# Dittrich and Thiemann challenge discreteness of LQG area etc operators!

#### marcus

Gold Member
Dearly Missed
This is very exciting. I have wondered about this merely as a spectator, because e.g. AFAIK the spinfoam formalism has not confirmed that about the geometric operators. What they say is that discrete spectrum HAS NOT BEEN PROVEN yet for the geometric operators, so it could go either way. Also I think there are versions of LQG which do not reproduce the discreteness result. This is sure to cause turmoil havoc and release a lot of creativity!

http://arxiv.org/abs/0708.1721
Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?
Bianca Dittrich, Thomas Thiemann
12 pages
(Submitted on 13 Aug 2007)

"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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#### Demystifier

2018 Award
For me, it is the most interesting result on LQG in last few years.
I have a great respect for scientists who are able to criticize their own theory that made them famous.

J

#### josh1

I have a great respect for scientists who are able to criticize their own theory that made them famous.
Yes, Demystifier, that's what makes them real scientists at heart.
What would've made them even better scientists is if theyd listened to the string community when they made basically the same points at the beginning of the lqg program!!! So much for DSR, quantum bounces and all the other unphysical ideas based on the false claims made by lqg people over the last two decades or so.

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#### Chronos

Gold Member
I agree to the extent that quantum spacetime is shaky. The most convincing evidence, IMO, is the 'unfuzziness' of distant [like quasars] v nearby objects at all frequencies. I have no doubt that particles 'commute' at discrete intervals, but spacetime does not appear to follow suit. Does that mean spacetime intervals are more 'fundamental' than particles? Possibly, but that is admittedly a wild guess. I think we are missing a 'background', not truly fundamental, but, more pervasive than we suspect.

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#### ccdantas

What would've made them even better scientists is if theyd listened to the string community when they made basically the same points at the beginning of the lqg program!!!
Would you point out the references which match the points now raised by Dittrich & Thiemann?

Listening is a always a good posture -- being a scientist or not. However, the point in question here refers to the fact that scientific research is mainly an activity that should develop with a good amount of freedom as well. And honesty. A good scientist should be prepared (and be deeply motivated to) scrutinize by him/herself his/her subject of investigation, and check for him/herself other proposals, arguments and results. Science proceeds in a healthy way when the same subject can be analyzed from different angles and possibilities exhausted as independently as possible.

Having said that, I am not sure how far the arguments raised by the authors in question exactly match the arguments raised by string theorists on the problems concerning the discreteness of the spectrum of geometrical operators in LQG. It would be interesting to offer a comparison of those arguments.

Christine

#### DisorderedBrain

I usually don't like the way Lubos express his points of view, but he indeed wrote about this in his site in 2004. The link is : http://motls.blogspot.com/2004/10/objections-to-loop-quantum-gravity.html

and here is the excerpt:

Clash with special relativity

Loop quantum gravity violates the rules of special relativity that must be valid for all local physical observations. Spin networks represent a new reincarnation of the 19th century idea of the luminiferous aether - environment whose entropy density is probably Planckian and that picks a priviliged reference frame. In other words, the very concept of a minimal distance (or area) is not compatible with the Lorentz contractions. The Lorentz invariance was the only real reason why Einstein had to find a new theory of gravity - Newton's gravitational laws were not compatible with his special relativity.

Despite claims about the background independence, loop quantum gravity does not respect even the special 1905 rules of Einstein; it is a non-relativistic theory. It conceptually belongs to the pre-1905 era and even if we imagine that loop quantum gravity has a realistic long-distance limit, loop quantum gravity has even less symmetries and nice properties than Newton's gravitational laws (which have an extra Galilean symmetry, and can also be written in a "background independent" way - and moreover, they allow us to calculate most of the observed gravitational effects well, unlike loop quantum gravity). It is a well-known fact that general relativity is called "general" because it has the same form for all observers including those undergoing a general accelerated motion - it is symmetric under all coordinate transformations - while "special" relativity is only symmetric under a subset of special (Lorentz and Poincare) transformations that interchange inertial observers. The symmetry under any coordinate transformation is only broken spontaneously in general relativity, by the vacuum expectation value of the metric tensor, not explicitly (by the physical laws), and the local physics of all backgrounds is invariant under the Lorentz transformations.

Loop quantum gravity proponents often and explicitly state that they think that general relativity does not have to respect the Lorentz symmetry in any way - which displays a misunderstanding of the symmetry structure of special and general relativity (the symmetries in general relativity extend those in special relativity), as well as of the overwhelming experimental support for the postulates of special relativity. Loop quantum gravity also depends on the background in a lot of other ways - for example, the Hamiltonian version of loop quantum gravity requires us to choose a pre-determined spacetime topology which cannot change.

One can imagine that the Lorentz invariance is restored by fine-tuning of an infinite number of parameters, but nothing is known about the question whether it is possible, how such a fine-tuning should be done, and what it would mean. Also, it has been speculated that special relativity in loop quantum gravity may be superseded by the so-called doubly special relativity, but doubly special relativity is even more problematic than loop quantum gravity itself. For example, its new Lorentz transformations are non-local (two observers will not agree whether the lion is caught inside the cage) and their action on an object depends on whether the object is described as elementary or composite.

J

#### josh1

Would you point out the references which match the points now raised by Dittrich & Thiemann?
Are we to believe that you are somehow unaware that string theorists have always had very specific reasons why lqg can never work? If you ask smolin or rovelli they'll tell you that string theorists did not write papers about their misgivings, but simply told them so verbally, and they still do.

...I am not sure how far the arguments raised by the authors in question exactly match the arguments raised by string theorists on the problems concerning the discreteness of the spectrum of geometrical operators in LQG...
I doubt that you're really all that interested in the answer to this question of 'how exactly these arguments match' beyond how much potential it has to annoy me.

#### marcus

Gold Member
Dearly Missed
this is an exciting moment for all the LQG-related QG approaches, one can sense the tension in how posters at PF are talking

we need to assemble some perspective and I can try to contribute a few pieces

first of course it hasn't been decided but IF it could be shown that the discreteness result is NOT true in canonical LQG, then this would liberate LQG from a major impediment and lead to a period of intense growth and development

one thing one would like to see is convergence with CDT and Reuter QEG along the lines of Reuter's vision at the Loops 07 conference (that by 2012 they would be proven to be equivalent with the number of quantization ambiguities in LQG corresponding to the dimensionality of the high-energy fixed point in QEG)---but CDT has no minimal length and Reuter QEG has only one paper suggesting the barest hint of a minimal length.

What CDT and QEG do is they trust nature and go to infinity---they rely on gravity being basically renormalizable and they let k-->infty and damn the divergences. In his Loops 07 talk, Reuter repeatedly appealed to Ambjorn's earlier CDT talk as exhibiting a kindred picture.

If it can be shown that canonical LQG does NOT imply a minimal length, then canonical LQG is free to take part in a multiple convergence of approaches.

However we should restrain our impulses to "jump the gun". It has NOT YET been shown that canonical LQG is free from the minimal length. IT COULD GO EITHER WAY. Dittrich and Thiemann have only shown that it is so far undecided!

I guess some of PF people here realize that Bojowald LQC is already prepared for it to go either way. He has a CONTINUOUS quantum cosmology model that behaves pretty much like the earlier discrete model.
The essential point is that LQC is INDEPENDENT from the parent theory and has to be verified or falsified on its own, by further CMB measurements, galaxy counts, and supernova studies. The proving ground for quantum cosmology models is cosmological observation.

There is a lot more to say. It could also have benefits if it turned out that the discrete geometrical spectra could be confirmed, in the canonical LQG context. But I'll stop here for now.

J

#### josh1

...IF it could be shown that the discreteness result is NOT true in canonical LQG, then this would liberate LQG from a major impediment...
So now you're saying that spin-networks have never been more than an albatross around lqg's neck? Maybe you should email rovelli to let him know that he should remove the chapters about spin-networks in his book. Oh wait, spin-networks are the subject of his book! :rofl:

#### marcus

Gold Member
Dearly Missed
For me, it is the most interesting result on LQG in last few years.
I have a great respect for scientists who are able to criticize their own theory that made them famous.
Demy, I agree with you in giving this Dittrich Thiemann paper high regard.

I'm not altogether convinced by it though, on reconsideration. BTW it has been substantially revised as of 17 August. It is now 13 pages and some of the statements have been changed.

As a side-aspect, if I had been paying closer attention I would have noticed a paper given at the Marcel Grossmann meeting in Berlin over one year ago---July 2006---showing that LQG does not have a smallest non-zero volume eigenvalue.

the paper was given by David Rideout and covered some supercomputer (CACTUS cluster) work funded by Renate Loll's European Random Geometry Network (ENRAGE). Rideout (London Imperial) and Brunnemann (Berlin) showed that zero is an accumulation point for the LQG volume spectrum. That is, there are values arbitrarily close to zero in the spectrum. THERE IS NO SMALLEST VOLUME in the LQG theory as Rideout and Brunnemann understand it.

http://arxiv.org/abs/gr-qc/0612147
Spectral Analysis of the Volume Operator in Loop Quantum Gravity
J. Brunnemann, D. Rideout
5 pages, 1 figure. Talk given by D. Rideout at the Eleventh Marcel Grossmann Meeting on General Relativity at the Freie U. Berlin, July 23 - 29, 2006
(Submitted on 22 Dec 2006)

"We describe preliminary results of a detailed numerical analysis of the volume operator as formulated by Ashtekar and Lewandowski. Due to a simplified explicit expression for its matrix elements, it is possible for the first time to treat generic vertices of valence greater than four. It is found that the vertex geometry characterizes the volume spectrum."

Page 4, conclusions: "This suggests that zero is an accumulation point of the volume spectrum. ...The complete results can be found in a
forthcoming paper [4].

Acknowledgments We thank Thomas Thiemann for encouragement..."

So already over a year ago a lot of people including participants in the major international MG 11 conference were aware that JUST FROM NUMBER CRUNCHING alone we can tell that LQG does not have a smallest non-zero volume.

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#### marcus

Gold Member
Dearly Missed
Carlo Rovelli's comment on the DT paper, posted today

http://arxiv.org/abs/0708.2481
Comment on "Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by B. Dittrich and T. Thiemann
Carlo Rovelli
6 pages, 1 figure
(Submitted on 20 Aug 2007)

"I argue that the prediction of physical discreteness at the Planck scale in loop gravity is a reasonable conclusion that derives from a sensible ensemble of hypotheses, in spite of some contrary arguments considered in an interesting recent paper by Dittrich and Thiemann. The counter-example presented by Dittrich and Thiemann illustrates a pathology which does not seem to be present in gravity. I also point out a common confusion between two distinct frameworks for the interpretation of general-covariant quantum theory, and observe that within one of these, the derivation of physical discreteness is immediate, and not in contradiction with gauge invariance."

Part of what Rovelli has to say can be understood from a simple example. Picture a system consisting of a pan of water on the stove. We turn on the burner for a certain interval of time and then measure the change in temperature. Within the context of the system there is NO WAY TO PREDICT THE LENGTH OF TIME we are going to heat the water for, but that interval can be MEASURED.
So there are independent variables or MEASURED INPUTS to a system which cannot be PREDICTED. This is such a simple obvious thing, and clearly this kind of measured but not predicted quantity must be included in any interpretation of quantum mechanics.

If it is a RELATIONAL quantum mechanics schema then obviously there will be MANY MANY possibilities for such "measured but not predicted" quantities. for one thing, a relational schema HAS NO OFFICIALLY CORRECT CLOCK, and more or less any quantity is eligible to be considered for the role of clock. Some quantities make lousier clocks than others, but pretty much anything can be considered. Once a clock is chosen then as in the case of the pot of water on the stove the time interval is an INDEPENDENT VARIABLE, it is a measureable input, not a predictable.

So when one looks at somebody's relational formalism, Rovelli might tell us, the first thing to ask is "Where are your measurable-but-not-predictable quantities in this formalism? what letter do you like to use for them?" One must have operators corresponding to measured inputs.
===============

right now the trouble I have with Dittrich's formalism is that I cannot see her treatment of these things. there is something irregular and ungeneral about how she handles the f, t, and tau. It doesnt seem possible to think of these things as simple measurements. It is not so straightforward.

It is pretty clear that in the Dittrich Thiemann paper it is Dittrich's work on relational observables that is the engine. Thiemann may be helping decide where to go but Dittrich is driving the car.

I think what has actually happened is something very valuable. I like Dittrich's work on relational observables but I don't think she has it quite right yet. So this paper helps by showing a contradiction. It may actually be that the contradictions shows a shortcoming of Dittrich's work and she can clarify her conceptual framework and improve it.

Here is the revised abstract:
"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 far from established. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators."

To show how Rovelli's paper meshes with this one I will supply imaginary rejoinders to the bold section of the abstract. Maybe someone would like to respond and argue the contrary position---but the main thing here is just to show the stand-off.
that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level.
that depends. It does survive in Rovelli's treatment.

Whether or not this happens depends crucially on how the gauge invariant completion is performed.
that may not actually be the important thing. what is more crucial is whether Dittrich's relational scheme is physically correct in the first place.

This indicates that 'fundamental discreteness at Planck scale in LQG' is far from established.
I like this because it would be exciting to have things shaken up and it would bring LQG closer to CDT and Reuter QEG if there were no minimal length. But maybe this Dittrich paper did not prove anything in that direction.

To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.
that could be wrong. that might NOT be the right way to confirm fundamental discreteness. the right way might be to get settled what is the physically correct setup for relational observables (see Rovelli's two options for that, numbers I and II)

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