Exploring the Generalized Kodama State: Properties and Physical Interpretation

[marcus]

https://www.physicsforums.com/showthread.php?p=1160863#post1160863

Andrew Randono is a grad student at U Texas Austin (where Steven Weinberg is, not to mention Jacques Distler). His advisor AFAIK is Richard Matzner (a Gen Rel expert at the Center for Relativity there). Randono spent the summer at Perimeter this year.
I think he has written a major paper generalizing the Kodama state. A two-part paper actually.

I put it on the links thread, but it could be useful to have a separate thread for discussion.
here are the abstracts:
http://arxiv.org/abs/gr-qc/0611073
Generalizing the Kodama State I: Construction
Andrew Randono
First part in two part series, 20 pages

The Kodama State is unique in being an exact solution to all the ordinary constraints of canonical quantum gravity that also has a well defined semi-classical interpretation as a quantum version of a classical spacetime, namely (anti)de Sitter space. However, the state is riddled with difficulties which can be tracked down to the complexification of the phase space necessary in its construction. This suggests a generalization of the state to real values of the Immirzi parameter. In this first part of a two paper series we show that one can generalize the state to real variables and the result is surprising in that it appears to open up an infinite class of physical states. We show that these states closely parallel the ordinary momentum eigenstates of non-relativistic quantum mechanics with the Levi-Civita curvature playing the role of the momentum. With this identification, the states inherit many of the familiar properties of the momentum eigenstates including delta-function normalizability. In the companion paper we will discuss the physical interpretation, CPT properties, and an interesting connection between the inner product and the Macdowell-Mansouri formulation of general relativity. "

http://arxiv.org/abs/gr-qc/0611074
Generalizing the Kodama State II: Properties and Physical Interpretation
Andrew Randono
Second paper in two part series. 18 pages

"In this second part of a two paper series we discuss the properties and physical interpretation of the generalized Kodama states. We first show that the states are the three dimensional boundary degrees of freedom of two familiar 4-dimensional topological invariants: the second Chern class and the Euler class. Using this, we show that the states have the familiar interpretation as WKB states, in this case corresponding not only to de Sitter space, but also to first order perturbations therein. In an appropriate spatial topology, the de Sitter solution has pure Chern-Simons functional form, and is the unique state in the class that is identically diffeomorphism and SU(2) gauge invariant. The q-deformed loop transform of this state yields evidence of a cosmological horizon when the deformation parameter is a root of untiy. We then discuss the behavior of the states under discrete symmetries, showing that the states violate P and T due to the presence of the Immirzi parameter, but they are CPT invariant. We conclude with an interesting connection between the physical inner product and the Macdowell Mansouri formulation of gravity."

 

[marcus]

To give some of the flavor of Randono's work, i will quote a few passages. Here is part of the Introduction of Paper A
===quote===
1 Introduction
Perturbative techniques in quantum field theory and their extension to quan- tum gravity are unparalleled in computational efficacy. In addition, because one can always retreat to the physical picture of particles as small field perturbations propagating on a classical background, perturbation theory maximizes the ease of transition from quantum to classical mechanics, and many processes can be viewed as quantum analogues of familiar classical events. However, the transparent physical picture disappears in systems where the distinction between background and perturbation to said back- ground is blurred. Such systems include strongly interacting systems, such as QCD, or systems where there is no preferred background structure, such as general relativity. In contrast, non-perturbative and background independent approaches to quantum gravity do not distinguish background from perturbation, and are, therefore, appropriate for modeling the quantum mechanical ground state of the universe itself that, it is hoped, will serve as the vacuum on which perturbation theory can be based. However, this is often at the expense of losing the smooth transition from a quantum description to its classical or semi-classical counterpart as evidenced, for example, by the notorious problem of finding the low energy limit of Loop Quantum Gravity.

The sticking point is that pure quantum spacetime may be sufficiently divorced from our classical understanding of fields on a smooth Riemannian manifold, that matching quantum or semi-classical states with classical analogues may be extremely difficult.

In this respect the Kodama state is unique. Not only is the state an
exact solution to all the constraints of canonical quantum gravity, a rarity in itself, but it also has a well defined physical interpretation as the quantum analogue of a familiar classical spacetime, namely de Sitter or anti-de Sitter space depending on the sign of the cosmological constant[1, 2, 3]. Thus, the state is a candidate for the fulfillment of one of the distinctive advantages of a non-perturbative approach over perturbative techniques: the former has the potential to predict the purely quantum mechanical ground state on which perturbation theory can be based. In addition, the Kodama state has many beautiful mathematical properties relating the seemingly disparate fields of abstract knot theory and quantum field theory on a space of connections[4].

In particular, the exact form of the state is known in both the connection representation where it is the exponent of the Chern-Simons action, and in the q-deformed spin network representation where it is a superposition of 2...

===endquote===

Further on, another sample exerpt from the Introduction section 1.2 (page 6)

==quote==
...When this operator is used in the Hamiltonian constraint, all of the states are annihilated by the constraint. In a follow-up paper we will then show that the generalized states are free of most of the problems associated with the original incarnation of the Kodama state, and we will discuss the physical interpretation of the new states and their relation to de Sitter space. We conclude with an intriguing relation between the physical inner product of two generalized states and the Macdowell-Mansouri formulation of gravity...
==endquote==

 

[marcus]

From section 4, Concluding Remarks, page 18
==quote==
4 Concluding Remarks
We have shown that the Kodama state can be generalized to real values of the Immirzi parameter, and the generalization appears to open up a large class of physical states. In the connection representation they are the exponent of the Chern-Simons invariant together with an extra term, so we might expect that in the spin network basis they may be expressed as a generalization of the Kauffman bracket. The states share many properties in common with the momentum eigenstates when the Levi-Civita is identified with the “momentum” parameterizing the family of states. Following this analogy, we have shown that the generalized Kodama states are eigenstates of a naturally defined Levi-Civita curvature operator with eigenvalues given by the curvature configuration parameterizing the state. This definition of the curvature operator places the full sector in the physical Hilbert space.

Naturally this operator must withstand a program of consistency checks to verify its viability as the Levi-Civita curvature operator, but we hope that we have laid the groundwork to begin such a program.

We set out to generalize the Kodama state in an attempt to resolve some of the known issues associated with the original version. We have shown that our generalization solves two of the known problems: reality conditions, and normalizability.

The problem of defining the reality conditions does not exist in the real theory, and we have shown that the states are delta-function normalizable under a natural inner-product. In the second paper of this two paper series we will show that the states are CPT invariant, and, by fine tuning of the coupling constants, they can be made to be invariant under large transformations. In addition, we will discuss the physical interpretation of the states.

==endquote==

 

[marcus]

https://www.physicsforums.com/showthread.php?p=1160863#post1160863
...
http://arxiv.org/abs/gr-qc/0611073
...
http://arxiv.org/abs/gr-qc/0611074
Generalizing the Kodama State II: Properties and Physical Interpretation
...familiar 4-dimensional topological invariants: the second Chern class and the Euler class. Using this, we show that the states have the familiar interpretation as WKB states, in this case corresponding not only to de Sitter space, but also to first order perturbations therein..."

http://en.wikipedia.org/wiki/WKB_approximation

Judging from the Wiki entry, it should really be called the "Jeffreys approximation" after the mathematician Harold Jeffreys who devised it in 1923.

Apparently the eponymous Wentzel, Kramers and Brillouin were not aware of Jeffreys' prior work.

"WKB approximation

In physics, the WKB (Wentzel-Kramers-Brillouin) approximation, also known as WKBJ approximation, is the most familiar example of a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be slowly changing.

This method is named after physicists Wentzel, Kramers, and Brillouin, who all developed it in 1926. In 1923, mathematician Harold Jeffreys had developed a general method of approximating linear, second-order differential equations, which includes the Schrödinger equation. But since the Schrödinger equation was developed two years later, and Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, Jeffreys is often neglected credit. Early texts in quantum mechanics contain any number of combinations of their initials, including WBK, BWK, WKBJ and BWKJ."

=======================

Here are some nice pictures of some mountain climbers in Colorado Rockies summer 2005
http://www.ma.utexas.edu/users/stirling/0508colorado/0508colorado.html
Andy is the one in shorts who always seems to be enjoying the trek.
they were climbing "Fourteeners"-----peaks in excess of 14.000 feet.
Stirling says he bagged 7 such peaks in 7 days. That kind of trip.
Andy started the PhD program at U Tex in 2002, so he is probably in candidacy now and
finishing his dissertation

 

[marcus]

It gets clearer, the more I read of it.

Randono's paper is a classic.

How about this italicized sentence on page 8 of paper B. This is in the section on parity reversal. How it affects the generalized Kodama states.
(Maybe we should be calling them "Randono states".)
==quote==

The net effect on the wave functions 18 is an inversion of the Immirzi parameter:

$$\Psi^\beta_R \rightarrow P(\Psi^\beta_R ) = \Psi^{-\beta}_R$$This is consistent with the general maxim with a growing body of evidence[3, 4, 5, 6, 7],
The Immirzi parameter is a measure of parity violation built into the framework of quantum gravity.

==endquote==

 

[selfAdjoint]

This is consistent with the general maxim with a growing body of evidence[3, 4, 5, 6, 7],
The Immirzi parameter is a measure of parity violation built into the framework of quantum gravity.

Borzhe moi! (Strikes forehead). Can this be it?

 

[marcus]

You and I have enough experience to know that it is NEVER it
You had me laughing for a moment uncontrollably. It never is, still, it does get closer!

============

BTW Baez was here yesterday evening reading a Kodama thread-----it was the other one ("Kodama mama") when I happened by.
he may be too busy to talk to us directly but I expect he will be putting some comment together before too long. Hope so.
If I have to be disillusioned, rather be by him. And if he likes it, so much the better!

 

[CarlB]

Having parity violation built into gravitation makes a lot of sense to me.

 

[selfAdjoint]

If I have to be disillusioned, rather be by him. And if he likes it, so much the better!

Yes, he would be the most preferred disillusioner, he or Lee Smolin. Then in order of preference:

• Witten rouses himself to diss it
• Distler posts another Trouble essay
• Lubos points out a flaw, and is correct!

 

[marcus]

Witten rousing himself would almost be worth the price of admission. I think Distler may be pinned down on this occasion by being in the same physics department at Austin.
Matzner, Randono's advisor, is nearly bald and has icy blue eyes. http://www.ph.utexas.edu/faculty/matzner.html
His faculty photograph does not have the usual jolly ingratiating smile.
He looks patient but not especially happy to have been interrupted at his work.