Spin Foam Models: A Perspective on Quantum Gravity

[Ambitwistor]

Originally posted by selfAdjoint
About that spectrum, everybody assumed after it was shown that length and area were quantized that they would come in discrete chunks. But of course without the eigenvalue spectrum they couldn't be sure of that.

Er, computing the eigenvalue spectrum of the area and volume operators is how people knew those observables were discrete.

Now Livine comes along and claims to have computed the spectrum, and gotten a continuous one for the geometry operators.

Livine has computed the spectrum for a different model than what other people have been considering. The debate is over what model is right.

 

[Ambitwistor]

Originally posted by eigenguy
I can understand surprise at the result that spacelike and timelike intervals in 3D LQG are continuous and discrete respectively, but so what? It's only the 4D case that matters, right?

After the work on 3D LQG, in his thesis (gr-qc/0309028) he went on to consider 4D models that have this property as well.

 

[selfAdjoint]

Thanks for that clarification.

 

[marcus]

Originally posted by Ambitwistor
After the work on 3D LQG, in his thesis (gr-qc/0309028) he went on to consider 4D models that have this property as well.

another remarkable thing, it seems to me anyway, is Asht/Lewand. used the compactness of SU(2) in an essential way to construct their measure on A----they took a limit of cartesian powers of Haar measure to make the celebrated AL measure. Now Livine is working with a non-compact group SL(2,C) and in what I've read it has always been described as a challenge to get a measure in the non-compact case, one of the outstanding problems.
He does this in clear, spare, elegant style. It takes 10 pages (Chapter 4, pages 47-57) and a theorem out of algebraic geometry---Rosenlich's theorem.
In my opinion those 10 pages are a thesis in themselves. I could easily be mistaken of course. He doesn't just wave hands, he let's you see just how he gets the measure---generalizing Ashtekar/Lewandowski to the non-compact case---and it has the columbus-egg simple-once-you-see-how feel. If there is some flaw in the proof, we can be sure that Jerzy Lewandowski will find it. Oh boy. (If Ashtekar and Lewandowski had seen how to extend to non-compact case you bet your booties they would have.) We must check out Rosenlich's theorem at PF. chortle chortle

 

[eigenguy]

Originally posted by Ambitwistor
After the work on 3D LQG, in his thesis (gr-qc/0309028) he went on to consider 4D models that have this property as well.

Are these 4d models as "tenable" as the conventional ones?

 

[Ambitwistor]

Originally posted by eigenguy
Are these 4d models as "tenable" as the conventional ones?

They are not yet well understood, as far as I know. I haven't read Livine's thesis, though.

 

[marcus]

Originally posted by Ambitwistor
They are not yet well understood, as far as I know. I haven't read Livine's thesis, though.

there is a thought-provoking discussion in Livine's thesis of what remains incomplete or fuzzy----starting on page 106 with section 8.4 "The alternative: a covariant Loop Gravity"

He is talking about constructing a more general framework from which both canonical LQG and the Barrett-Crane model can be derived
and he describes this as an "explicit linkage" (lien explicite) and later on page 109 as a "bridge" (pont) between the canonical formalism and covariant spin foams.

And he describes the preliminary or partial character of the result:

"I am only going to give a representation of the Dirac brackets at a finite number of points of Σ. Nevertheless in this partial quantization exactly the same kinematic structures are found as in the Barrett-Crane spin foam model which I described in Part IV. In this way, the quantization provides an explicit linkage between the canonical formalism and the covariant spin foam theories..."

On page 108: "...In effect, these are the only points where we know the normal to [the submanifold] Σ explicitly. The other points are rendered fuzzy in space time." (fuzzy = "flous")

On page 109: "So the theory is quantified at a finite number of points. I do not know how to explicitly quantify the whole, although it seems possible to do that using the algebra of cylinder functions described earlier, or the limit of cylinder functions under refinement. Nevertheless we obtain simple spin networks which diagonalize the area operator of a surface and the operator's spectrum is continuous. These same spin networks are equally the kinematic states of spin foam models and form, therefore, a bridge
between the canonical formalism and the covariant formalism of spin foams. Finally, one obtains a quantum theory (and a spectrum of the the area operator) which is independent of the Immirzi parameter g. This result is compatible with the implementation of the path integral realized in [70], which shows that it is independent of the Immirzi parameter..."

this stirs things up some, I'd say

the reference [70] was to a paper by Sergei Alexandrov and D Vassilevich
http://arxiv.org/gr-qc/9806001
and also Phys. Rev. D, called
"Path integral for the Hilbert-Palatini and Ashtekar gravity"

 

[amp]

I'm not so proficient with the math as you guys...but I have a question

When you mention the Lorentz topologies and the 10j numbers is there attractors strange or otherwise in some of the solutions? I got this thought in my mind as I was reading over the topic. If I'm out of my gourd ... just let me know. I was thinking if there were attractors is it possible they would make/cause localized properties manifest?

 

[marcus]

Originally posted by amp
When you mention the Lorentz topologies and the 10j numbers is there attractors strange or otherwise in some of the solutions? I got this thought in my mind as I was reading over the topic. If I'm out of my gourd ... just let me know. I was thinking if there were attractors is it possible they would make/cause localized properties manifest?
__________________
Knowledge is possessed only by sharing; it is safeguarded by wisdom and socialized by love.

Hello and welcome, amp, some of the others might see a link to the subject of strange (or other) attractors but given my limited knowledge and insight, I do not. I like your signature quote.

I will try to describe what this thread is about, in plain language. It may sound odd to those more used to technical language, and the attempt may fail. I only find out things by taking that kind of risk

 

[marcus]

I will try to describe what this thread is about, in plain language. It may sound odd to those more used to technical language, and the attempt may fail. I only find out things by taking that kind of risk

People only discovered in the past 100 years how to quantize mathematical models of the world.

Quantizing a (deterministic, classical) model is a way of allowing indeterminacy into it, but in a highly regulated way that was discovered in 1926 and has become habitual (because of its impressive track-record). But people have been trying for something like 70 years and so far have not succeeded in quantizing GEOMETRY.

Geometry is the oldest classical deterministic model of reality going back to the greeks or earlier, it is clearly on the agenda to quantize.

In 1915 Einstein's big step forward ('general relativity'= 'GR') was to model gravity by geometry. The GR message is that "Gravity is geometry." The distribution of matter (and energy) shapes space and the shape of space, in turn, tells matter (and energy) where to flow.

To quantize geometry (successfully) would be to quantize gravity as well, because the gravitational field and the shape of space are the same. Indeed some (myself included) think that space and time have no absolute existence apart from the gravitational field.

Therefore quantizing geometry, and most particularly, quantizing general relativity, has got to be on the historical agenda---on the critical path of human growth.

Unless GR is wrong and should be thrown out and replaced by some other model of gravity----but it has a long history of success as a theory.

Or unless Quantum Mechanics and the time-honored proceedure for quantizing mathematical models is a faulty proceedure! Hard to picture that---again a long run of predictive successes.

People say things like "QM and GR are the two major theories of the 20th century, so far they seem to be incompatible and nobody has been able to merge them into one theory, we need to put GR and QM together."

Loop gravity and spin foam models occur in people's attempts to quantize general relativity. I will try to say why and how they emerge in that theory-building effort

 

[marcus]

continuing on the "intuitive loop gravity" thread

the "intuitive loop gravity" thread would do as a place to
continue this since it is an introduction to the subject
for someone who just happens in and is curious about it.
so I'll continue there

 

[Ambitwistor]

Originally posted by amp
When you mention the Lorentz topologies and the 10j numbers is there attractors strange or otherwise in some of the solutions? [...] I was thinking if there were attractors is it possible they would make/cause localized properties manifest?

Right now we are nowhere near close enough to understanding the dynamics of spin foam models to understand issues like attractors. Of course, any complex system will tend to have some attractors. I don't know what this has to do with "localized properties", though.

 

[amp]

Thanks Marcus... Ambitwistor

What I mean be localized properties are - matter, fields(ie energy electric,magnetism,guage(?)) like the aggregation of clumps in a whirlpool.

 

[Ambitwistor]

Originally posted by amp
What I mean be localized properties are - matter, fields(ie energy electric,magnetism,guage(?)) like the aggregation of clumps in a whirlpool.

Well, right now, there has been very little treatment of coupling other fields to gravity in the context of spin foam models; mostly people talk about spin foam models of vacuum gravity.

 

[marcus]

q-BC spin foams (positive cosmological constant)

Since this is the spin foams thread I will mention a new development

yesterday on arXiv was posted
Girelli/Livine "Quantizing speeds with the cosmological constant"
http://arxiv.org/gr-qc/0311032

and this makes heavy use of a paper by Karim Noui and Philippe Roche that uses the quantum Lorentz group in Barrett-Crane spin foams
In other words it uses the q-deformed SL(2,C)
to do Lorentzian BC spin foam theory with a positive cosmological constant
and it turns out to be helpful
http://arxiv.org/gr-qc/0211109

It looks like Livine and others are "putting the pieces together" in a certain sense.
The quantizing speeds paper suggested that some effects might be observable in the CMB.
It also suggested a quantized law of momentum conservation, at top of page 4,
BTW I think what Livine calles the "rapidity" is the velocity multiplied by a factor of (1 - beta^2)^-1
and the effect would only be visible in situations where Λ is large-----i.e. very early universe for example during an inflationary epoch---today's Λ of 1.3E-123 (in Planck units) is way too small to allow any effect to show. The steps in "rapidity" or velocity are of the same size as the Λ thus inconceivably small.
But quantization of velocity can have left macroscopic traces from a period when Λ was large. So they are proposing this. Heady stuff.

Putting pieces together: L's thesis bridges between covariant loop gravity and Lorentzian BC spin foam.

Then Noui and Roche put positive Λ into BC spin foam and get the quantum deformed q-BC spin foam.

Now Livine and Girelli say: "In the present work, we wish to explore some "physical" implications of a non-vanishing cosmological constant in the spin foam setting, more precisely within the framework of the Barrett-Crane model."

 

[selfAdjoint]

The papers both make use of BF theory. Perhaps this brief description from the review paper by Baez, cited as 1 in the Noul & Roche paper will help.

In a certain sense this the
simplest possible gauge theory. It can be defined on spacetimes of any dimension. It is `backgroundfree',
meaning that to formulate it we do not need a pre-existing metric or any other such geometrical
structure on spacetime. At the classical level, the theory has no local degrees of freedom: all the
interesting observables are global in nature. This remains true upon quantization. Thus BF theory
serves as a simple starting-point for the study of background-free theories. In particular, general
relativity in 3 dimensions is a special case of BF theory, while general relativity in 4 dimensions can
be viewed as a BF theory with extra constraints. Most work on spin foam models of 4-dimensional
quantum gravity seeks to exploit this fact.

 

[marcus]

Originally posted by Ambitwistor
Well, right now, there has been very little treatment of coupling other fields to gravity in the context of spin foam models; mostly people talk about spin foam models of vacuum gravity.

true for a fact! I have seen only a few, and generally quite recent, papers putting matter into spin foam models. I guess it was "amp" who asked about this. here are some

Mikovic "Spin foam models of matter coupled to gravity"
http://arxiv.org/hep-th/0108099

Mikovic "Spin foam models of Yang-Mills theory coupled to gravity"
http://arxiv.org/gr-qc/0210051

Mikovic "Quantum field theory of open spin networks and new spin foam models"
http://arxiv.org/gr-qc/0202026

Louis Crane "A new approach to the geometrization of matter"
http://arxiv.org/gr-qc/0110060

Daniele Oriti and Hendryk Pfeiffer "A spin foam model for pure gauge theory coupled to quantum gravity"
http://arxiv.org/gr-qc/0207041

I'll mention another paper which includes fermions but unlike the others deals only with the 2+1 (three dimensional) case. It just happed to look interesting:

Livine and Oeckl "Three-dimensional Quantum Supergravity and Supersymmetric Spin Foam Models"
http://arxiv.org/hep-th/0307251

from Livine/Oeckl abstract: "...A main motivation of our approach is the implementation of fermionic degrees of freedom in spin foam models. Indeed we propose a description of fermionic degrees of freedom in our model. Complementing the path integral approach we also discuss aspects of a canonical quantization in the spirit of loop quantum gravity. Finally, we comment on 2+1-dimensional quantum supergravity and the inclusion of a cosmological constant."

 

[marcus]

Originally posted by Ambitwistor
Well, right now, there has been very little treatment of coupling other fields to gravity in the context of spin foam models; mostly people talk about spin foam models of vacuum gravity.

To judge specifically one might look at Daniele Oriti's thesis (Cambridge) that just came out.
"Spin Foam Models of Quantum Spacetime"
http://arxiv.org/gr-qc/0311066

Oriti has a good bibliography and his chapter 7 (pages 281-305) is devoted to "The coupling of matter and gauge fields to quantum gravity in spin foam models".

Prior to this the person with the most papers about matter in spin foam quantum gravity seems to have been, I believe, Mikovic. Oriti has references to a number of papers by Mikovic and others.

Anyway Oriti's thesis allows one to estimate how much work has been done on coupling matter to gravity in spin foams (a rather new line of research with or without including matter!)

BTW Oriti's thesis has some topics of interest in common with Livine's that appeared in September----Boucles et Mousses de Spin en Gravite Quantique. The two have collaborated on a couple of papers.

 

[meteor]

I would like to add that apart the Barrett-Crane model, there are other spin foam models, for example the Reisenberger model, the Iwasaki model or the Gambini-Pullin model