How do ‘quantum loops’ differ from ‘Wilson lines’?

[Dcase]

As I attempt to understand the conflict between Quantum Loop Gravity [QLG] and the various types of string theory, I noticed that quantum loops appeared to be very similar to Wilson lines, sometimes called Wilson loops.

Speculation: relating [QLG] to twistor string theory [TST].

1 - Consider this possible analogy involving the Schroedinger equation [SE]:

quantum loops are to helical strings
as
time-independent SE is to time-dependent SE

Time-dependent SE utilizes “i” somewhat like the Hawking concept of imaginary time.

2 - Substitute helicoids [springs or slinky-toys] for all loops in the Rovelli home page image from
http://www.cpt.univ-mrs.fr/~rovelli/loop_quantum_gravity.jpg

This adds a virtual fourth dimension representing time as helical trajectory.

The helix appears to be important in transmitting information in trajectory:
- ballistics due to rifling
- mechanics
--- planetary orbits are concentric relative loops when the sun is motionless
--- planetary orbits are concentric helices when the sun is in motion

These examples appear to make use of the Bars concept of 2T physics.
2T physics may involve the transition from one gauge to the next gauge.

One might also consider these examples a means of transforming Penrose spinors into Penrose twistors.

 

[f-h]

The Loops in LQG are Wilson Loops. The new thing is that due to diffeomorphism symmetry the Wilson lines now make a countable basis of your statespace.

 

[Careful]

The Loops in LQG are Wilson Loops. The new thing is that due to diffeomorphism symmetry the Wilson lines now make a countable basis of your statespace.

Well, I thought that H_diff was still non separable when averaging occurred over analytic or C^infinity diffeormorphisms. As far as I followed this, one expected seperability to come from averaging over homeomorphisms; has this been proven already?

Cheers,

Careful

 

[marcus]

The Loops in LQG are Wilson Loops. The new thing is that due to diffeomorphism symmetry the Wilson lines now make a countable basis of your statespace.

in one treatment the kind of symmetries are called by Rovelli "extended" diffeomorphism
they are allowed to have a finite number of points where the smoothness condition fails.*

Rovelli and Fairbairn argue that the orginal theory GR was already invariant under such extended diffeomorphisms. So no big deal to extend the class of diffeomorphisms and factor the space down to where it is separable.

http://arxiv.org/abs/gr-qc/0403047
Separable Hilbert space in Loop Quantum Gravity
Winston Fairbairn, Carlo Rovelli
14 pages, 3 figures.
J.Math.Phys. 45 (2004) 2802-2814

"We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a consequence of the fact that the knot-space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the fine-tuning of the mathematical setting. We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable."

* the paper is short and sweet, and my memory of it is imperfect, so instead of relying on my paraphrase you might do well to just check the paper directly. IIRC an extended diffeo is a regular diffeo except the C-infinity condition can fail for the map or its inverse at a finite number of points.

I think there might be some interesting mathematics thesis problems in checking this class of functions out rigorously, putting a function-space topology on such things etc. Some weirdness might happen, and make it fun. Winston Fairbairn would know if there are good problems.

BTW F-H, there are several different ways people have of boiling the hilbert space down to countable basis and IIRC there are several different classes of functions that are restrictions or extensions of the diffeos which come into the picture. It didnt seem clear-cut to me. Maybe Ashtekar would treat things differently from Rovelli. But Rovelli's and Fairbairn's style of proceeding looks pretty nice to me. I wish there were more research on it checking it out to make sure there are no surprises. The nice thing is it gets you right away to a basis of labeled KNOTS and knots are interesting and beautiful.

 

[f-h]

Yes, I think that's correct, and as far as I know it was proven here: http://arxiv.org/abs/gr-qc/0403047 (probably depending on the level of rigour you expect)

This is the way it's in Rovellis book and the way Smollin teaches it.

That is, if you take your fields to be smooth except at finitely many points, and diffeomorphisms likewise you get seperable.

Edit:
Ah, marcus you beat me to it...

*BTW F-H, there are several different ways people have of boiling the hilbert space down to countable basis* I wasn't aware of that. Do you have a reference/link to a thread?

 

[marcus]

*BTW F-H, there are several different ways people have of boiling the hilbert space down to countable basis* I wasn't aware of that. Do you have a reference/link to a thread?

NO on reconsideration, I have to disappoint you: the only way I like and feel I moderately understand is Rovelli. with the extended diffeo.

but one time I tried to understand what Ashtekar and Lewandowski were doing in
"Background Independent Quantum Gravity: a Status Report" (BIQG)
http://arxiv.org/abs/gr-qc/0404018
and I got bogged down and could not follow it---it seemed to me they were NOT using rovelli's extended diffeos. And earlier treatments by Smolin, he was not either and I don't understand how he managed. Maybe you just hoped that, even tho kinematic statespace isnt, the physical constraints will make it separable. Recent Smolin papers cite Fairbairn Rovelli, but before 2004 there wasnt any so they don't.

I don't know what people did before Fairbairn and Rovelli (they must have done various and sundry) maybe they didnt wear clothes and lived by picking fruit from the trees.

 

[Dcase]

Hi Marcus and f_h:

Thank you for your responses.
Your both seem to agree that LQG Loops are equivalent to Wilson Loops.

If this is so, why is there such conflict between these two camps?

Does this mean that quantum loops are time-independent,
while helical strings are time-dependent?

Does this mean using helicoids for all loops in the Rovelli image, referenced above, transforms QLG into TST?

 

[marcus]

...why is there such conflict between these two camps?

Do you really want to know? I ask with all due respect. Or would you prefer to explain your own theories to us? It is all right either way with me. I actually would not enjoy having to summarize the points of contention and difference between the two lines of research.

Francesca recently suggested someone read Rovelli's dialog, if they wanted to understand the string/loop friction. You could try that.

http://arxiv.org/abs/hep-th/0310077
A dialog on quantum gravity
Carlo Rovelli
20 pages
International Journal of Modern Physics D12 (2003) 1509-1528

"The debate between loop quantum gravity and string theory is sometimes lively, and it is hard to present an impartial view on the issue. Leaving any attempt to impartiality aside, I report here, instead, a conversation on this issue, overheard in the cafeteria of a Major American University."

 

[Careful]

**
This is the way it's in Rovellis book and the way Smollin teaches it.

That is, if you take your fields to be smooth except at finitely many points, and diffeomorphisms likewise you get seperable. **

Ok, I am still reading (page 9) but have some immediate remark (which may be totally wrong). It concerns Rovelli's claim that introducing fields which are not differentiable at isolated points does not affect the classical theory. The surprising thing is that the authors suddenly start to speak about diffeomorphisms on M = \Sigma * R while the issue of diffeo invariance is usually limited to \Sigma alone. The reason why I make this remark is that - generically - one would naively expect such non-smooth points to propagate in the classical theory so that the class of fields on *spacetime* which are not differentiable on isolated points just cannot provide new solutions (imagine a diffeomorphism of the form (a(x,y,z),b(x,y,z), c(x,y,z), f(t)) with f smooth and a,b,c smooth except on isolated points). Now, either you enlarge the class to account for singular worldlines or so (and effectively change the quantum/classical theory) or you simply say that classical evolution cannot even ``lift off´´ from this particular set of data which makes me wonder about this silly hamiltonian constraint (note : propagating conical singularities could be seen as point particles and hence contribute to the ``energy momentum tensor´´). Note: usually you weaken the C^2 property to C^1 lipschitz to keep the Einstein eqn's well defined in distributional sense.

EDIT : he adresses these issues for the area and volume operator for which there are obviously no problems and makes a brief handwaving remark about the Hamiltonian operator (which requires somewhat more rigor for my taste).

So thoughts are welcome...

Cheers,

Careful

 

[marcus]

...

Ok, I am still reading (page 9) ... he addresses these issues for the area and volume operator for which there are obviously no problems and makes a brief handwaving remark about the Hamiltonian operator (which requires somewhat more rigor for my taste).
...

I am glad to see you going over Fairbairn/Rovelli with a fine tooth comb.
Besides expressing pleasure and encouragement I cannot offer substantive assistance. Maybe someone else can.

what I think is that this is a key paper because other people took it up and cite it. Not everybody but some do go by this route now.

because the paper has this (not-very-visible-but-real) importance it would make a good research paper or thesis for someone to just go in and prove rigorously or find counterexamples to some central statements.

common sense suggests that there is someone in the Rovelli team already doing that, but I didnt hear of it so can't say for sure.

In my humble opinion, one of the statements that one should either prove rigorously or construct a counterexample is very simple namely that classical GR is invariant under extendeds.

the way one would attempt a proof is one would consider an extended diffeo f(x) with one rough point and smooth everywhere else. And one would exhibit a SEQUENCE of normal smooth diffeo f_n which CONVERGES in some suitable way to the extended f. then one would observe that Gen Rel is invariant under each of the f_n and one would argue that therefore it is likewise invariant under f. QED.

maybe this argument is given quickly and lightly in the F/R paper, I don't remember. but I don't think it is spelled out in explicit detail. I think it needs to be. And conceivably if one did this one might find a mistake, which would be interesting and stimulate the authors to be resourceful and come up with some alternative.

If one did that explicit proof and it turned out to be RIGHT then that would mean that Einstein's theory was not merely diffeo invariant but that it was also invariant under extendeds. So it was more invariant than Einstein knew, all along! That would be charming.

then as a mathematician one could legitimately study the extendeds as a new class of functions (unless it happened that some Pole or Romanian had already studied them, which is always possible)

Since I don't want to get down into the mud or chocolate sauce and struggle with you, all I can do is cheer you on. Anything you do, it seems to me, if done "carefully" enough to pass peer review, would be to the good.

 

[Chronos]

Too many 'ghosts' [unphysical results] in Wilson lines for my taste - not that 'Loops' are ghost-free. Some of the toys in both models work pretty well, but both fall short of reproducing GR in 3+1 dimensions. Until that happens, I can't work up much enthusiasm for either approach. Personally, I prefer Loll's simplical slice approach, but it too has run up against the same wall, IMO.

 

[f-h]

Dcase, Wilson loops are a tool from (often Lattice) Quantum Field Theory, nothing to do with Strings apriori.

Careful, if Einstein allows 4 diffeos with finitely many points "unsmooth", then this would induce 3 diffeos with finitely many points "unsmooth" on some slices of a foliation, right? In this sense if you convert the extended 4 diffeos into the hypersurface deformation algebra you would not expect the non smooth points to "propagate".

 

[Careful]

**
Careful, if Einstein allows 4 diffeos with finitely many points "unsmooth", then this would induce 3 diffeos with finitely many points "unsmooth" on some slices of a foliation, right? In this sense if you convert the extended 4 diffeos into the hypersurface deformation algebra you would not expect the non smooth points to "propagate". **

Hi, sure I thought about this (sorry for the late reaction, went on holiday's) but such line of thought is definately troublesome. Basically, what the paper says is the following: let psi_ i -> psi where psi is a diffeomorphism which is singular on a finite set of points, then
lim_i (psi_i)_* g is a solution of the ``generalised´´ Einstein equations if g is a solution of the EEQN's (right ?) - the limit here is to be understood in the C^0 sense. Let me jump ahead a bit: the final goal of the paper - or of LQG - is to define Laws of nature without any reference to ``smoothness´´. Therefore, one needs to have a dynamical law (Hamiltonian constraint) which is entirely combinatorial (+group theoretical) in nature. Classically, any generalized gravitational law is deterministic : now, you start out from a hypersurface \Sigma with isolated points where the initial data is non-smooth. Suppose, as you do, that one can propagate for finite time on a neighborhood O of \Sigma such that the solution is smooth everywhere on O - \Sigma, then you know that the solution will be smooth everywhere in the future unless you meet a physical singularity. My point is, that in this way, the initial hypersurface \Sigma is *preferred* in the sense that these innocent singularities are only found there. The only way to avoid this, would be that these singularities propagate and give a genuinely different solution (one would expect the particles to gravitate). Perhaps, one should not give a classical motivation to arrive at this point at all (such as causal set people do right from the start). Otherwise one should construct the classical generalized laws which determine the propagation of non-smooth initial data.

Cheers,

Careful