# Loop Quantum Gravity

marcus
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nonunitary, thanks much, both for your reflections on the area
spectrum (the ES, non-ES issue) and for the report on the conference. I am glad to hear it turned out well and is likely to be repeated!

marcus
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Originally posted by nonunitary
...
Can we live with the new value of the Immirzi parameter, SU(2) and some exclusion principle (as by Corichi and Swain)? Is supersymmetry relevant (as Ling and others suggest)?
...

in case anyone is interested in following up on the references
John Swain's original paper was posted May 2003
http://arxiv.org/gr-qc/0305073 [Broken]
"The Pauli Exclusion Principle and SU(2) vs SO(3) in Loop Quantum Gravity"

He expanded it for publication and reposted last month
http://arxiv.org/gr-qc/0401122 [Broken]

the article by Ling and Zhang is
http://www.arxiv.org/abs/gr-qc/0309018
"Do Quasinormal Modes Prefer Supersymmetry"

(this is not a recommendation of Swain's or Ling's ideas, but just in case a reader wants to see what nonunitary was referring to, the investigation of BH quasinormal modes has caused a ferment of ideas among which ES is only one contender, it has also brought new people into LQG: for example Swain is an experimental particle physicist who was drawn into LQG by this. As nonunitary indicates some (if not all) of the ES people are newcomers too)

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marcus
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Originally posted by nonunitary
Marcus,
...
As for the LQG meeting in Mexico I have heard that it was a big success. Lot's of progress in agreeing on several conceptual points regarding spin foams, the hamiltonian constraint, semiclassical issues and phenomenology. Also, lots af ideas of where to go and what to look at came out of the discussion. I think that after this first "NAFTA" meeting on LQG there will be more on a rotating basis.

Please let the rest of us know if there are any talks from the conference posted or any further reports are available.

About what you said on the subject of ES area spectrum:
may I translate the main objection you have
from loops to spin network states?
Is the objection then that
there can be edges of the spin network with zero spin (?)
and that one wishes they would not contribute area but they do contribute area because of the term (j + 1/2).
Right now I am not clear about the problem. Maybe one
defines spin networks so that the miniumum j is 1/2?

EDIT: Alejandro Corichi recently posted a clarifying 3page paper on this question:

http://arxiv.org./gr-qc/0402064 [Broken]
"Comments on area spectrum in Loop Quantum Gravity"
this paper may be said to settle the ES hash or
cook the ES goose

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About what you said on the subject of ES area spectrum:
may I translate the main objection you have
from loops to spin network states?
Is the objection then that
there can be edges of the spin network with zero spin (?)
and that one wishes they would not contribute area but they do contribute area because of the term (j + 1/2).
Right now I am not clear about the problem. Maybe one
defines spin networks so that the miniumum j is 1/2?

Marcus,

You are right. Maybe I was not clear enough. A closed loop is a particular case of a closed graph, and we can define a spin network there by assigning reps. of SU(2) to it, labelled by j. If one choses j=0, one has the trivial function (explained in my previous post). One then defines spin networks starting with j=1/2 and do not include the $j=0$ case. One could include it, but then everytime one has a statement, one would have to add something about the j=0 case. It is not convenient and might lead to confusion.

Now, if one had three-zillion edges with j=0, it is not that one wished that they do not contribute. The statement is much stronger:
if there is an operator that "sees" the j=0 edges, then it is not a well defined operator of the theory. This is a result that comes of the very precise and rigurous ways in which the Hilbert space of the theory is built. My favorite way of seeing this problem is by the C*-algebra argument I used, but I am sure that one can come up with more explanations.

There is an analogy in ordinary QM, where the Hilbert space is L^2 "functions". Actually they are equivalence class of functions where two functions define the same state is they are the same "almost everywhere". Suppose we want to define the operator that has the action "evaluate the wave funtion at the origin".
Clearly, the action of the operator on two functions of the same class that hapen to take different values at the origin will be different. The operator does not respect the equivalence classes and is therefore not well defined.

marcus
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another aspect of LQG to discuss

early on, around page 2 I think, Tsunami urged that this thread
"continue on in nerdy fashion"
and it has done so, becoming a thread for introducing and discussing interesting aspects of LQG
on page 3 we opened the can of worms of the LQG Area Spectrum
which a minority (nonunitary points out that they are newcomers also) wants to revise so that it is evenly-spaced (ES)
IIRC the first link was one Meteor posted on the "surrogate sticky" thread, and he also supplied one other ES link here.
Nonunitary showed where the roadblocks are to adopting the proposed ES spectrum.

Pages 3-5 of this thread contain discussion of the area spectrum and some Black Holery. Thanks to all who participated---it was interesting and we may get back to it.
-----------------------------------

Now a new topic. 2+1 quantum gravity.
Ordinarily one thinks of studying 3+1 dimensional gravity.
Can anything useful be learned from studying gravity in 2 spatial
and one temporal dimension----in 3D instead of 4D?

Some famous people have found it interesting enough to explore and write about

S. Deser, R. Jackiw, G. 't Hooft "Three-dimensional Einstein Gravity" (1984)

then a few years later (in 1988) Edward Witten "(2+1)-Dimensional Gravity As An Exactly Soluble System"

S. Carlip has a Cambridge monograph on it "Quantum Gravity In 2+1 Dimensions".
---------------------------
Apparently despite the encouraging title of Edward Witten's paper there are still good many problems to solve about even this dumbed-down or toy version of gravity. Maybe it should not be called toy. And there is a widely shared suspicion that successfully quantizing gravity in 3D will give lots of hints as to how to do it 4D.

Some of the people in LQG who are currently working on 3D(or have irons in the fire) in random order:

Laurent Freidel
Karim Noui
Alejandro Perez
David Louapre
Etera Livine

I think what we might to do is just check out a little of what they are doing to keep track of what is happening in the 3D quantum gravity
department.

Does anyone have other suggestions----they are welcome too.

marcus
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(2+1) Loop Quantum Gravity and allied approaches

to work by the LQG people just mentioned
in the 2+1 D direction

of particular interest I think is a series of 4 papers that is in the works, the first one is out
and David Louapre says to expect the second in a few weeks

1. L.Freidel and D. Louapre,"Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles"
http://arxiv.org./hep-th/0401076 [Broken]

2. L.Freidel and D. Louapre, “Ponzano-Regge model revisited II: Mathematical aspects; relation with Chern-Simons theory, DSU(2) quantum group and link invariant". To appear.

3. L.Freidel, E. Livine and D. Louapre, “Ponzano-Regge model revisited III: The Field Theory limit”. To appear.

4. L.Freidel and D. Louapre, “Ponzano-Regge model revisited IV: Lorentzian 3D Quantum Geometry”. To appear.

----------------------

Karim Noui and Alejandro Perez have one in the works called

"Three dimensional loop gravity coupled to point particles".

Ambitwistor referred to a talk by Perez at Penn State last fall about this. Louapre mentioned a more recent talk by Karim Noui.
Freidel/Louapre and the other two are probably getting results along some similar lines.

I want to read some in the Freidel/Louapre paper number 1. because it gives an overview of what they intend to accomplish in this series of papers.

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marcus
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BTW in case you looked at Freidel and Louapre's paper
and didn't recognize that unusual symbol on page ten
it is the Hebrew letter "daleth"
(I asked David to be sure)

marcus
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scattering amplitudes calculated in LQG?

Freidel/Louapre hep-th/0401076:
"In our paper, we consider the spin foam quantization of three dimensional gravity coupled to quantum interacting spinning particles. We revisit the original Ponzano-Regge model in the light of recent developments and we propose the first key steps toward a full understanding of 3d quantum gravity in this context, especially concerning the issue of symmetries and the inclusion of interacting spinning particles.

The first motivation is to propose a quantization scheme and develop techniques that could be exported to the quantization of higher dimensional gravity. As we will see, the inclusion of spinless particles is remarkably simple and natural in this context and allows us to compute quantum scattering amplitudes. This approach goes far beyond what was previously done in this context by allowing us to deal with the interaction of particles.

The inclusion of spinning particles is also achieved. The structure is more complicated but the operators needed to introduce spinning particles show a clear and beautiful link with the theory of Feynman diagrams [26]."

Marcus,

Are spinless particles one's with spin zero, or particles that are not expressed in this value?

Can you give short definition of quantum scattering amplitudes?

Thanks!

LPF

marcus
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my original post was mistaken, thanks to sA for the correction

there is an entry-level discription of "amplitudes"
in Feynmann's book "QED The Strange Theory of Light and Matter"
around page 78. Do you have a library where you could
borrow this book. It is very thin (150 pages) but costly

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Staff Emeritus
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They really mean that the math of spin does not apply to these particles. They are also called scalar particles, and they are frequent subjects in developing new ideas because they are easy to work with. Actually there are no scalar particles in today's physical theories except the hypothetical Higgs partical, and you bet these scal particles are not intended to model the Higgs.

---the math of spin does not apply to---scalar particles.

The math of spin does apply: spin-0 means lorentz scalar, which is where "scalar" comes from.

there are no scalar particles in today's physical theories except the hypothetical Higgs partical

There is the dilaton of ST.

marcus
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Hello all, I'm quoting some exerpts to give an idea of context and what the article's about. This first quote was from page 3.
Freidel/Louapre hep-th/0401076

"In our paper, we consider the spin foam quantization of three dimensional gravity coupled to quantum interacting spinning particles. We revisit the original Ponzano-Regge model in the light of recent developments and we propose the first key steps toward a full understanding of 3d quantum gravity in this context, especially concerning the issue of symmetries and the inclusion of interacting spinning particles.

The first motivation is to propose a quantization scheme and develop techniques that could be exported to the quantization of higher dimensional gravity. As we will see, the inclusion of spinless particles is remarkably simple and natural in this context and allows us to compute quantum scattering amplitudes. This approach goes far beyond what was previously done in this context by allowing us to deal with the interaction of particles.

The inclusion of spinning particles is also achieved. The structure is more complicated but the operators needed to introduce spinning particles show a clear and beautiful link with the theory of Feynman diagrams [26]."

The next exerpt is from page 20:

Freidel/Louapre hep-th/0401076

"...
It is well known that, at the classical level, three dimensional gravity can be expressed as a Chern-Simons theory where the gauge group is the Poincare group. The Chern-Simons connection A can be written in terms of the spin connection &omega; and the frame field e,
$$A = \omega_iJ^i + e_iP^i$$
where Ji are rotation generators and Pi translations.

Moreover, since the work of Witten [44], it is also well known that quantum group evaluation of colored link gives a computation of expectation value of Wilson loops in Chern-Simons theory. Our result
therefore gives an exact relation, at the quantum level, between expectation value in the Ponzano-Regge version of three dimensional gravity and the Chern-Simons formulation...

[footnote] &kappa; = 1/4G is the Planck mass in three dimensional gravity"

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marcus
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?

So why is the Planck mass different in 3 dimensions?

got kind of a glimmering
but has anybody thought thru this one and

In 4D---the world we have----the planck quantities are such and such.

But, according to this paper in 1+2 dimensions----their 3D world---the planck mass is
$$\frac{1}{4G}$$

thats what the footnote on page 20 says, that i just quoted
anyone want to comment or differ with this or explain?

marcus
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in 1+2D
newtons constant has dimensions of
inverse mass
http://arxiv.org/hep-th/0205021 [Broken]
s'what I thought cause of force falling off
of sq. recip

the mass unit in 3D is basically 1/G
order one coefficients like 4 or 8pi are
mostly a matter of convention (how you
write the einstein equation, the 8pi business)

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marcus
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lets put c and hbar in explicitly and see what the planck units
actually are in (1+2)D

thing to notice is that in 4D we have
$$GM^2 = \hbar c$$
because GM^2 has to equal the unit force x area (inverse sq. law)
and that equation defines the pl. mass in 4D

but in 3D GM^2 will equal the unit force x distance!
and that is the unit energy in the system: Mc^2, so we have instead

$$GM^2 = M c^2$$ which solves to

$$M = \frac{c^2}{G}$$

After that, easy, unit energy is
$$E = \frac{c^4}{G}$$

and unit freq is
$$\omega = \frac{c^4}{G\hbar}$$

That makes unit time
$$T = \frac{G\hbar}{c^4}$$

and unit distance
$$L = \frac{G\hbar}{c^3}$$

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I was meaning to elaborate on the reasons why the ES-are espectrum of Alekseev and colaborators is not well defined.
Too late, this guy must have read my posts and took part of them, added a new argument with graphs and posted:

http://arxiv.org/abs/gr-qc/0402064

I think I have to agree with him. What he didn't say though is that one might be abre to define a new operator that somehow "ignores" a j=0 edge, but there is some work involved in showing that it is possible.
Anyway, farewell to the ES-area operator of APS.

marcus
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marcus
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Originally posted by nonunitary
... this guy must have read my posts and took part of them, added a new argument with graphs and posted:

http://arxiv.org/abs/gr-qc/0402064

I think I have to agree with him. What he didn't say though is that one might be abre to define a new operator that somehow "ignores" a j=0 edge, but there is some work involved in showing that it is possible...

again you were prophetic, the guy has added a paragraph to his
conclusions and updated the preprint
(it is now a little longer and is dated 17 February instead of 13 February)
and the addition includes the case where the operator is
so the spectrum is ES except for a double-size space at zero.
the author does not like this case but he includes it (with a warning) presumably for the sake of completeness

I checked the Gour/Suneeta paper (gr-qc/0401110) and it did not
seem to disturb their calculation of BH entropy
I could not see any reason to accept or reject, it appeared (at least for now) to be just an arbitrary ad hoc fix.

marcus
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I am looking for previous discussion of BH entropy, BH area, in LQG context.
Sauron recently posed some questions about entropy and LQG in another thread and hopefully there is something relevant to that here.

------here's some of Sauron's post-------
I have a few generic questions/reflections about some of the themes LQG is addressing.

Let´s begin by the question of entropy. My deal is whether the concept of entropy makes sense in GR at all. At least in the same sense as in ordinary statistical mechanics.

I know about two main results. The one, of wich i have a reasonable understanding , about the black hole area behaving like entropy. I also have notice about (but no understanding at all) results of Penrose relating the Weyl tensor to entropy, at least in cosmological scenarios.

The question is that in the microcanonical device the entropy is related to the number of micro-states compatible with an energy. But in GR there is no a good (and less local) definition of the energy of the gravitational field....
--end of exerpt--