Coherent States and the Role of Classical Variables in Quantizing Gravity

[Federation 2005]

In classical gravity, a state may be described by the combination of the 3 elements:
(1) An n-dimensional manifold M
(2) The specification of which subbundle F of the linear frame bundle LM is the orthonormal frame bundle with signature (n-1) +, (1) -.
(3) A GL(n) connection A.
One can then prove that for a fixed F, the connection A reduces to a SO(n-1,1) connection W.

In the quantization of a classical system (i.e., in a quantum systerm that has said classical system as its h-bar -> 0 limit), the state space may always be taken as the same, but now with the addition of "transition probabilities". The limit, itself, may be described as the limit of the transition probability T(z1,z2) to the Kroenecker delta delta(z1,z2) for any 2 states z1, z2, as h-bar -> 0. This mode of quantization -- where the state space is quantized rather than the operator algebra -- leads to what are called coherent states. Thus, if z is a classical state then W_z is the corresponding coherent state, and Tr(W_{z1} W_{z2}) = T(z1,z2).

This mode of description also transcends the distinction between classical vs. quantum. Both pure classical and pure quantum systems can be described under this same umbrella, as well as everything in between.

A sector is then any subset of states that have 0 transition probability with the rest of the state space. The total state space decomposes into an orthogonal sum of sectors, called "coherent subspaces". The extreme cases are
* pure quantum system -- only one coherent subspace
* pure classical system -- one sector, essentially, for each state (i.e. every 2 states have 0 transition probability between them)
* hybrid -- each sector describing a "superselection mode" or "coherent subspace". The parameters that index the sectors then comprise the classical variables of the system.

The question, then, naturally avails itself, no matter what formalism is used to attempt to quantize gravity: what are the coherent states? More precisely, what is the coherent state W_{M,F,A} corresponding to the classical state (M,F,A)?

How many sectors?

In particular, how would one describe W_{M,F,A} and W_{M,F',A'} for 2 DISTINCT frame subbundles F and F' of TM? Going by the Unruh-Davies effect, one has that the state spaces of two frames lie in distinct coherent subspaces when the frames are mutually accelerating. Here, that would mean that F parameterizes between DIFFERENT sectors.

That is: F must be a CLASSICAL variable.

The same goes for the M parameter.

This brings the issue of background back to the foreground, big time (pun intended). It's one thing to make F part of the "dynamic" background (i.e., to make F, itself, subject to the overall system's dynamics), but it's an entirely different thing to make it part of the *non-quantum* background (i.e., an external or classical mode). However, there is a marked tendency in LQG and amongst those who work in LQG to confuse the two.

Just because you're quantizing the connection does not mean that either F or M are brought under the umbrella of the whole endeavor. There is nothing that says that M ought to be anything but an ordinary manifold. It means two completely different things to quantize a geometry (as in the manifold M, itself), versus to quantize structures (like A or even F) that are sitting ON TOP of the classical geometry M.

But trying to bring M (or even F) under the programme, you're biting off more than you can chew. The issue with the coherent states W_{M, F, A}, W_{M, F', A'} when F and F' are different, already shows that.

Rovelli has had a tendency to think he could evade these issues by going into denial about M. But that simply doesn't cut it. No matter how the theory is formulated, no matter whether it be classical, quantum or a combination of the two, there will somewhere down the line be SOME definition of the coherent states W_{M, F, A}. And it's at this point that the issue of the classical geometry (along with all the issues raised here) returns.

So, there is no denying M. Even if it's "not there", M still cannot be evaded.

 

[marcus]

Hello Fed, it sounds like you have your own idea of how Gen Rel should be quantized!

I am not sure how it makes contact, however, with the current work of Carlo Rovelli.

Have you looked at any of Rovelli's recent papers, say since 2005? I don't see any point of contact. Maybe you could designate a recent paper, since 2005, and point to something there, on some page, that we can look at----that is somehow related to what you say.

You say Rovelli is in denial about the manifold M. What is he denying about M?

I really need you to tell me a specific paper and point to a section of it, on some page, where he is in denial about some manifold.

 

[marcus]

I don't think anything you say connects with anybody else's approach to quantum gravity.
but it may be interesting to discuss in its own right!
You seem to have a distinctive personal idea of what quantized GR should look like and it seems to me a remarkable thing about it is that you require that the hilbert space have a semiclassical state that is in a sense "peaked" around some particular connection A.

I could, of course, be wrong. But that seems to me to be a highly unusual individual feature of your approach.

After all THE WHOLE CONNECTION IS NOT AN OBSERVABLE. In ordinary QM, observables are things like position or momentum. In conventional QG observables can be things like areas, volumes, densities, depending on what the approach is.

I don't think an experimenter can ever "see" a certain connection, or determine that the system is in the state specified by a certain connection.

So your notation W_{M,F,A} refers IMO to something bogus.
Something that doesn't exist in usual QG, and which doesn't have any operational meaning.

what the experimenter can see are aspects of a connection (like areas and volumes) which it can share with other connections to a certain extent.
So he can never tell what connection gave rise to the things he measures.

I could be wrong, as I say, but I think your approach to quantizing gravity---with this stringent individually chosen requirement----would, if it could succeed, lead to something quite different from anybody else's approach.

So it might be worthwhile your developing it.

 

[f-h]

Eh... the reality of M is a persistent feature of discussion in LQG (and Rovelli is, I think, not one of the main protagonists here). At the most simple level you can ask whether you want knotted spin-networks (embedded in a manifold) or unknotted (combinatorical) ones.

More abstractly, what information about the manifold does the path groupoid capture?

Anyway the two views might not be as contradictory as they might appear a priori (c.f. Thiemann and Giesels AQG)

 

[mhill]

My question is.. when you use a discrete model of space-time , how can you define (in Quantum terms) the classical Geodesics (minimum length) or the question that in a discrete space you can not define a metric g_ab . So i do not understand how you can obtain in the semiclassical limit the definition that particles move on Geodesic or that Space has curvature.

 

[marcus]

My question is.. when you use a discrete model of space-time ,

In LQG one uses a continuous model of spacetime. So this question does not directly apply to what Rovelli does. His approach starts with a smooth continuum. A differentiable manifold M x R.

how can you define (in Quantum terms) the classical Geodesics (minimum length) or the question that in a discrete space you can not define a metric g_ab . So i do not understand how you can obtain in the semiclassical limit the definition that particles move on Geodesic or that Space has curvature.

Well of course in the approach Rovelli uses (LQG, Spinfoam formalism) one is not working with a discrete space, so the question you ask is not appropriate to this thread.

But there are QG approaches which DO base things on a discrete space, like Sorkin's Causal Sets approach. So one could make a separate thread and ask how the semiclassical limit is achieved in some discrete approach like that, or even if it is achieved at all!

Perhaps the most interesting way to make sense of your question might be to forget about discrete approaches like Causal Sets, and just ask how the semiclassical limit---or for that matter how Newton's law----is achieved in Rovelli's version of QG.

He will be giving an online seminar talk 22 April at the ILQGS. Audio + Slides (PDF). You might like to listen, and scroll through the slides. I will get the link.

Here is the ILQGS schedule
http://relativity.phys.lsu.edu/ilqgs/schedulesp08.html
...
Apr 22 New developements in the definition of the spinfoam vertex, and the loop-spinfoam relation
Carlo Rovelli, Marseille

Here is a thread with announcements of interesting QG stuff going on this spring, including the Rovelli talk
https://www.physicsforums.com/showthread.php?t=214242

If you want an introduction to LQG, there is an online introductory course being taught at Perimeter by Simone Speziale.
He has been a close collaborator of Rovelli, especially on recent work (2005 and later) on deriving the graviton propagator for LQG.
This is essentially how the correct low energy limit is established. So if you are interested in this, there may be no better way to get acquainted with the subject than to watch Speziale's introductory lectures!

 

[f-h]

My question is.. when you use a discrete model of space-time , how can you define (in Quantum terms) the classical Geodesics (minimum length) or the question that in a discrete space you can not define a metric g_ab . So i do not understand how you can obtain in the semiclassical limit the definition that particles move on Geodesic or that Space has curvature.

Same as always, the spacing between eigenvalues of the operators becomes small in the large eigenvalues limit.

In the large distance limit, your expectation value for any region somehow determined physically (and thus with a discrete set of measurements) will be able to vary smoothly.