Exploring Coherent States in LQG: Understanding the Measurement Process

[sheaf]

I’ve been starting to look at how LQG works, and if I’ve understood it correctly, just thinking about the kinematics, the quantum state of spacetime is encoded in a spin network. If I want to determine the volume of a region of spacetime, the result will be an eigenvalue of some volume operator. For a classical picture of spacetime to emerge, where a region of spacetime has a well defined volume, isn’t it the case that spacetime must no longer be in a superposition, i.e. “measurements must have happened” ? Is there any agreed way that people heuristically think about the measurement process ?

 

[hamster143]

I’ve been starting to look at how LQG works, and if I’ve understood it correctly, just thinking about the kinematics, the quantum state of spacetime is encoded in a spin network. If I want to determine the volume of a region of spacetime, the result will be an eigenvalue of some volume operator. For a classical picture of spacetime to emerge, where a region of spacetime has a well defined volume, isn’t it the case that spacetime must no longer be in a superposition, i.e. “measurements must have happened” ? Is there any agreed way that people heuristically think about the measurement process ?

Consider a box with atoms. Each atom has a discrete spectrum of energy levels, and exists in a superposition of those, but the total energy in the box is, in some sense, "well defined".

 

[marcus]

... Is there any agreed way that people heuristically think about the measurement process ?

The spin networks are the eigenvectors of area and volume operators. The labels on the network determine a certain amount of volume for each node and a certain amount of area assigned to each link.

In order to have a meaningful measurement of volume there must be some physically defined region. Heuristically, think of a region defined in some covariant way, by a physical object or some definite events. Then there is an associated volume observable.

In the state specified by a particular spin network, the volume is just the sum of all the volumes belonging to the nodes of the network which are inside the volume. (I duck the question of borderline cases.)

Or there is some physically defined surface---a desktop?!, an horizon of some type?---and you look at the set of all the links in the network which are CUT by this surface, and add up all the area numbers which are associated with those cut links.

John Baez might have a "This Week's Finds" about this from some years back. He's good with heuristics. Or Rovelli if you find something of his written for beginners, where he wants to make it intuitive.

The nodes of the network can symbolize chunks of volume (that could be revealed by measurement) and the links can stand for flakes of area bounding the chunks. Intuitively if you specify all the possible areas and volumes that you could measure, then you have somehow determined the geometry of the universe. The state of geometry of the universe is somehow revealable by taking all possible area and volume measurements.

I guess there could be other combinations of different sorts of geometric measurements that would also determine a geometric state. There could be some other basis for the kinematic Hilbertspace. Eigenvectors of some other types of geometric measurement.

I'm not a loop gravity expert, but you asked for some heuristic and I can offer a bit of personal intuition. Have you tried Rovelli's book, an online draft is available free for download, or any of the various introductory articles. What level are you trying to go in at?

 

[tom.stoer]

Afaik neither volume nor area operator are diff.-inv. observables in LQG.

 

[marcus]

I must be wrong then! You are definitely more knowledgeable than I! I will leave my post uncorrected so that the questioner can see what i was trying to say, even if it was in error.
Can you offer an improved version of the heuristics?

I don't want to argue, I'll take your word on it. Should we say whether the spin networks are embedded or combinatorial ones (there are those two approaches)? I was thinking that in order to have some definite particular area (or volume) operator there must be some definite physical object. If you then want to talk about diffeo-invariance then if you re-coordinatize the space then the physical object gets re-coordinatized along with everything else.

 

[hamster143]

Afaik neither volume nor area operator are diff.-inv. observables in LQG.

That depends on how you look at it. If you want a volume operator that gives you a 3-volume of a region that's bounded by points with specific 4-coordinates, then of course it's not a diffeomorphism invariant observable, in ANY theory. If you fix the gauge by factoring out diffeomorphisms, what remains is perfectly observable.

 

[Orbb]

Concerning gauge invariance, my understanding is that area and volume operators defined at the kinematical level are in fact invariant under 3-diffeomorphisms. Of course, physical obversables, as with quantum states, have to be invariant under the full gauge symmetry and therefore are the subset of the kinematical part that also satisfy the Hamiltonian constraint.

 

[sheaf]

Thanks everyone, I guess where I'm coming from is I'm trying to see, given the tiny amount I know about LQG, how classical GR arises in some limiting case. Given that in GR volumes of regions are well defined, I was trying to understand how that would arise given that the universe may not be in an eignestate of the volume operators. But maybe what Hamster said is the right way to look at it - i.e. as an ensemble.

I'll definitely take a look at the Rovelli online draft though - I wasn't aware that was available.

 

[marcus]

I'll definitely take a look at the Rovelli online draft though - I wasn't aware that was available.

Google "rovelli" to get the website. Google "rovelli book" to get the link to PDF
www.cpt.univ-mrs.fr/~rovelli/book.pdf

Hanno Sahlmann has a recent (2010) introduction to LQG on arxiv.
There is a 1998 introduction by Rovelli and Upadhya on arxiv.
I might think of other stuff, you just have to sample and find what's right. Maybe Orbb, Hamster, Tom have some recommendations.

Just in case they might be helpful, Rovelli's 2008 review article:
http://relativity.livingreviews.org/Articles/lrr-2008-5/
and a video of the introductory talk he was invited to give at the Strings 2008 conference:
http://cdsweb.cern.ch/record/1121957?ln=en
the slides for that talk are also available separately at the cern website:
http://indico.cern.ch/getFile.py/access?contribId=30&resId=0&materialId=slides&confId=21917

 

[atyy]

The most common approach in LQG to try to get classical geometry is to use a coherent state. There's a reader friendly write up in the last section of http://arxiv.org/abs/1007.0402 .

I have never understood why things are supposed to be in coherent states.