Some basic questions regarding LQG

[tom.stoer]

I am a new member; I hope I didn't miss existing questions and answers.

One of my questions regarding loop quantum gravity / Ashtekar variables is: what is the relation between the familiar SO(3,1) Lorentz symmetry of special relativity and the local SU(2) symmetry of the vierbein-rotation introduced in Ashtekar's formulation. Are they more or less identical in the sense as SO(3,1) ~ SU(2)*SU(2) ? Or is it not allowed to think of the gauge symmetry in Ashtekar's formulation as Lorentz symmetry?

If this local symmetry is related to the Lorentz symmetry: why only Lorentz and not Poincare?

Thanks
Thomas

 

[tom.stoer]

Let me add one remark: my feeling is that the symmetries are not related. Reason is that in my opninion Lorentz and Poincare symmetry are not good symmetries of general relativity. They are only a symmetries of certain solutions.

 

[Coin]

If this local symmetry is related to the Lorentz symmetry: why only Lorentz and not Poincare?

This is something I have long been confused about as well, I would be interested in any explanation anyone might have.

Also as I understand: Poincare symmetry is just Lorentz symmetry plus translation, is it not? If that is the case, then what would it even mean to have a "local" Poincare symmetry?

 

[tom.stoer]

That goes into the same direction. Of course Lorentz-Symmetry is a global symmetry as well. Perhaps it's better to try to understand these global symmetries as special diffeomorphisms?

@Coin: yes, Poincare is Lorentz + Translation.

Usually I would think that H generates time-translations. This is trivial as H is weakly zero (modulo gauge) and therefore this is a good symmetry on the physical states.
OK, but what about the P_i for the space-translation? This should be again a special diffeomorphism. Are these symmetries then related to lapse and shift?

Explanations are welcome!

 

[eendavid]

The SU(2) indices of the Ashtekar variables are the SO(3) freedom that the triads have in a 3+1 formulation of general relativity. These are not related to diffeomorphisms, which act on the space indices (well it's a bit more complicated, but that's the idea).

 

[tom.stoer]

I know, that's the reason for my question.

There is a local SU(2) ~ SO(3) symmetry. I think that it has nothing to to with global Lorentz invariance. Am I right?

 

[eendavid]

I mean, GR is not globally Lorentz invariant. Maybe, if we want SO(3,1) in the story, I shoud rephrase what I said. Let us describe GR with tetrads, thus introducing a new SO(3,1)~ SU(2)xSU(2) degree of freedom. This freedom has to do with the choice of tetrads, not with what people talk about when saying GR is locally Lorentz invariant. The SU(2) we see in Ashtekars variables is one of the SU(2)'s of the SO(3,1) which appears in the tetrad degree of freedom, the other SU(2) disappears because one tetrad is gauge fixed in this formalism.

 

[tom.stoer]

To make a long story short - SU(2) in Ashtekar's formalism is not related to Lorentz symmetry. That was my feeling from the very beginning, because I could find no mechanism to derive global SO(3,1) from local SU(2).

Thanks

 

[tom.stoer]

Another question regarding loop quantum gravity is: what is the current status regarding a semi-classical / long-distance limit? Can it proven that general relativity (or any of it's solutions) follow from a certain semi-classical state of LQG?

Thanks
Thomas

 

[marcus]

Another question regarding loop quantum gravity is: what is the current status regarding a semi-classical / long-distance limit?...

One way to gauge the current status is to google "perini pirsa". This gets you the video and the PDF file of the slides of Perini's talk at Perimeter Institute that he gave in February 2009.

Perini is one of those working on the question you mentioned----the LQG long distance limit.
His research is on the graviton propagator.

Pirsa is "perimeter institute recorded seminar archive". So if you know the name of someone who gave a talk at Perimeter recently you can often find the talk this way.

In this case if you google "perini pirsa" you get
http://pirsa.org/index.php?p=speaker&name=Claudio_Perini
and that leads to his February 11, 2009 talk:
http://pirsa.org/09020023/

I'll see if I can think of anything else that would give an idea of current status.

There was a general overview by Rovelli given to last year's string conference but that was August 2008, already some time ago.
Rovelli's next major LQG overview talk will probably be in September at the Corfu school.

The abstract for Rovelli's series of lectures at the school is here:
"I present a new look on Loop Quantum Gravity, aimed at giving a better grasp on its dynamics and its low-energy limit. Following the highly succesfull model of QCD, general relativity is quantized by discretizing it on a finite lattice, quantizing, and then studying the continuous limit of expectation values. The quantization can be completed, and two remarkable theorems follow. The first gives the equivalence with the kinematics of canonical Loop Quantum Gravity. This amounts to an independent re-derivation of all well known Loop Quantum gravity kinematical results. The second is the equivalence of the theory with the Feynman expansion of an auxiliary field theory. Observable quantities in the discretized theory can be identifies with general relativity n-point functions in appropriate regimes. The continuous limit turns out to be subtly different than that of QCD, for reasons that can be traced to the general covariance of the theory. I discuss this limit, the scaling properties of the theory, and I pose the problem of a renormalization group analysis of its large distance behavior."
http://www.maths.nottingham.ac.uk/qg/CorfuSS.html

They seem to be making progress towards determining the long-range limit. Perini's talk is the most recent I know of. If I remember, I will post more links here as new talks and relevant papers appear.

 

[tom.stoer]

Thanks a lot! Sounds rather promising.

 

[tom.stoer]

... and here's my next question: has anybody checked or even implemented the BRST quantization program for LQG?

(should be straightforward for the Gauss law, but I have no idea how to proceed with the diffeomorphism constraint)

 

[tom.stoer]

Second trial: does anybody know about a research program to quantize LQG with BRST methods?

 

[marcus]

It's fun to try searching for relevant material even though I don't know the answers.
Here's a Spires search that got essentially nothing:
FIND DK QUANTUM GRAVITY AND DATE > 2000 AND DK TRANSFORMATION, BECCHI-ROUET-STORA

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+DK+QUANTUM+GRAVITY+AND+DATE+%3E+2000+AND+DK+TRANSFORMATION%2C+BECCHI-ROUET-STORA&FORMAT=www&SEQUENCE=citecount%28d%29

I'll try again saying "loop space" instead of "quantum gravity". Or "spin, foam".
One could also use the symbol K for keyword instead of DK for desy-keyword.

I got nothing again, using:
find dk transformation, becchi-rouet-stora and dk spin, foam
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+DK+TRANSFORMATION%2C+BECCHI-ROUET-STORA+AND+dk+spin%2C+foam&FORMAT=www&SEQUENCE=

Maybe someone else will show up who has an idea. I haven't got anything helpful on this one.

Wait. I replaced DK by K and got 94 hits on this one:
FIND K TRANSFORMATION, BECCHI-ROUET-STORA AND K QUANTUM GRAVITY
It is not LQG but at least we are getting some BRS and QG.
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+K+TRANSFORMATION%2C+BECCHI-ROUET-STORA+and+k+quantum+gravity&FORMAT=www&SEQUENCE=

Here is one, very old and not much chance of it being helpful:
http://arXiv.org/abs/hep-th/9409046
Algebraic structure of gravity in Ashtekar variables
P.A. Blaga, O. Moritsch, M. Schweda, T. Sommer, L. Tataru, H. Zerrouki
(Submitted on 9 Sep 1994 (v1), last revised 15 Sep 1994 (this version, v2))
The BRST transformations for gravity in Ashtekar variables are obtained by using the Maurer-Cartan horizontality conditions. The BRST cohomology in Ashtekar variables is calculated with the help of an operator $\delta$ introduced by S.P. Sorella, which allows to decompose the exterior derivative as a BRST commutator. This BRST cohomology leads to the differential invariants for four-dimensional manifolds.
Comments: 19 pages

 

[tom.stoer]

Thanks a lot.

from a first glance: it could be a new field of research - or a road to nowhere :-)

 

[tom.stoer]

Next question: is there a recent paper explaining the status of the Hamiltonian constraint?

All what I know is that the spinnetwork representation had some quantization (factor ordering ambiguities, SU(2) representation at new vertices and links, ultra-locality) that could be fixed by looking at the semiclassical / IR limit.

Of course it's promising that many results of LQG are generic in the sense that they are independent from the details of the Hamiltonian; nevertheless:w/o knowing what the Hamiltonian really IS it's hardly possible to talk about a fundamental theory of gravity (it's more a whole class of theories or a road towards a theory)

Is there some progress?

 

[marcus]

Next question: is there a recent paper explaining the status of the Hamiltonian constraint?
...

In my opinion the best recent overview is Rovelli's invited talk to Strings 2008 (CERN August 2008) together with the answers to questions from the audience.

It will sound paradoxical if I say that from my perspecive LQG has made rapid progress since 2006, but it has all been focused in the spinfoam version of the theory, nothing to speak of with the Hamiltonian.

Something you said in your post suggests what might be a fruitful research program:
Assuming the spinfoam version found by two groups in 2007 checks out---has the right large distance limit---then it determines a path integral dynamics. It may be possible to use this to decide what is the correct Hamiltonian in the canonical LQG version.

 

[tom.stoer]

Thanks.

I had a look at the talk and I downloaded two research papers from arxiv focussing on the graviton propagator. I need some time to understand them ...