Bert Schroer has a new version of his paper that was discussed here earlier this year, now with the amended title String theory and the crisis in particle physics (a Samizdat on particle physics). He claims that the version reflects a change in viewpoint due to his participation in this and other weblogs, and I believe he would like the opportunity to discuss this further here. There’s also a posting about this at the weblog of Risto Raitio.

**Update**: Schroer, agreeing with his critics that his paper had too many typos, has sent me a corrected version, which is available here, for use until the arXiv version gets updated. He also agrees that an “s” should be a “z” in Samizdat…

**Update**: Schroer has a new paper out, which contains a review of AQFT and a discussion of light-front holography, with further comments on the relation to the Maldacena conjecture.

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Active or passive, the bottom line seems to be that you allow to compute observables that are not diffeomorphism invariant, where everyone else says only diffeomorphism invariant observables are physical.

I tried to be very careful about diffeomorphism covariance and diffeomorphism invariance. The first property is the local covariance of Einstein and its very non-trivial recent adaptation to QFT (necessitating a very novel and surprising generalization of being forced to think about all globally hyperbolic manifolds at the same time even if for defining just one QFT in the new sense). Once this problem was solved the second problem of invariance (background independence) follows almost without additional sweat by converting the metric tensor into a quantum field (you only have to accept that by cohomological descend you get the operator Einstein-Hilbert like equations similar to the way you argue about the Maxwell operator equations).

It would be helpful if questions could be limited to things which are not carefully explained in the papers.

All I’m asking is for a clear explanation of why, if you allow to compute non-diffeomorphism-invariant observables, all but 2 ( d(d-3)/2 in d dimensions) of the components of the metric tensor still decouple.

I don’t see why the situation is any different from a gauge theory where you allow to compute non-gauge-invariant observables.

Anon

the remark about 4 constraints versus 6 propagating components was made in connection with a well-defined Cauchy problem for the classical Maxwell-Einstein equation. The purpose of that discussion was not to make analogies with gauge theories but rather to argue for the active interpretation of diffeomorphisms (in contrast to the passive interpretations in genuine gauge theories). This is basically the upshot of the Einstein hole argument

Norton, J.D.: The hole argument. Stanford Encyclopedia of Philosophy, http: //plato. stanford.edu/

entries/spacetime–holearg

The result is quite revolutionary recognition that The principle of general covariance forces one to regard spacetime points simultaneously as members of several, locally diffeomorphic spacetimes. It is rather the relations between distinguished events

that have a physical interpretation.

The adaptation of this principle to QFT (a very nontrivial step) led Brunetti, Fredenhagen and Verch to their paradigmatic change in the definition of what constitutes a QFT.

Concerning the analogy to gauge theory the following remarks are in order. The zero mass finite helicity class of Wigner representations permits to introduce pointlike field strength. For h=2 the field strength is a linear version of the Lorentzian Riemann tensor. But as for h=1 one cannot write the Wigner inner product in terms of an integral over a local density in these Field strength and last not least it is not possible to formulate ultraviolet-tame (or better “finite parametric classes” of) interactions with field strength, rather one needs “potentials” (for h=2 the analog of the metric tensor). Classically this is not a problem at all. Although I have not done the calculations (perhaps one finds this in the literature) I believe that the connection between the potentials and the field strength for the genral helicity case can be encoded into a gauge principle. For the QT the conceptual situation is very different since there is the unitarity (positivity) requirement. There are two ways to handle this problem: either you convince yourself that although pointlike-localized potentials are not possible, there are string-localized potentials (this is the way Mund, Yngverson and I propose in our joint work). In that case you face the new conceptual problem to adapt the Epstein-Glaser iteration to string-like fields. The standard way is to insist in pointlike potentials and forget about QT (the Hilbert space) in intermediate steps hoping that by some ghost extension with cohomological properties you can use all the advantages of the standard pointlike formalism and still return to physics at the end. This has worked for h=1. Whether it does so for interactions involving h-2 is the subject of present investigation. In such an approach the analogy to gauge theory in Minkowski spacetime is perfect.

This analogy is however obfuscated since GR requires an active interpretation of diffeomorphisms in the above sense of the new BFV notion of QFT demanding that the algebraic substrate of QFT is “living” on all isometrically equivalent spacetimes simultaneously! (this includes submanifolds of two given global ones which are isometric i.e. local diffeomorphism).

I ask you Anon, what is for you an observable in gravity? Is it a gauge invariant in the above setting of an h=2 particle theory? In that case the potential (metric tensor) would be gauge-dependent but the field strength (Lorenztian Riemann tensor) is gauge invariant hence observable in the sense of gauge interpretation. But according to the (always active) diff-interpretation the latter is presumably not observable.

I can of course answer your question in one sentence, but that answer would cause more conceptional damage than enlightenment.

Peter,

my response to Anon got again stuck in the spam filter

It seems to me that AQFT practitioners find the Lagrangian path integral method of QFT to be unreasonably successful. Well, if AQFT is indeed the way to do QFT, then the success (and limitations) of the Lagrangian path integral should be explainable within AQFT. Presumably the successful Lagrangian formulations are taking care automatically somehow the setting up of the unique hyperfinite type III_1 factor algebra, the modular positioning of a finite number of abstract monades, etc., etc.

Since “For higher than 3 spacetime dimensions the presently known descriptions still look somewhat concocted”, but we have highly successful 4 spacetime dimensions Lagrangian path-integral QFTs, presumably making the connection suggested above will enable us to see what a not-concocted example looks like.

IMO, this kind of demonstration is the only way to convince a large number of physicists that AQFT is actually useful.

I’m sorry to say that I find Schroer’s long and rambling responses impossible to follow. Peter Woit clearly thinks that Schroer’s work is of such importance that he posts something on his blog, every time Schroer posts or revises one of his papers on the arXivs. Since Peter is such a clear writer, maybe he could boost the physics content of his blog by posting a summary/explanation of this work by Schroer and collaborators.

It would be a great public service.

arun,

this is all deja-vue, we (maybe without you) discussed this in this blog in April/May.

From the phrasing of your question I sense that it is declamatorical, but nevertheless I will ignore this and answer it to the best of my knowledge

1) Funct5ional integrals are extremely limited; you cannot treat QFT in CST in this setting (that is why in mathematical physics presentations like those of Wald you wan’t find it) and as far as I remember i have explained why this is not possible. Question: do you think the QFT in CFT is not interesting?

2) you cannot solve any 2-dim. massive system, even in case it admits a Lagrangian presentation. Most of the systems do not even have a Lagrangian; they are characterized by a crossing symmetric unitary S-matrix (just like string theory, except that it is not metaphoric). And for chiral CFT even you would not use Lagrangians but rather the methods of AQFT (represent6ation theory of observable algebras, that is what AQFT is if you do not artificially limit it to 2 dimensions).

3) Most free fields cannot be characterized as being of Euler-Lagrange type (above spin 2 I do not know any) but their use is perfectly legitimate in causal perturbation theory (where you only use interaction polynomials in free fields) although the most important ones can

4) The Lagrangian approach where it works in your sense is an artistic device: you write something on paper in good faith, develop it in perturbation theory, see that it does not make sense (the integrals diverge) and use your hindsight and good physical sense to repair it (renormalization) and at the end you realize that you have a perfectly reasonable result which however (if you try to check whether it its into your original functional representation) fails to satisfies your original functional representation. There is nothing wrong with artistic arguments as long as one keeps some awareness about their nature. Imagine that Heisenberg would not have come up with QM in 1925. Then we would have presumable have learned clever artistic tricks (using our physical hindsight) to go somewhat beyond the H-atom and it would have been quite disastrous if we would have been satisfied with artistic explanation.

The artistic device of functional integrals (in QM it is mathematical) works because the whole idea of locality is so strongly incorporated that you cannot fail to extract the right result (if you have a good covariant formalism which was not available before 1949).

Anon

neither Peter nor I are willing to serve as your Nanny. Do you really expect that these subtle recent developments in QFT can be explained in a weblog. The only thing one can do is cite the literature and make some guiding remarks and that is what I did. The rest you have to do; there is nothing in my commentaries which is not in detail explained in the existing literature. But if metaphoric tales are more to your taste than autonomous mathematical physics, then you really should stay with string theory or something alike and join one of those globalized communities where you find people who are very experienced in telling you nice-sounding metaphors which have a higher entertainment value than what you can get from me.

OK, so give me a reference (e-print and page-number) where it is explained why all but d(d-3)/2 components of the metric decouple, despite the fact that you allow to compute non-diffeomorphism-invariant observables.

This isn’t a complicated question and, if it is answered in the literature, you could avoid writing paragraphs and paragraphs of gibberish by giving a detailed reference to the answer.

Anon,

That the propagation of the metric tensor according to the E-H equation leads to 4 non-propagating constraints (Cauchy problem) is in most texts on classical gravity (e.g. Wald’s book). Its use in connection with establishing the active interpretation of diffeomorphisms (in contradistinction to the passive interpretation of gauge transformation) you find in the introduction of

Fredenhagen, K., Haag, R.: Generally covariant quantum field theory and scaling limits. Commun. Math. Phys. 108, 91 (1987)

and the nontrivial adaptation to quantum matter in

http://br.arxiv.org/abs/math-ph/0112041

and a formal argument that background independence (diffeomorphism invariance) follows from diffeomorphism covariance automatically upon quantizing the metric potential (and fulfilling the operator form of the E-H equations) in

http://br.arxiv.org/abs/gr-qc/0603079

It would not be advisable to eliminate some g components (in gauge theory you would not work with the Coulomb gauge either), you rather prefer to find a BRST ghost extension and encode the elimination job into the BRST cohomological descend (or if you, like I, do not like ghosts in intermediate steps one works with string-localized metric potentials). Ergo in QFT in CST you may eliminate, in a full QFT including gravity you never do that.

The length of my previous contribution resulted only from trying to point out some subtle differences between gauge theories and background independent quantum gravitation since I had the impression you looked for a more intimate contact than a vague analogy.

10-4=6, not 2. I am asking how you propose to get from 10 down to 2.

Brunetti and Fredenhagen (gr-qc/0603079) do not address this issue. In fact, except for the very last page of their paper, they do not address the quantization of the gravitational field at all.

Nota, when they do turn to some tentative remarks on the quantization of the gravitational field, they say

“A possible obstruction could be that locally the cohomology of the BRST operator is trivial, corresponding to the absense of local observables in quantum gravity. Another problem is the fact that the theory is not renormalizable by power counting. Thus the theory will have the status of an effective theory.”

And they do not try to depart from the usual condition that only diffeomorphism-invariant (BRST-invariant) observables are to be computed.

You, on the other hand, say you can compute local, non-diffeomorphism-invariant, observables.

So, I’ll ask again, how is that compatible with the decoupling of all but d(d-3)/2 components of the metric? A reference to one of your papers, where this is explained (a page number, please), will suffice.

Anon

in any theory based on BRST you compute of course also non-diffeomorphism invariant objects, it is only after the cohomological descend that you find the physical objects (diffeomorphism covariance passes to diffeomorphism invariance). The elimination procedure you seem to like so much is a special treatment of constraints by solving them explicitly. This in QED would amount to the (non-covariant) Coulomb gauge approach and I have never seen any honest computation in that setting. The Arnowitt-Deser-Misner Hamiltonian framework faces this problems of constraints (but I know to little about it in order to say that what they do is an explicit elimination of components).

Could I ask you what is your interest in such horrible elimination procedures (I never advocated them, it is only in the famous classical “hole argument” that one implicitly uses them but you seem to have a monomanic interest in them).

The two helicity degrees of freedom are only manifest if you use our string-localized matrix g-potentials (in the BRST treatment they are masked through the presence of the BRST ghosts). String-localization is explained in

http://br.arxiv.org/abs/math-ph/0511042

We meanwhile know their two-point functions but we are still struggling with the conceptual problems of formulating a perturbation theory for string-localized fields.

The remark of B-F about the effective theory means that it is diff-invariant but cannot be used up to the Planck length (since the number of parameters increase with perturbative order).

Brunetti and Fredenhagen:

“A possible obstruction could be that locally the cohomology of the BRST operator is trivial, corresponding to the absense of local observables in quantum gravity.”

Are you saying that they (and not just they) are wrong, and that there are BRST-invariant local observables?

Or are you saying that you can extract sensible physics (e.g. achieve the desired decoupling) while working with BRST-non-invariant local observables?

In either case, could you give me a reference for the precise statement?

Anon,

I have no opinion whether the diff-invariant theory possesses still localizable flesh (localizable monades) or whether it only contains topological (or combinatorial according to once point of view) “bones” (type II_1 algebras, the ones Vaughn Jones uses in subfactor theory leading to the Jones polynomial. And I do not advocate a physical interpretation of BRST non-invariant objects.

The reason I am so much interested in the string-localized formulation of metric potentials is precisely to clarify this issue. In that case the g’s do act in the physical Hilbert space and the only thing which could go wrong is that they generate semiinfinitely extended spacelike strings which have horrible domain properties (e.g. create infinite energy states from finite energy ones). If this is not the case then the theory has in addition to E-H diff-invariant state other states in its Hilbert space. I am pretty confident that this picture is an alternative formulation to ordinary gauge theory but I would not make any bet in the gravitational case. String localized fields are not local observables in the strict sense, but neither are the highly useful fermion or anyon fields.

In any case if that “bone”scenario of B-F turns out to be true then the Maldecena conjecture in the sense of an AdS–CFT correspondence could not be true apriori, even if you go along with that weird idea that in addition to the known holography there is another gravitational one.

I told you that I believe that B-F are working on a BRST formulation, so we have to wait and see.

Doesn’t the AQFT approach rely on the existence of BRST-invariant local observables?

What can you hope to achieve in that approach, if there aren’t any (which is the conventional wisdom)?

“In any case if that ‘bone’ scenario of B-F turns out to be true then the Maldecena conjecture in the sense of an AdS–CFT correspondence could not be true apriori …”

To the contrary: the Maldacena conjecture relies on the fact that there are no local BRST-invariant observables in quantum gravity. If there were, then the Maldacena conjecture would most certainly be false. But I don’t see the point of discussing the Maldacena conjecture with you. There are plenty of experts on it, who could do that.

“I told you that I believe that B-F are working on a BRST formulation, so we have to wait and see.”

Above, you made some very grand claims for the AQFT approach to quantum gravity. In the absence of a proof that the conventional wisdom, about the absence of BRST-invariant local observables, is wrong, I think those claims are overblown.

The idea to use a BRST approach is not an idea emanating from AQFT, rather certain formal analogies to gauge theory (but not a naive identification of formalism). Since it has not been done it seems reasonable to do it before entering the blue yonder of amok running speculations. There are also other conservative ideas to formulate perturbative QG which are expected to lead to an operator version of E-H-like field equations on suitably determined states.

The nice aspect of the B-F argument (only on the level of string theoretic rigor one would call it a theorem) is that any implementation which achieves that is background independent i.e. background independence is a free ride and the hard work was done before while establishing diffeomorphism-covariance.

I appreciate your admission that you do not know much about the AdS–CFT correspondence and the Maldacena conjecture (your statement about a relation between BRST and the M-conjecture is rubbish) and that you prefer to leave that matter to experts.

Perturbative quantum gravity was studied exhaustively, starting with de Wit and Feynman, and followed by many others. The BRST approach to it was introduced by de Wit. Stating results found many decades ago is not “blue yonder of amok running speculations.”

As to Brunetti and Fredenhagen, as I said, the only “quantum gravity” in their paper is a page of speculations at the end. Most of the paper is about quantum field theory in curved spacetimes, a subject that was well developed when Birrell and Davies wrote their book. In their AQFT approach, B&F miss (gloss over?) almost all of the subtleties that make the subject an interesting one.

“your statement about a relation between BRST and the M-conjecture is rubbish”

No, my statement is correct.

All I’m saying is that, if you want to discuss AdS/CFT, you should do it with someone like Distler (who seems to have written quite an acute summary of Rehren’s proposal), not with me.

Egbert,

the concept of internal symmetries was since its inception (the SU(2)isospin of nuclear physics introduced by Heisenberg) one of the most mysterious proposals. Whereas it is natural to accept spacetime symmetries (since they accompanied us in the classical setting since the time of Newton) the understanding of internal symmetries is a mysterious concept in QT (in classical physics you can only get them by reading back QT concepts into classical physics i.e. they are classically unnatural). This problem was finally solved in the work of Doplicher, Haag and Roberts during 1970-1990

There idea was to abstract internal symmetries by taking a dichotomic view about QFT: local observable algebra

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continuation

The representation structure of interests coming from the DHR theory in low dimensions is richer, in this case the observable structure leads to representation sectors which carry a representation of the braid group (a generalization of the symmetric group) and instead of the field algebra you find something which does not permit a clear-cut separation into inner and outer (spacetime) symmetries.

Suppose now that you have gone through this analysis in a higher dimensional QFT and you compactify certain spatial dimensions and make them small. Do the correlation functions of such a QFT in the limit approach those of a QFT with smaller spacetime dimensions and a larger symmetry? Of course not. The observable algebra on such a brane becomes awkward (it develops those disagreeable physical properties which Rehren mentioned in his discussion with Distler) and its representation theory of physical interest will not develop new sectors which come from the small spatial coordinates.

What people mean by KK is a Fourier decomposition in the small coordinates in the classical theory and retaining only the lowest component; it is only there (i.e. before quantization) that KK works but it does so in a completely childish way.

What Lunsford attributed to Pauli is something slightly different (this was the reason for my interest). If I understood it correctly it says that you cannot apply KK to a higher dimensional Einstein-Hilbert theory (diff-covariant) into a lower dimensional E-H (lower dim. diff-covariance) and remaining gauge part. This sound very plausible, but it seems to be a classical statement.

My complete contribution to KK is now available at

http://kaluza-klein.blogspot.com/

I am mentioning this here because there was a mistake in my previous blog here which now has been corrected.