Krasnov action-start of a new spinfoam?

[marcus]

Krasnov action--start of a new spinfoam?

The Krasnov action has been offered as a possible basis for a new spinfoam model, in which the effective constants like G Newton would run with scale. This would differ from usual GR at extremely small scale (or in some other extreme regime) but might still have the usual largescale behavior.

This is discussed in a short paper posted recently:
http://arxiv.org/abs/0907.4064
Gravity as BF theory plus potential
Kirill Krasnov
7 pages, published in Proceedings of the Second Workshop on Quantum Gravity and Noncommutative Geometry (Lisbon, Portugal)
(Submitted on 23 Jul 2009)

"Spin foam models of quantum gravity are based on Plebanski's formulation of general relativity as a constrained BF theory. We give an alternative formulation of gravity as BF theory plus a certain potential term for the B-field. When the potential is taken to be infinitely steep one recovers general relativity. For a generic potential the theory still describes gravity in that it propagates just two graviton polarizations. The arising class of theories is of the type amenable to spin foam quantization methods, and, we argue, may allow one to come to terms with renormalization in the spin foam context."

 

[marcus]

To get a sense of what Kirill Krasnov is like, there's a 2006 pirsa video lecture. It is not a bad introduction to the present paper because he is talking about the initial steps of a research path that led him to the new action----he calls the action "BF plus potential", but I think if it turns out to be important people will call it "Krasnov action".

Here is the video lecture:
http://pirsa.org/06110041/

Here is an audio+pdf talk at the ILQGS:
http://relativity.phys.lsu.edu/ilqgs/krasnov032007.pdf
(He aims to prove analytically that the UV fixed point of Weinberg's AsymSafe approach actually exists. So far only the computer programs of Reuter, Saueressig, Percacci and others find this fixed point numerically. Krasnov looks for the reason it could exist, and approaches it analytically. See the last two or three slides.)
http://relativity.phys.lsu.edu/ilqgs/krasnov032007.aif

The current paper refers to a couple of earlier Krasnov papers:
http://arxiv.org/abs/0811.3147
"Plebanski gravity without the simplicity constraints"
http://arxiv.org/abs/0812.3603
"Motion of a 'small body' in non-metric gravity"

This pedagogical paper can help, since the new approach builds on Plebanski's formulation:
http://arxiv.org/abs/0904.0423
Plebanski Formulation of General Relativity: A Practical Introduction

Ingemar Bengtson's comment:
http://arxiv.org/abs/gr-qc/0703114
Note on non-metric gravity
9 pages
"We discuss a class of alternative gravity theories that are specific to four dimensions, do not introduce new degrees of freedom, and come with a physical motivation. In particular we sketch their Hamiltonian formulation, and their relation to some earlier constructions."

 

[marcus]

One sign that this new Krasnov approach is getting a strong reaction is that Laurent Freidel posted a paper about it in December 2008.
http://arxiv.org/abs/0812.3200

There is a review paper that I didn't mention yet:
http://arxiv.org/abs/0711.0697
Non-metric gravity: A status report
Kirill Krasnov
13 pages, invited brief review for MPLA
(Submitted on 5 Nov 2007)
"We review the status of a certain (infinite) class of four-dimensional generally covariant theories propagating two degrees of freedom that are formulated without any direct mention of the metric. General relativity itself (in its Plebanski formulation) belongs to the class, so these theories are examples of modified gravity. We summarize the current understanding of the nature of the modification, of the renormalizability properties of these theories, of their coupling to matter fields, and describe some of their physical properties."

It's interesting what happens with black holes in Krasnov gravity. He gives what looks like a semiclassical analysis in
and the model does not have a singularity----it has re-expansion out the back door.
See slides 24-32 of http://relativity.phys.lsu.edu/ilqgs/krasnov032007.pdf
(based on joint work with Yuri Shtanov.)

The Krasnov action is classical, in the sense of non-quantum, but different from GR---defining a larger class of solutions from which GR can be recovered. It suggests that one could quantize along spinfoam lines and get a new spinfoam vertex. The idea would be to define spinfoam quantum gravity starting from the new action rather than from a version of classical GR. It looks like the Plebanski formulation of GR, but less restrictive--a constraint having been weakened.

A joint paper by Krasnov and Rovelli on the black hole in LQG has recently been posted:
http://arxiv.org/abs/0905.4916

 

[MTd2]

Here is an audio+pdf talk at the ILQGS:
http://relativity.phys.lsu.edu/ilqgs/krasnov032007.pdf
(He aims to prove analytically that the UV fixed point of Weinberg's AsymSafe approach actually exists. So far only the computer programs of Reuter, Saueressig, Percacci and others find this fixed point numerically. Krasnov looks for the reason it could exist, and approaches it analytically. See the last two or three slides.)
http://relativity.phys.lsu.edu/ilqgs/krasnov032007.aif

Maybe, but it is not far away, this paper was uploaded today

http://arxiv.org/abs/0908.4476

Section V, p.12, specially the last paragraph of the section, on p.14.

Note that, on fig. 5, besides qualitatively reproducing the confinement properties of assymptotic safety, you can see the famous crazy "worm" like structures near plank scale emerging as oscilations as you get near and near this scale. Note that this is all done in the old Barrett-Crane model.

 

[marcus]

That Christensen Khavkine Livine Speziale paper you mention ( http://arxiv.org/abs/0908.4476 ) highlights another exciting development, using the supercomputer at UWO to study spinfoams, in this case the Barrett-Crane version. Thanks for the pointer, MTd2.

In connection with Krasnov's generalization of the Plebanski action, we should mention some papers by Smolin:
http://arxiv.org/abs/0712.0977
The Plebanski action extended to a unification of gravity and Yang-Mills theory

(that appears motivated both by Garrett Lisi's E8 work and by Krasnov's, remember Krasnov's the Perimeter seminar from 2006, on video.)

A follow-on paper by Smolin and Speziale was recently flagged by MTd2.
http://arxiv.org/abs/0908.3388
A note on the Plebanski action with cosmological constant and an Immirzi parameter

"We study the field equations of the Plebanski action for general relativity when both the cosmological constant and an Immirzi parameter are present. We show that the Lagrange multiplier, which usually gets identified with the Weyl curvature, now acquires a trace part. Some consequences of this for a class of modified gravity theories recently proposed in the literature are briefly discussed."

The "class of modified gravities" is a direct reference to papers by Krasnov and his predecessors (Bengtsson, Capovilla..). And this Smolin Speziale paper is preparatory to a unification paper that those two are co-authoring with Lisi.

 

[marcus]

There are several quite nice papers by Ingemar Bengtsson bearing on this topic. Part of an earlier development that in a sense prepared the way for what Krasnov is doing. Bengtsson occasionally lapses into a frank almost "down home" style. I could mention several passages in several papers but here is one:
http://arxiv.org/abs/gr-qc/9305004
"Although we will later make some comments on three-dimensional spaces, most of our results are peculiar to four dimensions, and they all hinge on the fact that the six-dimensional space W of two-forms on a four-dimensional vector space may be split into two orthogonal subspaces, with a scalar product given by...

Moreover it turns out that there is a one-to-one correspondence between all the ways in which W may be split into two orthogonal subspaces W⁺ and W^- and the space of conformal structures on the original vector space. Assuming that the triplet of two-forms Σ_αβi forms a basis for W+, it may be shown that these basis vectors are by construction self-dual with respect to the metric..."

This is why 4D is special in the Krasnov approach, and why it is based on two-forms.

 

[marcus]

Basically the Krasnov approach (and its progenitors like Capovilla Dell Jacobson*, and Bengtsson's) are two-form gravity. Gravity is 4D geometry and the traditional way to think of geometry defined on a manifold is not as a two-form but as a metric----a distance function is what describes the geometry, the distances angles areas, etc. So in 1915 Einstein GR the gravitational field was simply the metric, or an equivalence class of metrics.

So the basic intuition we come with is that of metric gravity.

Now these people, starting with Plebanski, pull a switch on us and say don't think of the geometry as given by a metric, think of it as given by a triple of 2-forms. Or a 2-form valued in some group algebra.

So what is your gut intuition of the most basic thing that a 2-form is? For me it is a FLUX measure of a field.
You give me two vectors and I use them to describe a little parallelogram and I tell you the flux of field-lines flowing thru that parallelogram doorway. For every doorway or window, described by two vectors, I tell you a number (or in more complicated situations something more complicated than a number).

So I am telling you a bilinear functional on the tangent space. You give me two vectors and I tell you the flux number and it is a linear relation in both arguments. If you double one of the vectors you will get (approximately) twice as much flux, because the door will be twice as large. So twice as much will be going thru.

So by having this bilinear form/functional, or bilinear function of two tangent vectors, I can describe all the flux in the picture in all directions thru any window you want, framed by any two vectors you pick. So a two-form tells the whole flux story.

Krasnov calls the basic two-form by the letter B.

And this is the fundamental handle on the gravity field. It is no longer the metric g that we are so used to.

And so we need to get a new batch of intuition because this business is a strong development.

Now Krasnov says that when you have a two-form B you can consider connections A which are compatible with B. Meaning that if you take the directional derivative of B by this connection, you get zero. D_A of B = 0.
That is, the connection is such that B doesn't change, whichever way you go.

And then he takes the curvature of this connection. How weird do the contortions have to be, as you move across the face of this manifold, just so that you won't notice any difference in the two-form B?

So it all comes out of B. If you have a two-form, of the kind he prescribes, then it has a compatible connection A, and the connection has a curvature, which he calls F. And F is also a two-form. And he Krasnov, and his forefathers Plebanski Capovilla Bengtsson etc etc, will presently write down a LAGRANGIAN involving the twoform B and the curvature twoform F.

Ingemar Bengtsson likes the fact that the particulars of this construction with twoforms only work in 4D. It gives us something special about 4D to appreciate. He has a kind of down-home earthy sensibility (as well as being a grandmaster relativist). I like the frankness. Other people don't always mention that this approach is special to 4D.

And this will be a gravity theory that doesn't directly use the metric function as a handle.

*and Plebanski before them.