Frank Wilczek has a new Reference Frame piece in this month’s Physics Today. It’s about the question of whether the parameters of our fundamental physical theory are uniquely determined by abstract principles, or “environmental”. He gives two reasons for suspicion about the idea that these parameters are calculable from a fundamental theory:
1. They have complicated, “messy” values and, despite much effort, no one has come up with a good idea about how to calculate them (an exception is the ratio of coupling constants in a supersymmetric GUT). He writes:
Could a beautiful, logically complete formulation of physical law yield a unique solution that appears so lopsided and arbitrary? Though not impossible, perhaps it strains credulity.
2. Some of the values are fine-tuned to make complex structures and thus life possible:
It is logically possible that parameters determined uniquely by abstract theoretical principles just happen to exhibit all the apparent fine-tunings required to produce, by a lucky coincidence, a universe containing complex condensed structures. But that, I think, really strains credulity.
Personally I don’t see the same degree of believability problems that Wilczek sees here. On the first point, it seems quite plausible to me that there are some crucial relevant ideas we have been missing, and that knowing them would allow calculation of standard model parameters, by a calculation whose results would have a complicated structure.
On the second, it’s not at all clear to me how to think about this. Sure, the fact that our universe has highly non-generic features means that it is incompatible with generic values of the parameters, but there’s no reason to expect the answer to a calculation of these parameters to be generic. I guess the argument is that there would then be two quite different ways of getting at some of these parameters: imposing the condition of existence of life, and a fundamental calculation; and if two different, independent calculations give the same result one expects them to be related. But the question is tricky: by imposing the condition of the existence of life in various forms, one is smuggling in different amounts of experimental observation. Once one does this, one has a reason for why the fundamental calculation has to come out the way it does: because it is has to reproduce experimental observations.
Wilczek avoids any mention of string theory, instead seeing inflationary cosmology and axion physics as legitmating the idea that standard model parameters are fixed by the dynamics of some scalar fields, or something similar. This dynamics may have lots of different solutions so:
We won’t be able to calculate unique values of the parameters by solving the equations, for the very good reason that the solutions don’t have unique values.
The fundamental issue with any such anthropic or environmental explanation is not that it isn’t a consistent idea that could be true, but whether or not it can be tested and thus made a legitimate part of science. It’s easy to produce all sorts of consistent models of a multiverse in which standard model parameters are determined by some kind of dynamics, but if one can’t ever have experimental access to information about this dynamics other than the resulting observed value of the parameters, why should one believe such a theory? It is in principle possible that the dynamics might come from such a simple, beautiful theory that this could compel belief, but the theories of this kind that I have seen are definitely neither simple nor beautiful. If you want me to believe in a complicated, fairly ugly theory, you need to produce convincing evidence for it, some sort of testable predictions that can be checked. Wilczek does believe that multiverse theories may provide such predictions:
Of course, the very real possibility that we can’t calculate everything in fundamental physics and cosmology doesn’t mean that we won’t be able to calculate anything beyond what the standard models already achieve. It does mean, I think, that the explanatory power of the the equations of a “theory of everything” could be much less than those words portend. To paraphrase Albert Einstein, our theory of the world must be as calculable as possible, but no more.
One can’t argue with this: if a model make distinctive predictions, and these can be compared to the real world and potentially falsify the model, one can accumulate evidence for the model that could be convincing. Unfortunately I haven’t seen any real examples of this so far. The kind of thing I would guess that Wilczek has in mind is his recent calculation with Tegmark and Aguirre that I discussed here. I remain confused about the degree to which their calculation provides any convincing evidence for the model they are discussing.
Unlike many theorists, Wilczek personally seems to be an admirably modest sort of person, and perhaps this has something to do with why the multiverse picture with its inherent thwarting of theorist’s ambitions to be able to explain everything has some appeal for him. Over the years during which particle theory has been dominated by string theory, Wilczek has shown little interest in the subject, perhaps partly due to its immodest ambitions. But I see two sorts of dangers in the way his article ignores the string theory anthropic landscape scenario which is what is driving the interest of much of the theory community in these multiverse models. As his advisor David Gross likes to point out, accepting this scenario is a way of giving up on the perhaps immodest goal he believes theorists have traditionally pursued, and one shouldn’t give up in this way unless one is really forced to. None of these models is anywhere convincing enough to force this kind of giving up.
The second danger is that what is happening now is worse than just giving up on a problem that is too hard. The string theory landscape anthropic scenario is being used to avoid acknowledging the failure of the string theory unification program, and this refusal to admit failure endangers the whole scientific enterprise in this area.
Update: It has been accurately pointed out to me that Wilczek does mention string theory briefly at one point in the article (“Superstring theory goes much further in the same direction”), and alludes to it at another place (when he talks about a “theory of everything”).
Are you sure that you are talking about professional condensed matter physicists?
Yes, I am quite sure. Of course, no one is claiming to have worked out completely how to compute things properly. That is the point.
The point is that AQFT is one attempt to codify the mathematical structures of QFT. There are others (I can think of at least three mathematicians who have set out to define QFT, two of whom have Fields medals). To date, I think it’s safe to say that none of them completely capture what we know ought to be true about QFT. So, maybe you’ve chosen the right way or maybe not. Beats me. But I hardly think it’s fair at this point to say that the AQFT formalism must be the right answer when there aren’t any realistic examples and, furthermore, it seems to ignore (at least judging by your previous conversations with Mentos) much of the modern Wilsonian understanding of the subject.
How about something easier? Does AQFT have anything interesting to say about TQFTs? In two dimensions, at least, the axiomatization in terms of categories has shown a lot of success. In 4D, the twisted N=2 TQFT that gives Donaldson invariants has proven to be extraordinarily fruitful. Can you see the AQFT structures in this context, and do they say anything new or useful about Donaldson and Seiberg-Witten invariants? Recently, a topological twisting of N=4 SYM has proven to describe much of the mathematics in the geometric Langlands programme. Can AQFT say something there? The categorical structure a la Atiyah, Segal et al definitely is apparent.
The first step would be to have convincing experimental agreement with those quantum numbers. If those are related to e.g. braid group statistics this would be interesting, since statistics is an almost kinematical property (and the only known property by which low dimensional are structurally different from higher dimensional ones!). I find it perfectly conceivable that e.g. a large statistical (or quantum-) dimension could account for a high temperature in high T_c, but I am not an expert.
You did not answer my question about Vaughn Jones.
You did not answer my question about Vaughn Jones.
Sigh. All right, then. He seemed well, and did a lot of windsurfing, as usual. But I can’t say I really know him personally.
Can you see the AQFT structures in this context, and do they say anything new or useful about Donaldson and Seiberg-Witten invariants?
This is a very good question. AQFT should be able to shed light on these invariants, if it has any computational power.
“But I hardly think it’s fair at this point to say that the AQFT formalism must be the right answer when there aren’t any realistic examples and, furthermore, it seems to ignore (at least judging by your previous conversations with Mentos) much of the modern Wilsonian understanding of the subject.”
Sorry Aaron, this apodictic manner is not my style of dialog and if I really said that, it was a lapsus linguae. In fact the situation is quite the opposite: exactly because we know that we are on such slippery ground we are trying so hard to control the existence of 4-dimensional QFT (those successes in d=1+1 are only successes in the sence of a theoretical laboratory, but we have already learned some messages and at least there is a new strategy). Our aim is to make QFT like any other area of theoretical physics.
Concerning the Wlson Renormalization group I once asked that question to Fredenhagen after he finished his work with Duetsch on the AQFT-inspired version of the Petermann-Stueckelberg renormalization group. He told me that this can be done and the result will be written up. Fredenhagen has never claimed anything which he was not able to deliver and I attribute the fact that nothing has appeared yet as due to his very demanding job of leading such a big institute in the midst of a German buerocracy and politicial leadership which come with new directives almost every week.
Concerning those impressive results about Donaldson invariants, Seiberg-Witten, the Langlands program etc. I think that the relation to particle physics is of a more metaphorical nature i.e. they use a language which is taken from particle physics but it is not really particle physics. The setting tends to be Euclidean instead of real time (I am not so sure about Langlands) and certainly I do not know how to get a real time into those mathematical structures. I also confess that I have never seen any mathematically controllable real time operator theory with a electric-magnetic duality although I know order-disorder variables in real time QFTs in d=1+1.
The relation of TQFTs to reality is irrelevant here. They certainly appear to be examples of quantum field theories. For example, QFT arguments were used to invent Seiberg-Witten invariants which were previously unknown to mathematicians. It’s hard to believe that these things simply are not QFTs given their success. If your formalism has no room for them, then I don’t see why I should take is seriously as an axiomatization of QFT especially given that TQFTs are easier than the real thing.
Let me ask about anomalies. There are very beautiful ways of understanding gauge anomalies from the path integral point of view. Does AQFT in any way incorporate this geometric understanding of anomalies?
The setting tends to be Euclidean instead of real time (I am not so sure about Langlands) and certainly I do not know how to get a real time into those mathematical structures.
The question was really: how would AQFT tackle the problem of smooth 4D invariants? In other words, what is a combinatorial description of them? As for ‘real time’, many elements of SW theory have twistor theoretic descriptions, and these could possibly be of AQFT type. If so, then these invariants are really, really interesting physics – not just mathematics.
AQFT, Donaldson-SW
Temperature duality, Coupling duality
Holography, Cobordism
Monade, Monad
Braids, Dehn surgery
…and you might even convert a few thousand Landscapologists to the true path!
Anomalies have been treated and identified in the Duetsch-Fredenhagen AQFT setting of perturbation (naturally it is in real time). Euclidean field manifolds and functional integrals are metaphoric instruments, they are very suggestive and after a lot of massaging leads to correct results but they do not exist mathematically apart from superrenormalizable (and not very interesting) models of the Glimm-Jaffe type. In a previous blog I described how I went through all this in the middle of the 70s looking at the Euclidean integration aspects of the Schwinger model, finding by generic functional integration (the Schwinger model is the only case where this can be done generically i.e. without specializing to instanton configurations) learning everything about the Atiyah-Singer index theory in order to generalize that wonderful Schwinger model relation between zero spinor modes and winding number of the gauge field (partially in collaboration with N. Nielsen and also with Swieca) some years before Alvarez-Gaume, Witten etc. I think I have some credentials in that kind of functional geometry. But I left after a couple of years because it was far away from my conception of real time quantum physics and because I think that artistic manipulations which have no mathematical existence even if they are highly successful are not my favorite passtime. Just imagine what would have happened to quantum physics if people would have been complacent about the quite successful Bohr-Sommerfeld old QT. If you like metaphorical arguments which lead to consistent results that is fine for me, but I find it absurd to be asked (30 years after I left this metaphorical use of QFT precisely for the indicated reason) to produce results which I do not consider to be part of particle physics. Neither QFT as an instrument of particle physics nor its AQFT extension can achieve what you are asking for; however some gifted particle physicists can be inspired by its setting and derive wonderful mathematical results. Not even Graeme Segal, who was inspired by QFT to abstract his nice euclidean setting of gluing via cobordism claims that he is doing what particle physicists call QFT. There is a difference between particle physics and “that what physicists are doing” (quotation from Hrzebruch after giving an account of the Atiyah-Witten work).
All right, then. How would you calculate the Standard Model parameters? Is that physics?
I just want to make sure I understand you. Is it your belief that TQFTs are not examples of QFTs?
As for the rest, quantum mechanics explains and encompasses the old Bohr-Sommerfeld rule (slightly modified). As best I can tell from you, AQFT neither explains nor encompasses the geometric understanding of anomalies.
I’m mostly going through this, BTW, to show to the various readers here that the reason the vast majority of the field does not pay attention to AQFT is not just prejudice and groupthink or whatever. It is because it does not seem to capture any of the things that are generally understood to be fundamental and useful features of QFT.
And I have a question: why should a physicist accept Connes’ vision of what is really geometry? He’s just a mathematician.
Kea,
I have no aswer and I do not think that anybody else has. Sometimes I think that we had to pay a very high prize for that relatively easy group theoretical entry. We are now in a labyrinth, the best (in the sense of phenomenological success) quasiclassical straight jacket imaginable (remember that the gauge principle is a classical selection principle for L-invariant interactions) i. e. I do not think we will get out of the present mess without a very significant additional conceptual investment. One should be modest and admit that not every problem can be solved at any time. Solving such problems in a diretissima is mostly not possible. We should patiently go ahead and pressing more the inner logic of QFT; in no way did we reach yet the inner core of QFT. But I expect that as in previous cases (except string theoru) the revolutionary aspect may be less in a change of principles but more on the conceptual side of their implementation.
A small piece of cherry on the cake which AQFT-inspired arguments can deliver is that if one starts with massive vektormesons in zero order than perturnative consistency requires the introduction of an additional physical degree of freedom whose simplest realization is a scalar field (Higgs but without that vacuum condensate). So something like the Higgs seems to be a perturbative necessity. But this is probably something expected in any case this has no bearing on your question.
I may have created the wrong impression that AQFT is a community thing like string theory or LQG. It is not, rather it is a shared belief of some pragmatic-minded individuals whose only distinction from others is that they insist that the requirements on the searched for theory should be clearly formulable without invoking “classical crutches” i.e. without quantization and if possible without highlighting particular field coordinatizations (apart from conserved symmetry currents).
Yes Aaron, TQFT is associated to quantum field theory but it is not itself QFT. You can extract from localizable QFT an algebra of the topological bones, but TQFT has no autonomous interpretation because all the interpretation of QFT, I repeat all the physical interpretation, goes through causal localization. Even particle momenta are in first place not Fourier transforms but rather encode geometric relations between events (clicks in counters). One forgets these aspects if one computes Feynman integrals in momentum space but in scattering theory and much more so in curved spacetime QFT one is painfully reminded of this elementary facts of life. Physicists manage to invent a Lagrangian description (e.g. Chern Simons) but this does not make them QFT, they remain at best TQFT. You may not see this directly in the action (unless you really make the Osterwalder Schrader test) but the algebraic content of such a theory is radically different from that of a localizable QFT (they are similar to those algebras Vaughn Jones uses in his subfactor theory). A derivation of e.g. the TQFT containing the mapping class group from a full chiral QFT on a circle was given e.g. in the appendix of the pre-electronic paper I wrote with Fredenhagen and Rehren. Since the original chiral theory was localizable, you know the concrete spacetime interpretation of the knots and mapping class group operators only by pointing to those operators in the QFT algebra from where they originated. The only memory about localization they carry is a distinction between right and left (a property of the braid group). I do not know any construction which allows to reconstruct a QFT from a TQFT. The auxiliary Riemann surface pictures one makes are strictly metaphorical, they have absolutely noyjing to do with the physical localization.
I happen to think that the present formalism of gauge theory does not meet this test (see my remarks I made earlier, giving support to Thomas Larson in Fantastic Realities, probably more radical than what he had in mind).
Bert, my key observation about gauge symmetries is quite radical: not all gauge symmetries need to remain gauge after quantization, due to gauge anomalies. There is a strong, ideological resistance against this (“a gauge symmetry is a redundancy of the description”), despite the fact that several well-known examples of consistent anomalous gauge theories exist; the free, subcritical string (D < 26) is the canonical example (no-ghost theorem).
In fact, it is impossible to combine a trivial action of local gauge transformations with nonzero charge, provided that you embed the algebra of gauge transformations into its natural completion containing also divergent, superselection-changing transformations. I made this observation, which is both obvious and trivial, in math-ph/0603024.
From this perspective, the key difference between string and YM theory is whether you allow divergent gauge transformations. In string theory, you allow generators which diverge when z -> infinity (L_m with m > 1), but in YM theory you forbid generators which diverge when r -> infinity.
Aaron,
you were a great sparring partner yesterday night. From FAQ’s to the standard prejudices you hit the whole scale of arguments and sometimes I had the impression that you played the devil’s advocate (this is what I sometimes do in order to enhance the informative flux). Since I have reservations about proselyting, this is a good way to conduct a dialog and I tell you that I stayed on the PC up to midnight in order not to loose this ideal opportunity.
I hope that I convinced you that the extension of standard QFT (Lagrangian quantization) called “algebraic” is basically the execution of a step towards intrinsicness which in geometry you already excepted a long time ago, namely independence of coordinates (in the case at hand independence of field coordinatization which requires to go beyond Lagrangian quantization) although, as in geometry, it is not forbidden to use coordinates. Due to the inexorably singular nature of sharply localized quantum fields this step is naturally more sophisticated than its differential geometric analog.
I also hope to have convinced you that TQFT, despite this name, is not QFT, because the most important ingredient since the time of Faraday and Maxwell namely localization (the meaning of fields) is missing. The impreciseness of nomenclature, hugely enhanced by string theory, would have tragic confusing consequences if we fall prey to our own bad semantic creations.
I tried to explain that AQFT achieves the separation of topological bones from the localizing flesh by extracting “kinematical” subalgebras with tracial states (they are never localizable or transportable). This is an extremely important technical step in the DHR theory of superselection sectors. The latter is a phantastic achievement: the reconstruction of the full QFT (including statistics, inner symmetries…) only from its observable shadow (another reductionist construction of AQFT). This is one of these inverse constructions of the kind which Marc Kac characterized in a more acoustic context as “how to hear the shape of a drum”. The idea is that we have very good intuitive insight into local observables, but we enter a conceptual high risk zone if we arrogantly claim that we can pull the structure of the full QFT with all its “unobservable” (i.e. its not directly accessible charge- and halfinteger spin- carrying fields) out of our hat or head.
This extraction of topological (“kinematical” in a certain extended sense of the word) bones in the old DHR work (see Haag’s book) has been extended to the low-dimensional realm of braid group statistics in the appendix of the mentioned FRS work, and I already mentioned that in this richer braid-knot-mapping class group context the tracial states coincide with the famous Markov traces of Vaughn Jones subfactor theory. The latter is a bone theory par excellence, although the bones which in the Chern-Simons-like
“backreading” into a Witten kind of Lagrangian jacket (but outside the Osterwalder-Schrader reflection positivity which is related to localization) take on the appearance of topological bones, are more combinatorial bones in Jones’s subfactor setting.
Another bone framework outside of localization is the LQG. I know too little about its algebraic nature to sav what kind of algebras it leads to but there are definitly no monades (see earlier blogs) around.
I saw many instances of extraction QFT —> TQFT but I am not aware of a single case were localizable meat was put onto topological bones. This is why I am a bit sceptical about the relation of LQG to localizable ordinary quantum matter; but on the other hand I have no argument why such a thing cannot work. Maybe Lee Smolin or somebody else from LQG has an idee how to achieve that.
I will vanish from the radar screen for at least one day and I of course hope that my contributions to this weblog do not only create animosities but are also a little bit helpful in a positive sense.
Is it plausible that the right concept of real-time Minkowski-background 4D field theory is disconnected from that of Euclidean and/or topological field theory?
Taking the risk of sounding like a broken record (at least I won’t be the only one), let me say this:
The FRS result on RCFT (->) shows precisely the opposite. The full understanding of rational conformal field theory has only been obtained after realizing how it splits into a topological part and a part knowing about the conformal background structure.
And I think we all agree that RCFTs are of concrete physical relevance. They describe stuff people measure in laboratories.
Furthermore, the FRS theorem solves a problem (namely that of understanding what full 2D RCFT is) which cannot even be formulated with present AQFT technology.
That’s only in part due to the fact that AQFT chooses (by way of axioms) to restrict attention to Minkowski background, while we need Euclidean backgrounds here.
But there are more reasons. It is not known to date how to describe field theories with general boundary conditions using AQFT, let alone field theories with defect lines.
(Defect lines are for instance generated by the ‘t Hooft operators that implement the Hecke transformations in twisted SYM, the way Kapustin and Witten explain (->)).
Klaus Fredenhagen has some first ideas on how boundary conditions might be modeled in the AQFT context. There was supposed to be a project concerned with investigating this question in the new String Theory Research Center in Hamburg (->) – but the referees canceled this particular project.
Anyway, my impression is pretty much the same that Aaron expressed:
AQFT is one attempt out of several for extracting (guessing, really) the right mathemtical structures (the right axioms) from the semi-heuristic physical understanding of field theory. Given the results obtained from these axioms, it does not really look like the AQFT axioms make closer contact with observable physics than, in particular, Atiyah-Segal formulations do.
Moreover, as the example of that Stolz/Teichner paper was supposed to illustrate, where necessary the Segal formulation (which asserts that QFT = representation of cobordims categories in Vect) incorporates useful insights obtained from AQFT.
So, for me, the conclusion is this:
Clearly, AQFT has axioms which are a plausible first guess for the axioms that a real QFT should satisfy. But just as clearly, something about these axioms is not yet flexible enough. Something is still missing. Something of course is clearly right about them.
Surely, as long as the AQFT axioms apply not to a single physically non-trivial theory, they are in need of modification.
What is good about axioms is that they allow people to unambiguously work out their consequences. This is happening using AQFT axioms, and a couple of mathematically interesting insights have been obtained.
However, I see no evidence that Segal-like formulations of QFT, with their motivation in topological field theories, are farther removed from the QFT existing in nature than the AQFT formulation. On the contrary, things like the FRS theorem seem to tell me that the opposite is the case.
Suggesting that Atiyah-Segal is “just metaphorics” only because it is not formulated in terminology of AQFT sounds misleading to me.
I’m just posting this to say that Urs pretty much said what I was going to say and to reiterate that any definition of QFT that does not include TQFT (of which many are not obtained by first starting with a non-topological theory) seems severely wanting to me.
That it does not encompass the geometric insights regarding anomalies and seems to allow perverse theories as are obtained in Rehren duality doesn’t help things either.
Kea wrote:
There is an obvious covariance condition in the axioms of AQFT. You assign algebras (of observables) to subsets of spacetime. You want these algebras to behave nicely under diffeomorphisms of spacetime.
As about any covariance condition, this can be rephrased as a natural transformation of some functor.
As far as I can tell, this observation does not give rise to something qualitatively new.
Heh.
Let’s not all pile on at once.
[everything urs and aaron said +]… unable to handle quantum field theories with degenerate vacua (see our previous discussion of Seiberg-Witten theory), does not incorporate renormalization-group behavior a la Wilson, apparently has trouble dealing with supersymmetry, …
In short, it is pretty much silent about the past 30 years of developments in quantum field theory. And when it does speak up (eg, Rehrens Duality) , it says something silly.
Were it not for the efforts of this blog owner (hosting various documents for Prof. Schroer, and enthusiastically promoting his arxiv postings), few would ever have heard of AQFT. And for good reason.
Bert, Urs, Aaron, Kea,
The (quantum field) theories you are discussing look very distant from ordinary quantum mechanics of particles that served us so well in low-energy physics. Is there an important reason (a no-go theorem, or something) that forbids application of simple rules of QM (the Hilbert space, the Hamiltonian, the wave function, etc) to high-energy phenomena and requires the complete shift of the paradigm from quantum particles to quantum fields? This shift of the paradigm looks especially strange if one takes into account that predictions of QFT are usually limited to small (radiative) corrections to the QM results, e.g., the Lamb shifts.
Surely, when energies are high, we cannot limit ourselves to the Hilbert space with fixed number of particles, because particles can be destroyed and created in accordance with Einstein’s formula E=mc^2. Thus, instead of the fixed-particle-number Hilbert space we need to consider the Fock space where the number of particles can change from zero to infinity. Then creation and annihilation of particles can be described by simply writing the interaction Hamiltonians as polynomials in creation and annihilation operators in the Fock space. However, this doesn’t require introduction of a radically new formalism, such as QFT.
My question is: why, in your opinion, this simple-minded approach doesn’t work? Are there important reasons (besides historical) to consider fields, rather than particles, as fundamental ingredients of nature? Thanks.
My question is: why, in your opinion, this simple-minded approach doesn’t work?
Oh, but it does work…if you try to turn what Bert is talking about into higher category theory and raising and lowering operators become associated with the concept of categorification and decategorification…and Minkowski space gets turned into twistor geometry so that it’s true topos theoretic nature can be identified.
Eugene
How can Bert and Urs/Aaron both be right? There is only one way. Urs/Aaron need to accept the possibility that 30 years of String theory has missed something important. Then they can look for it in the higher category theory language that they use. On the other hand, Bert needs to think about a very, very small request – the possibility of reformulating AQFT, without losing any of its structure, in a different mathematical language.
Oh, I forgot to mention…that conference that Vaughn Jones was organising? For some reason we had a lot of speakers talking about TFTs.
I think we should call vacuums elephants. Don’t you, Who? The word vacuum just has too many connotations of dirty housework and dust mites.
Urs/Aaron need to accept the possibility that 30 years of String theory has missed something important.
What does any of this have to do with string theory? This has been a discussion about whether AQFT is the right way to think about field theory.
For Eugene, you can look at why QFT developed in the first place, I suppose, but a straightforward question is how do you propose to do QCD?
What does any of this have to do with string theory?
All right. Sorry. I was just using the term as a short hand for the conventional physicist’s use of SYM etc.
…but a straightforward question is how do you propose to do QCD?
The SU(3) confinement comes in at the tricategorical level when one is forced to break the Mac Lane pentagon and use premonoidal structures.
I just got back so before I go to sleep I will at least try to answer a few of those questions.
Let me start with Eugene because it is the most physical question and I have a eference for such straight physical questions
The transition from QM to QFT is indeed a total paradigmatic shift, to exaggerate a bit in order to get this point across: the only thing they share is the Planck h.
Let me explain this by looking back at history. Whereas the paradigm changing QFT was discovered by Jordan, Dirac had a better accepted entrance into this issue by placing multi-particle quantum mechanis (leading up to Fock space) into the centre of the stage. For the case of the Schroedinger theory, the both points merges rapidly. Together with the discovery of the Dirac equation, Dirac developed hole theory and some of the first textbooks which contained low order calculation (no loops) worked quite nicely (viz Heitler). But later it was seen that the particle-based hole theory is inconsistent: nobody was ever able to do renormalization theory based on it and when you massaged it so that you could do it, you lost the particle base and slipped into QFT. Another observation which indicated that there was a paradigmatic change was that of Furry and Oppenheimer who realized to their great surprize that an interacting field applied to the vacuum does not create a particle state but rather a state which had a nonvanishing component to the one-particle subspace but in addition the (in infinite order you get a “cloud” which involves infinitely many particle-antiparticle pairs) vacuum polarization admixture. The modern way is to demonstrate this paradigmatic change by a rigorous structural (model independent) theorem which says that if in a compact spacetime region you find any operator localized in that region which creates a polarization-free one-particle state, the theory is necessarily interaction-free (in other words you now have an intrinsic quantization-independent way of seeing the presence of an interaction). The first region where this breaks down is the noncompact wedge region, in that case so-called PFGs ((vacuum)-polarization-free-generators) exist even though the theory is not free. This theories which arise in this way are precisely the d=1+1 factorizing model and the Fourier transforms of the PFGs fulfill the Zamolodchikov-Faddeev algebra relations i.e. the positive/negative frequency components are still close to particle creation/annihilation operators. Although these models have scattering without particle creation (and in this sense they are close to what you wanted in your question), they have very complicated vacuum polarization clouds which prevent any quantum mechanical particle picture for compact regions! It is true that the generators of the noncompact wedge region still look like relativistic particles, but even on that level the ground is somewhat slippery since that elastic S-matrix realizes the full “nuclear-democracy principle” saying that there is no genuine particle hegemony between elementary an bound (though there is still a charge hegemony between fundamental and fused charges) every particle is formed from all the other particles inasmuch as charge conservation allows this. So already in this relatively simple class of interacting theories (where you still can save some of the particle concepts on the level of wedge localization) the paradigmatic change is obvious.
This has very grave consequences. For example those arguments in Susskinds and Weinberg’s work you often find the terminology relativistic QM instead of QFT. Unfortunately these are not just words, in computing those cosmological vacuum expectation of the energy-momentum tensor these authors fill levels as if it would be quantum mechanics and get their absurdly large cosmological constant values. This is conceptually totally wrong (it contradicts the local covariance principle which according to Kea does not contain any new information) and was profoundly criticized in a paper by Hollands and Wald (with a very nice title, you find it in one of my old contributions to this weblog). Of course saying that does not mean that you can easily do a correct calculation (the cosmological reference state is not well kown and in gravity you rarely deal with invarianr vacuum states). If this is the basis for anthropical ideas, it is an extremely flimsy basis indeed.
In this context it is interesting to mention another point. The modular methods which we developed recently (in the paper with Mund and Yngvason which I already mentioned several times) permitted us to solve an old problem from the Wigner representation theory namely what are the fields for those massless infinite spin representations. We showed that they are semiinfinite spacelike string-localized. This solves an old problem which generations of particle physicists tried to understand. Weinberg in his book mentions the infinite spin Wigner representation and then dismisses it by saying that “ Nature does not make use of them”. But the main job of a theoretician is of course to investigates its physical manifestations and decide afterwards whether it should be dismissed. After all the massless matter separates into two families the neutrino- photon… family and the much larger infinite spin (better helicity tower) family. It is true that this quantum matter has very unusual properties (already Wigner noticed a very strange thermal behavior). I think nowadays with the black matter around, one would think twice before dismissing it with those words.
.
…it contradicts the local covariance principle which according to Kea does not contain any new information
I never said that.
Sorry Kea, I saw it somewhere, but it is not in your blogs. I applogize.
It is natural that there are TQFT at Jones’s department because type II subfactors are (in some generalised sense) TQFT. But Jones knows very well (through his work with Wassermann) that QFT needs localization. In fact I am sure that he deepy appreciates the recent construction existence proof (and construction) of the minimal series by Kawahigashi Longo Pennig and Rehren . This is quite an achievment and this cannot be done by Frobenius algebras and sewing (you cannot construct QFT which has localization by sewing bones). It seem that some people have forgotten what a proof is.
Kea, remember I said that without localization you cannot talk about QFT and not that TQFT is a second rate enterprize. or anything like this mathematics does not need the blessing of QFT in order to be brilliant. The entire subfactor theory of Vaughn Jones was done in a context without localization and he would not even dream to use the word QFT for those constructions (although he knows perfectly how to to subfactor theory in the localizable type III setting)
Urs you said
“Klaus Fredenhagen has some first ideas on how boundary conditions might be modeled in the AQFT context. There was supposed to be a project concerned with investigating this question in the new String Theory Research Center in Hamburg (->) – but the referees canceled this particular project”.
Urs I can perfectly understand that you are quite angry at those referees for having jeopardized this collaboration with the group of which you are a member.
Now we are getting closer. I could have told you already before you lost this collaboration which you desired so much, that whenever string theory related particle physicists get to evaluate QFT projects they would wreck that bit of reasonable research which still exists in Germany because their horizon is extremely limited.
I was a student of Harry Lehmann who together with Kurt Symanzik and Wolfhard Zimmermann, through there discovery of how one can even in the presence of that inexorable vacuum polarization (see the answer to Eugene’s question) extract pure particles and their scattering matrix (lehmann received the prestigious Heinemann prize), brought a bit of brilliance to postwar German particle physics.
Two years before Harry Lehmann died I had a conversation (on one of my frequent visits to Hamburg) with him which is engraved in my memory. He asked me what is happening in Berlin, why does such an important position at the Humboldt u]University go to a string theoretician, an area whose contribution to physics was smaller than any pre-assigned epsilon. I told him that neither I nor my colleague Robert Schrader had any say in this; nobody ever asked us. He thought this was the beginning of a downpath in German particle physics.
Once I attended a seminar at the HU given by Sidney Coleman. Maybe younger people do not know, but Sid was the critical conscious of particle physics, a worthy successor of Pauli in keeping particle physics lively and healthy. He looked a bit frail and his behavior was changed from what I remembered from earlier encounters. What really shocked me (just because it did not fit his earlier critical image at all) was that he supported string theory. Later Robert Schrader told me that Sid was unable to continue his critical role because he suffered from an early onset of the Alzheimer disease. Only then I understood in retrospect why things had changed so much.
Well, Urs, I am not surprized that things go that way. Though I cannot hep you, I am sure that now, after the more QFT part of your joint venture was thrown out, you will see certain things my way.
Aaron, here is one last attempt to get you away from that proximity of Mentos.
The first derivation of the Hawking radiation in a collapsing star was done by Fredenhagen and Haag using the setting of AQFT (it was the problem which Hawking would have liked to solve, but as a result of conceptual complications he had to settle for the stationary case, see the book of Wald on this subject). I could continue this list, and in addition I already told you that everything which QFT can do AQFT can also do because the first one is included in the second. But there is a whole list of results which you can only obtain from AQFT because the inclusion is strict (not an equality). In my earlier blogs I have mentioned some of these results.
What I however cannot do is provide a vaccine against the string virus which like the bird flew virus is absolutly deadly, but it only kills the mind and the head is then filled with strings.
…remember I said that without localization you cannot talk about QFT…
Er…hello? I have been agreeing with you about that.
“I already told you that everything which QFT can do AQFT can also do because the first one is included in the second.”
Provided, of course, one uses a suitably narrow definition of “QFT,” which excludes theories with degenerate vacua, theories with nontrivial WIlsonian RG behavior, supersymmetric theories, TQFTs, …
In short, throw out every development in “QFT” of the past 30 years, and AQFT has you covered.
Peter Woit, for instance, will be disappointed to learn that his beloved Chern-Simons Theory is not a QFT.
Mentos, you seem to honestly think that my role here is to please the owner of this weblog, well at least you are a honst guy.
“The SU(3) confinement comes in at the tricategorical level when one is forced to break the Mac Lane pentagon and use premonoidal structures.”
” Sid was unable to continue his critical role because he suffered from an early onset of the Alzheimer disease.”
Guards! Guards! Major crackpot invasion in the Woit sector!
Sid was unable to continue his critical role because he suffered from an early onset of the Alzheimer disease.
He could have just been faking it, having got tired of arguing against Superstring theory.
“The first derivation of the Hawking radiation of a collapsing star was done by Fredenhagen and Haag using the setting of AQFT. It was the problem Hawking would have liked to solve…”
Bert, why is this a better calculation than Hawking’s? I assume you mean the paper: “On the derivation of the Hawking radiation associated with the formation of a black hole”, Commum. Math. Phys. 127, p273 (1990).
But this comes quite a bit later than Hawking’s so they did’nt do it first. I have not seen this but the archive of Comm. Math. Phys. is available free online, I think at “projecteuclid.com”, so will look it up out of curiosity.
Correction, it is “projecteuclid.org” for anyone interested in the archived math journals there.
Chris: the calculation in the nonstationary situation is conceptually much more demanding than that in the stationary envirement of the Schwarzschild spacetime and it is not surprizing that this came quite a bit later.
I did in no way intend to build up a case AQFT against Hawking. I only mentioned this in connection with the ineredicable prejudice which affects some participants of this blog. This serious mental incapacity seems to affect mostly blog contributers who permitted there mind to be run by string theory and I certainly did not have you in mind.
When I entered this blog in April I had the illusion that one can change prejudices by rational arguments and facts, but now I realize that there are limitations. It is extremely difficult to argue with people who desperately insist to hang on some unfortunate premature terminology (before the time when the use of big Latin Latters became popular) and insist to take it litterally and defend their physical confusion-creating semantics with claws and teeth (a new breed of string millenium fundamentalists.)
Sorry, it was Stevem; but it could have been also Chris.
Eugene,
in order to enjoy some distraction, let me return to an old blog in which you surprized me by your familiarity with the Coester-Polyzou relativistic particle theory of “direct particle interaction”. I now understand the origin of your recent question about the paradigmatic relation relativistic QM—-QFT. I do not think that there are many people in this weblog who really know about the existence of a relativistic multiparticle theory (without the property of being “second quantize representable”) which fulfills all the requirements one can formulate in terms of pure particle concepts (including the very nontrivial cluster factorization). But you probably agree with me that it is not what we consider as “fundamental (I do not mean the hegemonic string theory interpretation of this word)” since it lacks vacuum polarization (although you could think of manufacturing something which approximates this by adding channel couplings between particle states with different particle number). But I think that even you would not try to understand the Lambshift or the cosmological vacuum problem (involving the energy-momentum tensor) in such a framework; you would rather make this big paradigmatic shift into QFT, would’t you?
I looked at some of these other approaches you mentioned, but I have the intense impression (I don’t have the time ti make the necessary lengthy calculations) that those fail precisely on this cluster issue. With other words I think that any relativistic particle theory has to look like C-P + more complicated channel couplings.
I do appreciate Bert’s “unorthodox” perspectives on physics (in comparison to today’s “string” orthodoxy). I’ve been looking at AQFT on and off over the last two decades or so, though I don’t have a complete understanding of it.
I bought into the string hype back in the mid 1980’s when I was young and impressionable. In recent years, I was also still largely a string supporter until all that anthropic silliness started. (Though I could change my mind again).
“I do appreciate Bert’s “unorthodox” perspectives on physics (in comparison to today’s “string” orthodoxy).”
That’s “string,” in the same sense Kea used above, shorthand for standard QFT, as understood by 99% of contemporary high energy theorists?
Wow, the level of activity here is quite something. It’s a holiday weekend so I’ve been very busy not working. Even if I was working and near a computer I don’t think I’d have time to carefully read everything posted here, much less figure out how to properly moderate it. Discussion has certainly gotten off-topic, but at least it’s interesting…
A couple quick comments of my own:
About the idea that I’m the prophet and promoter of AQFT: this is silly. Like many serious ideas out there about how to make progress on QFT, there are things about AQFT I find interesting, others I’m not enthusiastic about. I’m no expert on the subject, but happy to learn more about it from those who are. It’s not obviously the best way forward, but then again nobody has an obviously best way forward at this point. It’s a research program that has been pursued for many years by a very serious group of people, any such alternative program deserves attention these days.
About Sidney Coleman. His health problems are relatively recent, and undoubtedly keep him from commenting now on the current state of the subject, which is a loss for the field. By the way, at the party for Wilczek held here in New York last month his wife Betsy told me that they often see Coleman and he is in better shape than it seemed at the recent conference in his honor, something I was glad to hear.
I don’t know exactly what Coleman’s attitude towards string theory was during the 80s and 90s, as far as I know he made no public comment on it, but also chose not to work on string theory problems. All this was long before he ran into health problems.
Mentos,
Yes.
Whether AQFT or any other alternative can replace it in the near future, I doubt it. What would impress me would be something which could calculate the n-loop m-point function (n neq m, in general) in a few lines, without having to crank out zillions of feynman diagrams.
“What would impress me would be something which could calculate the n-loop m-point function (n neq m, in general) in a few lines, without having to crank out zillions of feynman diagrams.”
Then I suppose you are intrigued by the Witten/Svrcek/Cachazo/Spradlin/Volovich/… twistor-inspired reformulation of (super)Yang-Mills perturbation theory.
I think the goals of AQFT (or any other attempt to give a rigorous account of QFT) do not include an efficient reformulation of perturbation theory.