe and God and the rest of the world just go by it.

  13. Chris Oakley says:

    Well, Karthik, for an antidote I suggest that you get a copy of Peter’s book. It’s not out in the U.S. until September, but even so thus far 24 out of 27 people approve of my positive review. Oh, and Peter, my Swiss bank account is #0103242345 (UBS Zurich).

  14. Michael says:

    Peter, could it be that it is you, rather than the path integral approach to QFT, that is hiding some very real problems? The problems people have with path integrals seems to be inversely proportional to their scientific accomplishments.

  15. Science is increasingly becoming like religion where the scientists (priests) interpret the Universe and God and the rest of the world just go by it.

    It might seem that way when one looks at theoretical physics, but it is not the case for the majority of science. Applied fields of science, physics, chemistry, biology, etc, etc have made huge leaps in the last decade.

    Look at how much the world has changed in even the last ten years. Mobile communications are commonplace, the internet is widely accessable, medicinal drugs continue to improve, we are gaining a better understanding of our histories through archeology and new research, we have mapped the globe down to the metre levels.

    Science is having an impact on people’s everyday lives, ironically at the same time that people feel science is becoming more and more distant from everyday life.

  16. JC says:


    Are there any easy non-Euler-Lagrange ways to treat theories with Wigner spins above 2, besides the free particle cases not coupled to anything (ie. Fierz-Pauli equations)?

  17. Chris Oakley says:


    As far as I remember, higher spin inconsistencies arise because physical massless higher spin fields derive from gauge potentials in much the same way that F_{\mu\nu} derives from A_\mu for spin 1. The free field actions are invariant under gauge transformations in these potentials, something which either must be preserved when one introduces interactions, or generalised (e.g. Yang-Mills). This turns out to be impossible for s > 2 because one cannot write down a interaction that is gauge invariant for all the fields in the coupling (one is presumably not allowed to use the gauge-invariant field strengths here). If one is not using the action principle one would still require gauge-invariant couplings, so I do not see that one would be better off.

  18. Feynman integral is just a dirac delta’ to guarantee the extremality (say, the minimum) of the Lagrangian, but h-regulated. The point that quantum mechanics is the consequence of forbidding the h->0 limit in differential operations was guessed first by Pauli in 1923 ( ) upon contemplation of landé j(j-1) formulae and this in turn inspires Matrix Mechanics.

  19. Tony Smith says:

    JC asked about high-spin particles, and Chris Oakley replied, mentioning that techniques like gauge invariance don’t work for spin greater than 2.

    Weinberg, in Volume I of his work The Quantum Theory of Fields (Cambridge 1995) at pages 243, 244, and 253 describes the high-spin situation in a bit more detail:
    “… It should be mentioned that from time to time various difficulties have been reported in the field theory of particles with spin greater than or equal to 3/2. Generally, these are encountered in the study of the propagation of a higher spin field in the presence of c-number external field. Depending on the details of the theory, the difficulties encountered include non-causality, inconsistency, unphysical mass states, and violations of unitarity. …
    … there are good reasons to believe that the problems with higher spin disappear if the interaction with external fields is sufficiently complicated. For one thing, there is no doubt
    about the existence of higher-spin particles, including various stable nuclei and hadronic resonances. If there is any problem with higher spin, it can only be for ‘point’ particles, that is, those whose interactions with external fields are particularly simple. …
    … both higher-dimensional ‘Kaluza-Klein” theories and string theories provide examples of a charged massive particles of spin two interacting with a electromagnetic background field. …
    … in order to construct a theory of massless particles of helicity +/- 2 … gravitons … that incorporates long-range interactions,it is necessary for it to have something like general covariance. As in the case of electromagnetic gauge invariance, this is achieved by coupling the field to a conserved ‘current’ theta ^ mu nu , now with two spacetime indices, satisfying d_mu theta ^ mu nu = 0. The only such conserved tensor is the energy-momentum tensor,
    aside from possible total derivative terms that do not affect the long-range behavior of the force produced.
    The fields of massless particles of spin j greater than or equal to 3 would have to couple to conserved tensors with three or more spacetime indices, but aside from total derivatives there are none, so high-spin massless particles cannot produce long-range forces. …”.

    Tony Smith

  20. knotted string says:

    I love the Lawrence Berkeley National Lab bubble photo of ‘universes sprouting off’ at

    My baby niece can blow lots of bubbles like that. So I wonder if Scientific American will do a feature about her model of the multiverse? 😉

  21. Bert Schroer says:

    Here is a summary answer to some questions concerning higer spin.
    The important aspect of higher spin particles is neither the Lagrange setting not the equation of motion (E-L or not), but rather the formula which expresses pointlike covariant fields in terms of Wigner creation/annihilation operators (in momentum space!) and u-v interwiners (interwining the canonical unique m,s Wigner representation with the infinitely many covariant spinorial representations). Looking at such a formula one knows exactly in what physical Fock space this fields lives. Any such field can be used to define a polynomial interaction within the setting of causal perturbation theory (and if you have done this with one spinorial choice you can transform that interaction into any other choice of linear (in Wigner operators) field “coordinatization”). Fields of this type which in addition arise from field equations (i.e. the generic solution of the field equation obeys that formula with the intertwiners ) are known for any spin e.g. the Bargmann-Wigner equations for (m,s) representations for m>0. The short distance behavior of the two-point function for such pointlike field becomes more singular with increasing spin. This can be overcome by not insisting in pointlike fields but allowing spacelike semiinfinite string-localization (not string theory!). In that case the intertwiner functions are more complicated and although only one physical Wigner spin enters, there are infinitely many spinorial representations (namely all which belong to the same physical spin) which enter (all this is contained in a paper which will appear soon in CMP, see also (math-ph/0511042). In terms of localized fields this amounts to a field which depends simultaneously on x and on a spacelike direction e (a point in de Sitter space of one dimension lower). In that case the field A(x,e) fulfills the causal localization of a string starting at x and going in e-direction to spacelike infinity. It fluctuates in both x and e and the e-fluctuation is the reason why the fluctuation in x can be milder than in the pointlike presentation of Wigner (m,s) particles. In fact the short distance behavior in x for this string-localized description does nit increase with spin! This makes it very interesting to look for a perturbation theory of string-like higher spin fields (work in progress). All these ideas are outside of Lagrangian quantization but inside the setting of modular localization.
    Even more fascinating is the modular localization approach for zero mass. I am very time pressed, but I will come back to this point (meanwhile you can have a look at the above reference where all these cases are treated in detail.)

  22. a philosophy student says:

    Science is increasingly becoming like religion where the scientists (priests) interpret the Universe and God and the rest of the world just go by it.

    Applied fields of science, physics, chemistry, biology, etc, etc have made huge leaps in the last decade.

    Science is having an impact on people’s everyday lives, ironically at the same time that people feel science is becoming more and more distant from everyday life.

    There’s a relative huge gap between science and applied science – technology. People see technology gradually evolving and by this are not amazed at new applications appearing, and excisting ones getting better, more sophisticated. Inherently technology is close to everyday life.

    ‘Real’ science, through time by its very nature probing deeper and deeper into the fabric of reality, has got involved with issues far away from everyday life. Nanotechnology, genetic manipulation an such still have an imaginable connection with future everyday life, but allready generate an awe feeling. Things as neuroscience and consciousness research, and multiverse theories even more. By claiming some deeper and fundamental knowledge, especially when bearing on excistence and human life, scientists, one can imagine, could evoke religious-like feelings, for this is what religion is all about.

  23. Bert Schroer says:

    In previous notes I explained the difference between (interpretive) autonomous and metaphoric arguments in an example taken from QFT. Perhaps another example of more immediate interest is the way how chiral QFT serves to solve some problems in higher dimensional massive QFT and how it enters as a basic structure in the construction recipes of strings which brings me back to the main theme of this Weblog.
    In 4-dimensional QFT it is the holographic projection which recycles the original ambient spacetime indexing of quantum matter into a new indexing which associated to the causal horizon of the latter (the algebraic substrat is preserved but its spacetime indexing is radically changed). In the latter the original bulk matter appears as a transversely extended chiral matter (and in general the inversion will not be unique without additional information). The ambient quantum matter fluctuates (vacuum polarization) at the causal horizon of the ambient localization (half the lightfront in case of a wedge, the lower mantle of a double cone in case of a double cone localization) wheras the holograpic projection fluctuates only in lightray direction at the boundary of the lightray extension. If one’s main interest is to study extensive quantities (energy, entropy) of the localized bulk (caused by vacuum polarizations at the boundary of the bulk) it is easier to calculate after the holographic projection instead of doing this directly with the bulk matter. What remains of the original bulk fluctuation is the vacuum polarization at the end of a lightlike interval. Some results can be found in hep-th/0511291. The role of chiral theories in the recipe for string construction is completely different. It is not appearing in form of a holographic projection but it is there from the beginning in form of a source space of a QFT whose “field-value space” (target space) is the arena of the quantum matter, except that it lacks the most important property of direct target quantum fluctuation. The only fluctation is that of chiral source theory (factorially) repeated 24 or 10 times (note that in the QFT holography the chiral theory is part of the spacetime-indexed original ambient matter). In this way deep and very interesting mathematics of modular forms is processed into metaphorical physics. It is more or less evident that a mathematical formalism whose physical interpretation is metaphorical instead of autonomous may activate a messianic expectation. It is therefore not surprising that in a recent article of a mathematician (hep-th/0601035) one finds the following remakable statement: “The rising number of string theorists is a good indicator of confidence that string theory (or, at least, the existence of supersymmetry) will be confirmed in accelerator experiments and in astronomical observations and that this can happen pretty soon”. It is also not surprising that the foot-soldiers of such a metaphoric science become fundamentalistic. The academic version of cutting off the head of an infidel seems to renounce the hospitality to anybody whose critical remarks put some of the metaphors into question
    ( This also shows plainly why John Horgan’s terminology of calling the post string era of metaphoric physics “ironic science” does not fit this new reality. The foot soldiers of this new particle physics fundamentalism cannot be accused of any ironic attitude whatsoever.

    Recently I sometimes asked myself whether Witten had any idea what strange monoculture his string theoretic creation would develop into. The only final logical step for a seemingly irrevocably metaphorical theory is that of Susskind and Polchinski and it would be interesting to know what holds Witten (and Gross) back from taking that ultimate anthropic step.

  24. JC says:


    In principle, can the algebraic qft (aqft) framework derive an exact non-perturbative S-matrix solution for theories represented by, say:

    – real scalar fields with a phi^4 quartic interaction in d=1+1 spacetime dimensions?
    – a Thirring model with a four-Fermi interaction in d=1+1 spacetime dimensions?

    (It would be impressive if this can be done easily in practice for other theories like phi^4 theory and Yang-Mills in d=3+1, or any number of spacetime dimensions).

    At this point I don’t know whether I should be impressed that aqft can derive exact S-matrix solutions for some special cases in d=1+1, corresponding to elastic scattering with no particle production.

    In your paper hep-th/0003243, you show that theories in greater than 2 dimensions have a trivial S-matrix if tempered polarization-free generators (PFG) are used, and that theories in d=1+1 have no particle production if tempered PFG’s are used. For a theory representing real scalar fields with a phi^4 quartic interaction in d=3+1, what special properties would the PFG’s have to possess such that the S-Matrix is nontrivial and agrees with the ordinary perturbative qft results?

  25. Aaron Bergman says:

    If one produced a genuine anything for phi^4 in 3+1D without a cutoff, that would be worrisome. As I remember it, that theory has a Landau pole.

  26. Bert Schroer says:

    The simplicity of the family of theories which correspond to tempered vacuum-polarization-free generators of wedge-localized algebras is their algebraic Zamolodchikov-Faddeev algebra structure. The S-matrix appears simply as coefficients in that algebra. The double cone localized operators have already the full infinite vacuum polarization clouds (visible in the explicitly computed formfactors). The correlation functions are already extremely rich and complicated. Yes you should be impressed; no other approach has been able to achieve this. The massive Thirring model (Sine-Gordon) is among these models. But in general this approach does not start with Lagrangian names. It does not know what the name phi to the fourth power means because it is conceptually totally different and that name has no intrinsic meaning. Perturbation theory indicates that the latter has particle production and is not factorizing.
    Outside factorizing theories wedge generators have a much more difficult structure and the aim would be to come to a perturbative understanding. This perturbation theory is expected to show the true frontier between sensible (alias renormalizable) and nonsensic theories. One expects that any theory which was renormalizable in the standard perturbation theory of pointlike Lagrangian field will also remain sensible in the new setting.

  27. Brett says:

    This discussion has raised an intriguing question in my mind. Normally, in QFT, there is an extremely unattractive division of how the fundamental parameters that describe the theory are introduced. Most free parameters are introduced in the Lagrangian, but that alone does not determine the physics, except in trivial cases. In sufficiently “nice” theories (e.g. anomaly-free gauge theories), the regulators that can be used are severely restricted, and those available all give equivalent results. (In contradistinction, in an anomalous gauge theory, there are generally no good regulators to use.) In this case, the regulator can be (almost) banished from the physics, and this is just perturbative renormalizability.

    In phi^4 theory, much the same thing holds perturbatively, but nonperturbatively there’s a problem. It would be very interesting in AQFT gave a nonperturbative definition of this theory, with a cutoff, that did not separate the interaction and the regulator the way we ordinarily do. As a test of whether this kind of thing may be possible, I suggest looking for theories in 1+1 dimensions that look like the chiral Schwinger model, because that model is exactly soluble, but there is a regulator dependence in the final spectrum. If the regulator parameter could be encoded in the same way as the coupling in this theory, it would be quite interesting.

  28. Bert Schroer says:

    From a perturbative approach for wedge generators one does not expect anything new in case the models were renormalizable in the standard sense. But it could be that some of the models which one presently discards as nonrenormalizable may actually lead to perturbative expressions with a finite number of physical parameters. In any case these are presently pure speculations since such a “on-shell” perturbation theory has not been formulated and hence the approach based on modular localization remains limited to factorizing models.
    I would like to return to the issue of functional integration since there seems to be a widespread misconception. Of course there are many quantum mechanical models for which the path integral representation has a mathematical meaning in terms of rigorous measure theory. However this does not mean that one can perform explicit analytic calculations. Take the hydrogen atom. Even though the functional representation makes sense it is not known how the important property of integrability (which makes the problem operator-solvable) is encoded in the integral. Since one knows the result from standard quantum mechanics one can of course add recipes to the functional integral which will lead to the known result; but this can hardly be called mathematical physics and is just a game of personal entertainment.
    The fact is that besides interaction-free systems and camouflaged free systems (as the Schwinger model) there is no known case where you can base exact analytic calculations on functional integrals. Nevertheless the functional integral is useful for quasiclassical approximations and for perturbation theory and (as a result of its superficial geometric classical appearance) has a strong intuitive appeal.
    In a previous remark I said that the functional integral in QFT is a bit deceiving because when you want to compute explicitly renormalized correlation functions the formal elegance of that functional expression on a piece of paper is of not of much use; you have to make your hands dirty and add a lot of additional wisdom and tricks which the elegant formula did at most “morally” (I used the word artistic) contain but not factually. The algebraic approach is more honest and more equilibrated. You list the properties you want (taking account of the singular nature of pointlike fields) and what you get at the end fulfills precisely your initial requirements. In the functional approach to QFT this is not quite so (as I previously explained).

  29. JC says:


    (Silly question).

    Is there a well defined “limit” in the aqft approach which reproduces the functional integral for a system like the Thirring model (for example)?

  30. Bert Schroer says:

    Not that I know. For the Euler-Lagrange equation of the Thirring model (massless or massive) the nonlinear term can be defined in terms of a Schwinger point-split limiting procedure. But a Lagrangian action is not really part of the quantum world (it is conceptually somewhere between classical and quantum) and it is unclear what such point-split formulation (related to operator product expansion) means in such a measure-theoretical classical setting.

  31. Robert says:

    I have only glanced over Schroer’s notes briefly, but there seems to be a point that in the algebraic setting (causal perturbation theory to be precise) one can do the perturbation at the algebraic level and only as a second step consider the representation theory in terms of states (i.e. linear, positive, normed functionals on the algebra of observables). Does Schroer claim this is not possible in the path integral setting? I always thought, the state (important especially in a curved background where you don’t have a prefered Minkowsky vacuum) is encoded in the path integral in the boundary conditions of the fields you integrate over. But as always, I may be wrong.

  32. JC says:


    In the aqft framework, how exactly is the S-matrix a “purely” quantum object (as opposed to being between classical and quantum)? Do you mean that the aqft derived S-Matrix is the unique solution of a well defined inverse-scattering problem?

    Are there any rigorous proofs which show that for a particular set of asymptotic scattering data (ie. incoming and outgoing particle states, crossing symmetries, number of spacetime dimensions, etc …), there exists a unique S-matrix in the aqft framework?

  33. Bert Schroer says:

    functional integral representations require for mathematical reasons a Euclidean setting, but generic QFT in CST is totally outside a Osterwalder-Schrader setting. If it would not be outside then the state information would indeed be expected to reside in the boundary condition because this is the only freedom at one’s disposal.

  34. Bert Schroer says:

    it is only for factorizing models that the bootstrap construction of the S-matrix can be separated from QFT (which then can be constructed in a second step via the wedge generators). In the general case the S-matrix and the generators have to be constructed together in an iterative manner (as a scenario this was explained in m]one of my papers) but there are yet no concrete results. The reason why the S-matrix plays such an important role in nonperturbative construction is that in addition to its well-known role in scattering theory it determined the modular localization structure of wedge algebras (this is relatively new).
    The inverse scattering problem for a given S-matrix with all the crossing and unitarity requirements has indeed a unique associated QFT if one assumes that formfactors fulfil the crossing property (this is true for factorizing models but it has not been derived in the generality one needs it from the principles of QFT).

  35. For AQFT, I was very afraid that after the retiring of Haag it was going to be a dead field; Araki was active but very busy into politics of the IAMP sociery, and note that also Doplicher did a small jump to noncommutative spacetimes via his energy bkackholing relationship, so if he did not left the field, at least he put it in the backburner place. In Gottingen, Borchers did some continuity but very restricted to 1+1 field theory.

    When student, one day I sneaked into Luruper Chausse theoretical library and sit in a small working table. There scattered on the table there were the manuscripts of all the papers Haag had received for editorial task in Comm Math Phys and similar journals; some assistent was surely ordering them for final archival.

    In later years I have remembered this as a signal of exhaustion of the field too, to my own regret. I think a resuscitation will not be possible without (until?) reaching a depper understanding of the C* geometric work: non commutative manifolds, tangent groupoids, Hoft thingies, etc…

  36. Bert Schroer says:

    Alejandro Rivero,
    The scientific situation is not quite as bleak.
    There are new and very exciting ideas in the offing. If everything goes well we can hope for a second cataclysm (the first one was of course renormalized perturbation theory). As was the case already with renormalization theory, it will strengthen, deepen and significantly extend the old principles. The name “field” in QFT will more refer to its historical origin than to the new conceptual content.
    The high mathematical and conceptual barriers you are mentioning are not necessarily an impediment. They maintain high standards and prevent that sociological effect which results if too many physicists are undergoing a bose-condensation along the lines of a monoculture. I am by no means elitist and anti-democratic but if too many researchers work on one problem this tends to create confusion and regress instead of progress just as you see if you follow the blogs on string theory in this weblog. Mathematics is of course very important but is should never be allowed to run amok; the tuning between particle physics and mathematics is a very fine one and certainly I would be the last to plead for a return to a mathematical stone age in physics. Physical concepts should be always in the helm and the best situation is if they tell you what kind of mathematics is best for them.
    Naturally if you work under such conditions there are very strong restrictions, not only from experiments but also (and this is particularly serious for theoreticians) with past concepts and principles. It is very far away from the “everything goes” maxime of string theory. This kind of conceptually-geared work needs more time and is certainly in a strong tension with the high impact index attributed to the fashionable subjects.
    The question of whether there will be a future of particle physics which is worthy of its past depends presently very much on whether young intelligent, courageous and innovative young theoreticians in particle physics will find enough time to test their innovative ideas on past achievements or whether they will be “abgewickelt” (to borrow a German word which was very popular when the “Wessies” took over the academic institutions of the GDR) before they can accomplish this task and replaced by string theorists with a higher production efficiency.

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