LQG how do you get the Quantum of Space-time

[Klaus_Hoffmann]

That is the question, if you assume the space-time is discrete then it will be an smallest amount of volume line surface and so on, but how do you know how big this amount is ? for example in usual QM we have that energy is quantizied so the avaliable energy levels are eigenvalues of the Hamiltonian.

Then the question is how do you know what 'allowed' (quantization) volume, area and so on of your theory (of course depending on the metric model quantization will be different), in a mathematical way.

* only volumens that are eigenvalues of a certain operator V are allowed

* only areas (of surface) that are the eigenvalues of a certain operator S are
allowed

 

[marcus]

you don't get a "smallest amount of spacetime" in typical LQG

(there are quite a few different non-string QG approaches, so one can have confusion---e.g. the Causal Sets approach is based on the relations between events and you could say that one event (one node in the causality web) is a "smallest bit of spacetime", but I wouldn't say that, I would just say that there IS NO SPACETIME in the causalset approach at a microscopic level, and that space and time are a macroscopic illusion that emerges from the web of causal relations)

but you asked, I think, about the most common sort(s) of LQG, not the whole non-string menu.

In LQG the AREA OPERATOR HAS DISCRETE SPECTRUM and it has been calculated

and the VOLUME OPERATOR HAS DISCRETE SPECTRUM but it has not been calculated so far, if I remember right.

quantum operators correspond to making real measurments (in principle at least) and you cannot assume any simple relation between the spectrum of possible values you get by measuring area and the precise possible values you would get from measuring a volume.

nature is somewhat touchy about letting us make measurments, sometimes doing them in different order you get different answers, or she let's you only do one and not the other.

so you can't reason that just because the area observable values are such and so, that the volume measurment spectrum is going to be this and that.

if fact you can't infer that space or spacetime is made of little "atoms" of space or spacetime.

indeed, in the end as more is found out about LQG it might be discovered that, like in the case of causal sets, space and time do not exist at a fundamental level and they only EMERGE on a macroscopic scale by the massive conspiracy of more fundamental degrees of freedom.

I hope this confuses you as much as it does other people

Dan Oriti is putting together a book with chapters on all different QG approaches, to be published by Cambridge University Press, called
Approaches to Quantum Gravity: Towards a New Understanding of Space Time and Matter

 

[marcus]

that was a kind of general answer to the "smallest bit of spacetime" idea, but that said there is also a technical content to your question which is HOW DO THEY CALCULATE THE SPECTRUM OF THE AREA OPERATOR corresponding to some physical object like a desktop.

there are 1990s technical papers about the area operator spectrum, and more recent textbook level treatments. Should I find some links for you?
If you want to look at it on level of technical detail, the key thing (as i see it)
is constructing the kinematical Hilbert space of quantum states----and showing that the spinnetworks form a basis. Once you have the Hilbertspace and the basis then you can simplify everything down and just look at ONE quantumstate namely one spinnetwork and one particular material desktop

there's a mathematical argument showing that the usual idea of area corresponds to adding up the spinlabels of each of the graph edges that crosses thru the surface whose area. the total of every thing that "punctures" the surface corresponds to its usual conventional area----there is some algebra---there are some coefficients: it is not quite that simple but it still isn't very complicated.

if you live near a university library, check out Rovelli's book, or else I or someone else could get a link---if you want to inspect the technical side.

 

[jal]

I hope this confuses you as much as it does other people

Darn!...I thought that I was getting to understand... Then here comes Takashi Tamaki and drops a monkey wrench into the calculations in my blog.
I'll put him in my blog but I'll wait before changing my blog.
http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0341v1.pdf
Considering boundary conditions for black hole entropy in loop quantum gravity
Takashi Tamaki
04 July 2007
We argue for black hole entropy in loop quantum gravity (LQG) by taking into account the interpretation that there is no other side of the horizon. This gives new values for the Barbero-Immirzi parameter ( = 0.367 • • • or 0.323 • • •) which are fairly larger than those considered before ( = 0.261 • • • or 0.237 • • •). We also discuss its consequences for future experiments.
----------
jal

 

[Klaus_Hoffmann]

But how do you get the expression for AREA OPERATOR and VOLUME OPERATOR ? since without assuming any extra hypothesis you only have the Lagrangian of GR, the metric and similar.

Also what are the 'states' (Eigen functions /wave functions) of the Hamiltonian Constraint ??

 

[marcus]

But how do you get the expression for AREA OPERATOR and VOLUME OPERATOR ? since without assuming any extra hypothesis you only have the Lagrangian of GR, the metric and similar.

Also what are the 'states' (Eigen functions /wave functions) of the Hamiltonian Constraint ??

area and volume are defined on the kinematic Hilbert space of states---they don't involve dynamics: Hamiltonian constraint doesn't enter into it.

Rovelli goes into their definition at length in his book, a draft version is available free at his website (by arrangement with Cambridge Press, so it's
legit ) Sounds like you might be interested in technical issues and could enjoy reading some chapters. It has a good index and table of contents so not hard to navigate. That would be the detailed answer to your main question of how do you define the area and volume operators.

In brief, the way the subject is developed, they do the geometric operators like area and volume FIRST, before worrying about the Hamiltonian constraints and how to construct the physical Hilbert space.

BTW, although this does not involve area and volume, the whole business about the dynamics is interesting. what I see happening is that conventional LQG is looking to LQC and spinfoams for guidance.

In cosmology they work with simplified models and they get the dynamics to work, and then in Bojowald recent papers they PERTURB AROUND the simple solutions and get dynamical results in some sectors of the full theory. So they gradually remove restrictions like homogeneity and extend results...Ashtekar has some ideas of "improved dynamics" that have grown out of LQC. I think Thiemann may also have been influenced by comparing the full theory with LQC.

The other line of research that is feeding into it is spinfoams. Rovelli has published a bunch of papers in the past 2 years defining n-point functions, gravitons, using spinfoam formalism----and improving spinfoam dynamics by discovering a new vertex amplitude. the idea is to make the dynamics work in a "path integral" approach---a sum over spinfoam histories---and then use that to see how the dynamics should be in the canonical (LQG) formulation.