# Kodama Mama: the mother of all quantum states of gravity

1. Nov 15, 2006

### marcus

The Kodama state has been the "If-Only" of post-string Quantum Gravity for several years, since Smolin's 2003 paper "Quantum Gravity with a Positive Cosmological Constant" if not before.

As in----if only the Kodama state was normalizable, if only we had the Kodama state then we would have a nice classical limit and all our problems would go away or at least be much much easier!

You might want to call it the Holy Grail of background independent QG, but the thing about the Holy Grail was that a whole lot of people went optimistically out to look for it. But in 2003 people were kind of scared off the Kodama because of a paper of Wittens which was taken as explaining why it would never work in QG. So it was a holy grail sitting around in plain view that almost everybody tried to pretend wasn't there.

since Witten condemned the Kodama, it has taken the audacity of a brass monkey even to contemplate it, and not many researchers have.

So there is some historical background that I should gather here, and some links to previous Kodama research. Smolin co-authored with several people----Laurent Freidel, Stephon Alexander. there were a couple of people at Uni British Columbia. And now handy Andy the grad student at University of Texas.

these people form, for me, a small honor roll. brave souls all

and I could be completely wrong---it looks to me like Andy, unlikely as that seems (since he doesnt even have his PhD yet) has cured the Kodama of its earlier lameness by generalizing it (using a real Immirzi number). And if I am wrong then I look stupid and fall flat on my face.

but either way we should have some background detail about the Kodama state in QG, which please contribute to if you wish and I will try to assemble here in this thread.

I guess I better give links to the two new Kodama papers by Randono, in case anyone hasnt got them already
http://arxiv.org/abs/gr-qc/0611073
http://arxiv.org/abs/gr-qc/0611074

Last edited: Nov 16, 2006
2. Nov 15, 2006

### marcus

I just attended 3 talks by Edward Witten (monday, tuesday, wednesday---the third was a couple of hours ago) and was freshly impressed by his impressivness.

so I have some subjective feel for the weight with which he put the quietus on the Kodama. just the slightest word---the barely perceptible shake of the head. that's part of the story, i suppose.

Here is Smolin's key paper, bear in mind that this 2002 paper was what he used as his TEXTBOOK when he taught a 25 lecture introductory QG course this spring 2006 semester---that you can watch video at Perimeter media. that is how central this paper is in Smolin view of QG.

http://arxiv.org/abs/hep-th/0209079
Quantum gravity with a positive cosmological constant
Lee Smolin
59 pages

"A quantum theory of gravity is described in the case of a positive cosmological constant in 3+1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the Wheeler-deWitt equation, Planck scale, computable corrections to the energy-momentum relations for matter fields. This may lead in the next few years to experimental tests of the theory.
To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary Chern-Simons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation.
The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout. "

In 2003 Smolin did a couple more about Kodama with various co-authors.
Smolin, Freidel
http://arxiv.org/abs/hep-th/0310224
Stephon Alexander, Justin Malecki, Lee Smolin
http://arxiv.org/abs/hep-th/0309045
It played a role in this comparative survey of stringy and non-string QG
http://arxiv.org/abs/hep-th/0303185

But the main "proceed at own risk" warning sign posted on the trail was
http://arxiv.org/abs/gr-qc/0306083
A Note On The Chern-Simons And Kodama Wavefunctions
Edward Witten
10 pages

"Yang-Mills theory in four dimensions formally admits an exact Chern-Simons wavefunction. It is an eigenfunction of the quantum Hamiltonian with zero energy. It is known to be unphysical for a variety of reasons, but it is still interesting to understand what it describes. We show that in expanding around this state, positive helicity gauge bosons have positive energy and negative helicity ones have negative energy. Some of the negative energy states would have negative norm. We also show that the Chern-Simons state is the supersymmetric partner of the naive fermion vacuum in which one does not fill the fermi sea. Finally, we give a sort of explanation of 'why' this state exists. Similar properties can be expected for the analogous Kodama wavefunction of gravity."

Stephon Alexander had a couple of more Kodamarelated things after 2003
http://arxiv.org/abs/hep-th/0503146
A Quantum Gravitational Relaxation of The Cosmological Constant (not directly about, but mentioned IIRC)
http://arxiv.org/abs/gr-qc/0503062
Fermionic sectors for the Kodama state
Stephon Alexander, Kristin Schleich, Donald M. Witt
4 pages
SLAC-PUB-10841

"Diffeomorphisms not connected to the identity can act nontrivially on the quantum state space for gravity. However, in stark contrast to the case of nonabelian Yang-Mills field theories, for which the quantum state space is always in 1 dimensional representation of the large gauge transformations, the quantum state space for gravity can have higher dimensional representations. In particular, the Kodama state will have 2 dimensional representations, that is sectors with spin 1/2, for many topologies that admit positive scalar curvature. The existence of these spin 1/2 states are used to point out a possible answer to certain criticisms raised recently in the literature. "

This last was with a couple of people at UBC Vancouver. It was explicitly groping for some way to counter the criticism from Witten.

Last edited: Nov 15, 2006
3. Nov 15, 2006

Staff Emeritus
Marcus, "Andy the Grad student" has done some remarkable work, but so have you here. This is terrific living history, and you are the man on the spot. Blogging from the trenches, as it were. Keep it up.

BTW we need some technical background on Kodama, WKB states, and all that. I am going to explore wiki and google and see what I can find in more surveyable form than reading all those papers.

4. Nov 15, 2006

### marcus

So what is wrong with the Kodama state?

Andy Randono takes up this question in the first of the two papers he posted yesterday on arxiv.

Andy seems to have cured Kodama ailments by generalizing it, but before he talks about his generalization, right in the introduction he talks about what is wrong with the traditional Kodama.

==quote Randono paper A page 3==

1.1 Problems
Despite all of these positive attributes of the Kodama state, the state is
plagued with problems. Among these are the following:

Non-normalizability: The Kodama state is not normalizable under
the kinematical inner product, where one simply integrates $|\Psi|^2$ over all values of the complex Ashtekar connection. The state is not known to be normalizable under a physical inner product defined by, for example, path integral methods. Linearized perturbations around the state are known to be non-normalizable under a linearized inner product[7].

CPT Violation: The states are not invariant under CPT[8]. This is particularly poignant objection in view of the CPT theorem of perturbative quantum field theory, which connects CPT violation with Lorentz violation. It is not known if the result carries over to non-perturbative quantum field theory, but it has yet to be demonstrated that the Kodama state does not predict Lorentz violation.

Negative Energies: It has been argued by analogy with a similar non-perturbative Chern-Simons state of Yang-Mills theory that the Kodama state necessarily contains negative energy sectors[8]. If the energy of one sector of the state is strictly positive, the CPT inverted state will necessarily contain negative energy sectors.

Non-Invariance Under Large Gauge Transformations: Although the state is invariant under the small gauge transformations generated by the quantum constraints, it is not invariant under large gauge transformations where it changes by a factor related to the winding number of the map from the manifold to the gauge group. However, it has been argued that the non-invariance of the Kodama state under large gauge transformations give rise to the thermal properties of de Sitter spacetime2[9]. Thus, non-invariance under large gauge transformations could be a problem or a benefit, but it is deserving of mention.

Reality Constraints: The Lorentzian Kodama state is a solution to the quantum constraints in the Ashtekar formalism where the connection is complex. To obtain classical general relativity one must implement reality conditions which ensure that the metric is real. It is an open problem as to how to implement these constraints on a general state. Generally it is believed that the physical inner product will implement the reality constraints, but this could change the interpretation of the state considerably.

1.2 Resolution
...
...
==endquote==

selfAdjoint, thanks for the kind words! hope some others weigh in on this

Last edited: Nov 15, 2006
5. Nov 16, 2006

### Chronos

The non-invariance under large gauge transformations is the most troubling issue, in my mind. I think this is the most fundamental question raised.

6. Nov 16, 2006

### marcus

Hi Chronos,
this is off-topic but did you see this Zippy the Pinhead comicstrip?

Thanks for highlighting this aspect of the Kodama discussion!
Naturally what one wants is to see how Randono's NEW Kodama states do in this department :-)

His generalization of the Kodama seems (at first sight) to have taken care of all the problems with the old Kodama that he mentions. So let's check out how the new Kodamas do regards large gauge transformations.

Last edited: Nov 16, 2006
7. Nov 17, 2006

### Chronos

I have now, hehe. Zippy likes to hedge his bets, as do I. I guess it is a matter of taste. All four issues are interesting and could raise different issues with different people. I recoil in horror at the prospect of negative energy, but, find it more palatable than gauge invariance violations.