Bert Schroer has sent me a very long and interesting comment for posting here. I’ve put it into a separate web-page. It includes both a lot of history and many different ideas. Unfortunately I don’t have the time right now to write much about it in response, but just will make one point about the part that he explicitly addresses to me.
Schroer claims that geometric methods in QFT have so far only been useful in dealing with free fields in a fixed background gauge field or metric. This is largely, but not completely true. Most of our reasons for believing the standard model are based on perturbative quantization of gauge fields, and for this it’s true that geometrical methods are not strictly necessary. But for QCD, we need a non-perturbative quantization of the gauge fields, and here lattice QCD is the best we’ve got. It is based upon discretizing a geometrically formulated path integral, preserving as much of the gauge field geometry as possible. My own guess is that there is still a lot to be learned about non-perturbative quantization of gauge fields, based upon the geometrical formulation of the problem given by the path integral approach and I have been working on speculative ideas of how to do this (some of which even involve gerbes and algebraic geometry…). This is still work in progress, maybe I’ll someday find it really can’t work, but for now I’m quite optimistic.
There’s also a new survey paper on QFT from Fredenhagen, Rehren and Seiler. The authors discuss the current state of understanding of QFT, with some points of overlap with Schroer. Like Schroer, they also discuss string theory in detail, and are critical of the inability of the string theory research program to come up with precise statements about what the theory is supposed to be. They are however, much less forceful in their criticisms than Schroer.
And at the end the finger points to Weinberg. Indeed he left the field jumping into the astrophysics wagon. And sure it seems to favour the “effective” approach to QFT, but this is a blame on QFT in general, not on AQFT only. Hmm.
Schroer’s letter is fascinating stuff. Aside from the geometric issue (or maybe you regard it as part of the same story) is his criticism of functional integral methods. Frankly I think the rush to use path integrals here there and everywhere has been a downer for particle physics. If algebraic methods can do the calculations and be rigorous too, that would seem an advantage.
These attempts to return physics to the 1930s or even further to the past are completely pathetic. Path integrals have rightfully become the dominant way to describe physics of quantum fields and their strength turned out to be even more obvious in theories with non-Abelian gauge symmetries (Yang-Mills symmetries much like conformal symmetries on the worldsheet etc. – we can’t really live without these things today). It would be extremely awkward to do physics without the FP-ghosts and path integral in this case which is very important today.
In the same way, it is impossible to undo the insights of the Renormalization Group from the 1970s – considered by many wise people to be the most important insights in physics in the second half of the 20th century. Quantum field theory is an important stuff. Some of them may be UV complete but even if they are UV complete, they should be used as effective field theories describing physics at some energy scale. Eventually we know that all QFTs break down at the Planck scale or earlier.
Complaining about QFTs being used as effective QFTs with respect to an energy scale, and complaining about the use of path integrals in physics of 2006 is a soft type of crackpotism that only differs from crackpotism that rejects the theory of relativity by a factor of 3 (counted by the number of years of developments in physics that the crackpots decide to ignore).
The whole concept of AQFT has become an unproductive line of reasoning because the basic philosophy of AQFT contradicts many things that have been demonstrably successful in the last 40 years: cutoff dependence and effectivity (non-exactness, in a sense) of QFTs, path integrals methods including ghosts. These methods had nothing to say about physics above 2 (or 3?) spacetime dimensions and the success of similar reasoning in 2D should be thought of as the maximum that can ever be extracted from this type of reasoning.
I think that Schroer has a number of interesting things to say about the problems with path integrals, and to a large extent, I agree with him. I too learned from Roman Jackiw to be very careful when dealing with functional integrals. However, I think that functional integrals can be fabulously useful tools. Their weakness is that they can easily fool us into thinking that we are working with mathematical well-defined and unambiguous objects, when in fact, we are not.
People have, of course, tried to fix functional integration’s problems. This would mean placing it on a firm mathematica footing, most probably in algebraic geometry. Then the usual “functional derivatives” would be reinterpreted as functors, whose representations could be studied, and so on. The problem with this idea is that it’s proven to be impossible thus far (although people I know are still working on it). A rigorous formulation of path integrals would automatically entail a rigorous formulation of renormalization–already a tremendously difficult task. And the very fact that these problems are so difficult is a significant reason that many people, such as myself, choose not to work on them.
I am somewhat mystified by Schroer’s idea of what “algebraic” QFT is. With “axiomatic” QFT, it’s reasonably clear what that means. And note that in axiomatic QFT, one cannot calculate any cross-sections. Schroer says one can do this in algebraic QFT, by means of ordinary Feynman diagram perturbation theory. However, such perturbation theory cannot be used in an “axiomatic” approach, because the resulting power series are asymptotic, and no rigorous estimates of their rates of divergence exist. In other words, this is an uncontrolled approximation, something that cannot be sensibly used in an “axiomatic” framework.
So my question is: What exactly does Schroer’s “algebraic” QFT encompass? Where does it end? Many (maybe most) papers in QFT do not explicity make reference to any path integrals but merely use the perturbation series as a starting point. What disqualifies these papers as “algebraic” QFT?
I said above why I think few people work on axiomatic QFT–because the really interesting problems are prohibitively difficult. The tractable questions are just much less interesting. Blaming one stupid commenter who made a bad impression on Weinberg just seems bizarre, and I think Schroer does the physics community a disservice with this (essentially ad hominem) accusation.
Schroer: “Isn’t it ironic that string theorists accuse the AQFT of the use of gratuitous mathematics when they themselves are using structures like gerbes and algebraic geometry?”
Oh no! Doesn’t he even realise that rigorous non-Abelian localisation leads that way?
Just noticed gr-qc/0603079 by Brunetti and Fredenhagen.
I strongly agree with the following Schroer’s statement:
“…Lorentz covariance, quantum theory and the cluster property do not lead to QFT. .. These so-called “direct particle interaction” models fulfill the cluster-decomposition property yet they do not have a
representation in terms of a second quantization setting…With other words this claim by Weinberg … is incorrect”
There are few examples which do not fit within usual local QFT:
1. “direct interaction” theories mentioned by Schroer (see papers by W. N. Polyzou)
2. non-local QFT: M. I. Shirokov, “On relativistic nonlocal quantum field theory”, Int. J. Theor. Phys. 41 (2002), 1027
3. A beautiful series of papers by H. Kita in 1966-1973. The last paper in this series is “A realistic model of convergent quantum mechanics of interacting particles” Prog. Theor. Phys. 49 (1973), 1704 where you can pick up the other references if interested.
Dear Prof. Schroer:
In any case considerable progress to find analoga of those structures which one has in chiral theories (energy-momentum virasoro structure, current algebras) are well on their way in recent articles of Todorov et al.
In recent years the only papers of Todorov that I have read are his biographies of Einstein and Hilbert, and Heisenberg.
The mathematical problem of generalizing the Virasoro algebra to higher dimensions was settled a decade ago. The diffeomorphism algebra in N dimensions has two non-trivial cocycles by the module dual to closed (N-1)-forms, found by Rao and Moody (Comm. Math. Phys. 159 (1994) 239–264) and myself (J. Phys. A 25 (1992) 1177–1184), respectively. It may be worth pointing out that Bob Moody is somewhat famous, e.g. for the codiscovery of Kac-Moody algebras. Note that when N=1, a closed 0-form is a constant function, so the Virasoro extension is central when N=1, but not otherwise. In fact, all extensions of the diffeomorphism algebra were classified by Askar Dzhumadildaev (Z. Phys. C 72 (1996) 509–517); I spent a considerable time to understand his paper, which resulted in the review math-ph/0002016.
It is of course widely recognized that the standard axioms in LQP, in particular, the strong emphasis on the Poincare algebra, cannot hold in quantum gravity. The Poincare group has two properties in flat space:
1. it is the group of isometries.
2. it is the spacetime group which acts on the physical fields alone.
It is unclear to me if the isometry property plays any useful role in QG; it seems more appropriate for QFT in curved space, which is another problem. The second property in QG is subsumed by the spacetime diffeomorphism group. This is nothing but background independence, which is widely regarded to be the key conceptual lesson from GR.
A digression on LQG. I agree with the LQG people that manifest background independence (the formalism should not depend on any reference background structure, be it a flat metric or a preferred foliation) is necessary in QG, but I don’t think that the LQG implements this deep idea correctly. In particular, LQG methods apparently fail when applied to the harmonic oscillator (hep-th/0409182), which in my opinion is a show-stopper. There are theorems which show that background indendence is incompatible with Fock quantization (in 1D, the only Virasoro rep with c=0 is the trivial one), but a no-go theorem only shows which axioms must be relaxed (here: anomaly freedom).
The generalization of the Virasoro algebra to 4D is thus the branch of mathematics which deals with background independence, quantum mechanically. The key insight from its representation theory, initiated in the Rao-Moody paper above and understood geometrically in math-ph/9810003, is that all fields must be expanded in a Taylor series around an operator-valued curve prior to quantization. This operator-valued curve, which can be identified as the observer’s trajectory in spacetime, plays a crucial role; e.g., it enters into the relevant extensions. Without it, it is impossible to deal with gauge or diffeomorphism symmetry on a genuinely quantum level.
The paper I mentioned in my previous comment was a first attempt to reformulate QFT to incorporate this insight. That paper contains (at least) one error and one omission already for the harmonic oscillator: there is an overcounting, and no inner product was defined. These problems have now been solved.
I want to say, sorry for overstaying my welcome, since I bowed out, but in light of the fudged WMAP data, to see Schroer and Larsson and others really talk physics – it does my heart some sorely needed good!
Thanks again, Peter.
Lubos concludes his latest post: “If you don’t care about the details of the Yukawa coupling, the content of the text above can be replaced by GoogleFights. ;-)”
which links to a Google comparison between his hits and those of Peter, naturally fixed…
Curiously, when Edward Witten and Lubos Motl are the contenders, Lubos still wins the fight!
Watch out Edward, you’re not the king of string anymore!
Not Even Wrong wipes out the Reference Frame.
I just now typed a 4 page letter answering most of the questions, but as a result of a wrong klick on my laptop I lost almost two hours of work; I am too frustrated to continue, maybe I will get to the sceen on the weekend again.
Schroer ends his response with
“Isn’t it ironic that string theorists accuse the AQFT of the use of gratitious mathematics when they themselves are
using structures like gerbes and algebraic geometry? I cannot think of more useless mathematical structures for
quantum theory (which deals with operators and states) than those. Hypocricy? Hubris? Double standards?”
Like it or not, physics is based on advanced mathematics. While I believe that string theory is based on faulty physics, one cannot fault the mathematics. Robert Hermann called it “experimental mathematics”, learn some new mathematics and see if you can apply it to physics. Who knows what will work until we try it? I beleve the problem with AQFT is paradigm paralysis, keep trying the same thing. The impossibility of combining the Poincare group with internal symmetries should have led to a rejection of the Poincare group in favor of the de Sitter group or something bigger. I use U(3,2).
Now that I’m back, I’ve turned back on comments here in case anyone wants to continue this discussion.