Argument Against LQG

[BenTheMan]

From the arXiv: http://arxiv.org/abs/0708.1721

 

Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?

 

One of the celebrated results of Loop Quantum Gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area and volume operators. This is an indication that Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present, because the afore mentioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli's partial and complete observables provides a possible mechanism for turning a non gauge invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that ``fundamental discreteness at Planck scale in LQG'' is an empty statement. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.

 

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The big realization seems to be in this paragraph:

 

In summary the geometrical operators play a very important role for the structure and further developement of LQG. However, so far the discreteness of their spectrum is a result at the kinematical level only. With ’kinematical level’ we mean that the geometrical operators defined so far are gauge dependent, i.e. they are not invariant under spacetime difeomorphisms. True observables have to be gauge independent, in the canonical formalism such observables are known as Dirac observables. Physical measurements are described by Dirac observables and not by kinematical observables.

 

In other words, the analogue of gauge invariance in GR is space-time diffeomorphisms (coordinate transformations). If one has a discrete spectrum for an area or length or volume operator, the eigenvalues are not gauge invariant---they depend on the choice of coordinates. Because observables must be gauge invariant, the discreteness of these operators is not an observable.

[Martin]

I'm glad to see you spotted the Dittrich Thiemann paper. This is an exciting development which will cause a big stir!

 

I think I mentioned that the discrete spectra result was in canonical LQG which people worked on a lot in the 1990s

whereas what people in the community have tended to work on more recently is the SPINFOAM approach, and group field theory, (and then there is more distant relatives like Reuter QEG and Loll CDT which are drawing attention). and these do NOT have the discreteness result!

Spinfoam and GFT do NOT duplicate the business of geometrical operators having discrete spectra.

It is questionable whether Reuter QEG has a minimal length. Loll and Ambjorn say flatly that CDT does not (or they did in 2005 and I dont think that has changed).

 

So I have been wondering about the discrete spectra result for quite some time.

 

What Dittrich and Thiemann say is IT COULD GO EITHER WAY. they point out that it has only been proved at a kinematic level and might or might not be provable at the real physical level (diffeo invariant states as you say)

 

I am looking forward to some lively action about this.

 

Rovelli, one of the co-inventors of the older canonical LQG approach, has been working only in Spinfoams for the past 3 years or so. It will be interesting to see what he says.

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If you are interested in my take----I happen to be a fan of Dittrich, one of the authors. I have read her papers and listened to talks etc over the past couple of years and have formed a very high opinion. She thinks deeply about things and does not move until she is sure.

 

Thiemann, of course, is one of the main authorities in LQG community. He is still quite young. But he has a 600 page book coming out next month. Cambridge University Press

 

"Modern Canonical Quantum General Relativity"

 

Thiemann has reformulated the old LQG in the past 2 or 3 years, with something called the Master Constraint program and a new approach called AQG (algebraic quantum gravity). I do not know if AGQ has the discrete spectra result of the older canonical version.

 

At the Zakopane QG school, and again at Loops 07, Thiemann was the speaker invited to give the main survey lectures about LQG proper (as opposed to spinfoam, CDT, QEG, Smolin's new approach etc).

 

So it is an interesting situation. The number of people working in the field seems to have roughly doubled since 2004. the field is totally in ferment. It is just beginning to hold an annual or semiannual conference. Large amounts of new research grant money have come in from ESF in the past couple of years. And the guy they get to give the official survey talks about LQG proper is Thiemann who has reinvented the approach and who is now challenging a major 1990s result (the discreteness)

 

Fun to watch

 

I still say you should read Reuter!

 

If you want a paper, the best paper to look at is this June arxiv posting:

http://arxiv.org/abs/0706.0174

It is absolutely great! maybe I should start a thread about it.

[BenTheMan]

What Dittrich and Thiemann say is IT COULD GO EITHER WAY. they point out that it has only been proved at a kinematic level and might or might not be provable at the real physical level (diffeo invariant states as you say)

 

Martin---

 

I cannot claim to have read and understood that paper (I am working on this one : http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.1651v1.pdf, which ALMOST scooped a project I am working on), but it seems that this scentence:

 

We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed.

 

says that the answer depends on the regularization scheme. This is pretty troubling, because one would hope that a fundamental theory didn't depend on the regularization scheme (like QFT answers do).

 

Further, it seems that the stronger statement:

 

This indicates that “fundamental discreteness at Planck scale in LQG” is an empty statement. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.

 

directly contradicts the quote from Smolin's ``Invitation'', no? (He claims that the discreteness is a concrete prediction of the theory.)

[Martin]

Hi Ben, I was curious about the paper you said you were reading because it almost scooped a project you were working on, so I fetched the abstract. I usually find the abstract more helpful than the full pdf text, when it's a paper I am not planning to read. Here it is in case anyone else is curious.

 

http://arxiv.org/abs/0707.1651

Local SU(5) Unification from the Heterotic String

W. Buchmüller, C. Lüdeling, J. Schmidt

38 pages

(Submitted on 11 Jul 2007)

 

"We construct a 6D supergravity theory which emerges as intermediate step in the compactification of the heterotic string to the supersymmetric standard model in four dimensions. The theory has N=2 supersymmetry and a gravitational sector with one tensor and two hypermultiplets in addition to the supergravity multiplet. Compactification to four dimensions occurs on a T^2/Z_2 orbifold which has two inequivalent pairs of fixed points with unbroken SU(5) and SU(2)xSU(4) symmetry, respectively. All gauge, gravitational and mixed anomalies are cancelled by the Green-Schwarz mechanism. The model has partial 6D gauge-Higgs unification. Two quark-lepton generations are localized at the SU(5) branes, the third family is composed of split bulk hypermultiplets. The top Yukawa coupling is given by the 6D gauge coupling, all other Yukawa couplings are generated by higher-dimensional operators at the SU(5) branes. The presence of the SU(2)xSU(4) brane breaks SU(5) and generates split gauge and Higgs multiplets with N=1 supersymmetry in four dimensions. The third generation is obtained from two split \bar{5}-plets and two split 10-plets, which together have the quantum numbers of one \bar{5}-plet and one 10-plet. This avoids unsuccessful SU(5) predictions for Yukawa couplings of ordinary 4D SU(5) grand unified theories."

 

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I am not sure you have the time or the will to learn about current developments in the main lines of research rivaling string. I think I've warned against going back to Smolin 2004 paper called "Invitation".

 

You seem very concerned to talk about the canonical LQG approach which essentially nobody AFAIK works on. It was a focus of attention back in the 1990s.

Just to take an example, Rovelli, Smolin, Ashtekar are the 3 most obvious picks for the "founders" of canonical LQG in the 1990s. None of them does canonical LQG.

Rovelli does spinfoam-----graviiton scattering, the classical limit, lately an improved spinfoam vertex

 

Smolin does spinfoam----unification with the various generations of the standard model of particle physics (getting matter out of the spinfoam model of spacetime geometry)

 

Ashtekar is doing loop quantum cosmology----analytical and numerical modeling the universe in many different cases, removing the singularity etc...

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If you know anyone who actually wants to get a realistic picture of the competition, and whose interest (like yours) is in unification, you might pass along this suggestion.

 

here's the Loops 07 program (I sampled Strings 07, there may be string folk who want to sample Loops 07)

http://www.matmor.unam.mx/eventos/loops07/program.html

 

and here is Smolin's talk about spinfoam+matter unification

slides:

http://www.matmor.unam.mx/eventos/loops07/talks/PL5/Smolin.pdf

audio:

http://www.matmor.unam.mx/eventos/loops07/talks/PL5/Smolin.mp3

 

the title of the talk is:

"Chiral excitations of quantum geometry as elementary particles"

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Again, the field has changed enormously in the past 2 or 3 years and simply because you may have heard Smolin's name and NOT heard of Reuter you should not necessarily take Smolin's research as representative of the main body of research being carried out.

 

Dittrich and Thiemann work is interesting but it bears on 1990s CANONICAL LQG which essentially nobody is working on AFAIK, this is the sector of work that has the result about discrete spectra of geometrical operators.

 

Canonical LQG (the Hamiltonian approach) is not very interesting to talk about, but I can only repeat that, when people get around to checking, the end result could go either way. If it turns out that canonical LQG does not have discrete spectra that would make it MORE LIKE the rest of non-string QG and, I would guess, facilitate convergence with the spinfoam approach and other approaches, like Reuter's.

 

So the Dittrich and Thiemann paper is a hopeful development, I think. But we will just have to see what comes of it.