Personal, incomplete, and hugely biased comments on the review article "Knot theory and a physical state of quantum gravity", by Tomáš Liko and Louis H Kauffman [Ref. 1]. I will accentuate some portions from the text with bold.
 

In my opinion, Tomáš Liko and Louis Kauffman have produced an outstanding review. They haven't address just one question: is it possible in quantum gravity -- at least in principle -- to define any physical inner product on a ‘kinematical’ Hilbert space?

Nope. It seems to me that the physical Hilbert space cannot be kinematical in principle, hence we cannot define any physical inner product on the ‘kinematical’ Hilbert space in principle. Put it differently, the task of finding the physical inner product requires some brand new ‘dynamical’ Hilbert space.

If true, how can we get it?

Replace the so-called external time parameter with a new 'time parameter' that will not break the general covariance [Ref. 1]. Recall that Bill Unruh has been advocating for an "explicit (but unmeasureable) time", the so-called 'mysterious time'. Fine, but we must also cure the Hilbert space from all remnants of 'double standards' in quantum gravity.

The task of finding the elusive dynamical Hilbert space is then redirected to the following puzzle: how come there is no 'problem of 3-D space' in canonical quantum gravity? Read about it at

http://www.God-does-not-play-dice.net/Weinstein.html

Also, recall that no Dirac observables are known for General Relativity: the set which is supposed to encode all the gauge invariant information about the system cannot be found, because, in the first place, the very dynamics of GR is 'red herring'. We just don't know it.

To sum up, it is nice to see that the research effort by Tomáš Liko and Louis Kauffman has been supported by DARPA and Air Force Research Laboratory [Ref. 1]. This whole issue is very important, and may have practical applications in the foreseeable future -- provided we find some physical inner product on some physical Hilbert space. Which in turns requires understanding the Hilbert space of the non-relativistic QM: why is it incompatible with STR?

Please 'flip a quantum coin' here, and cast your vote.
 

D. Chakalov
February 14, 2006
 
 

Tomáš Liko and Louis H Kauffman, Knot theory and a physical state of quantum gravity, Class. Quantum Grav. 23 (2006) R63-R90; hep-th/0505069 v3.

p. R85: "Having discussed the achievements of the approach to quantum gravity presented in this review, we will now turn to a brief discussion of several key problems that are still open. These include:

• The physical Hilbert space: time evolution is governed by the Wheeler–de Witt equation Sˆ? = 0, where Sˆ is the operator corresponding to the complicated Hamiltonian constraint in (86). The theory should express four-diffeomorphism invariance but the canonical decomposition requires a choice of external time parameter, thus breaking general covariance. The states described here form the ‘kinematical’ Hilbert space; that is, these states only solve the quantum constraints G Iˆ and Vi . What is needed is a set of functionals that will solve the Wheeler–de Witt equation and define the physical Hilbert space, and on it a physical inner product. A possible candidate for the Wheeler–de Witt operator has been defined by Thiemann in [68], but it is not obvious that this operator in
the classical limit describes general relativity at all.

• Reconstruction problem: any candidate theory of quantum gravity should in some limit reproduce a classical background metric that satisfies the Einstein equations. Furthermore, we expect that a good theory of quantum gravity coupled to matter should in some low-energy limit reproduce quantum field theory in curved spacetime. This is, however, a very non-trivial problem to address. Matter has been coupled to gravity via the self-dual variables and has been quantized [69–73]. The problem has only been solved on the kinematical Hilbert space. As mentioned above, an inner product for physical states is currently lacking in loop quantum gravity. Therefore the dynamics of matter coupled to gravity, as for example transition amplitudes to describe scattering processes within this
framework is currently still out of reach.
...

p. R86: "We conclude with a few remarks on the background dependence of matter quantum field theories. There it is always assumed from the outset that the background spacetime is continuous to arbitrarily small scales. In the non-perturbative quantization of pure gravity we generically end up with a discrete space with length, area and volume eigenvalues. So the initial assumption of quantum field theory is false in this context, and may in fact be the origin of the ultraviolet divergences that plague the quantum field theories.
...

p. R87: "Much of his effort was sponsored by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement F30602-01-2-05022."
 
 

===========
Subject: ... it is necessary to let the clock tick once.
Date: Tue, 05 Apr 2005 16:20:39 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Louis H Kauffman <kauffman@uic.edu>
CC: witten@ias.edu, pullin@phys.lsu.edu, c.isham@imperial.ac.uk

Dear Dr. Kauffman, 

Let me share with you my thoughts on your very intriguing paper on
non-commutative worlds [Ref. 1]. 

In the latest version of your quant-ph/0503198, you wrote [Ref. 1, p.
5]: 

"Consider the act of observing X at a given time and the act of observing (or obtaining) DX at a given time. Since X and X' are ingredients in computing (X' - X)/t, the numerical value associated with DX, it is necessary to let the clock tick once.

"Thus, if one first observe X and then obtains DX, the result is different (for the X measurement) if one first obtains DX, and then observes X. In the second case, one finds the value X' instead of the value X, due to the tick of the clock. 

"1. Let X'X denote the sequence: observe X, then obtain X'. 

"2. Let XX' denote the sequence: obtain X', then observe X. 

"The commutator [X, X'] expresses the difference between these two orders of discrete measurement."

Now, instead of considering Brownian walk [Ref. 1, p. 6], let's examine the Buridan donkey walk, 

http://www.God-does-not-play-dice.net/points.html#Buridan_donkey

We don't have *scalar* quantities anymore but a "knotted" dependence of the state of a given donkey on {the rest of the donkeys}, in line with the rules of 'relational reality'. 

I wonder if you can write down the donkian Hamiltonian. 

It seems to me that the dynamics of all donkeys will be frozen, because *before* we "let the clock tick once" for any donkey, we need to fix the state of {the rest of the donkeys}, but it order to fix the state of {the rest of the donkeys}, we need to "let the clock tick once" to fix the state of the initial donkey. 

Hence the dynamics is frozen on *logical* grounds; see the Crocodile Paradox at 

http://www.God-does-not-play-dice.net/ReflexiveParadoxes.html

More on the dynamics of 'relational reality' at 

http://www.God-does-not-play-dice.net/Miller.html#note

A penny for your thoughts! I will keep them private and confidential. 

I very much hope that you will receive this email, since some distinguished colleagues of yours have reported my email communications as "spam", 

http://www.God-does-not-play-dice.net/Sepunaru.html#note

Looking forward to hearing from you, 

Sincerely yours, 

Dimi Chakalov
--
http://www.God-does-not-play-dice.net
http://www.God-does-not-play-dice.net/download.html
 

Reference 

[Ref. 1] Louis H. Kauffman, Non-Commutative Worlds - A Summary, quant-ph/0503198 v2, 3 April 2005,
http://xxx.lanl.gov/abs/quant-ph/0503198
 
 

Note: Let me again stress that the metaphysical 'principle of locality' and its mathematical presentation with commutativity of spacelike separated observables (e.g., Steven Weinberg, The Quantum Theory of Fields, Volume 1, Foundations, Cambridge: CUP, 1995, p. 198) is not violated, because the bi-directional "talk" of all donkeys does not take place in the local mode of spacetime but in the global mode. Hence any donkey can freely move along a perfectly continuous trajectory in the local mode of spacetime, and can realize, with the benefit of hindsight, that its elementary timelike displacement  had been influenced by {the rest of the donkeys} from the global mode of spacetime.

More on this atemporal bi-directional "talk" in the global mode of spacetime can be read from John Cramer and Milo Wolff here. All we have to do is to discover the mathematical presentation of this form of retarded causality (known since the time of Aristotle) and solve the puzzle of relativistic "collapse".

To sum up, and taking the risk of being again deeply boring, let me stress that the holistic effect of 'relational reality' is vanishing small only at the scale of tables and chairs, hence in classical mechanics we can apply Leibnitz' rule of differential calculus and derive the dynamics of a body from its purely kinematical snapshot at any given "point" from its trajectory; read more from David Bohm here. In other words, if we toss a ball, the feedback from {the rest of the universe} is vanishing small, and although it does influence the ball's trajectory in the "dark gaps" of the global mode of spacetime, this influence can and should be ignored. Perhaps it is safe to say that we can ignore these "dark" holistic effects up to bodies with the size of our solar system, but if we move further, we're inevitably struck with the "dark effects" of {the rest of the universe}, such as "dark matter". I think these "dark effects" set the applicable limits of Einstein's GR, hence if we want to address the large-scale "dark matter" effects of {the rest of the universe} and the puzzle of "dark energy", we shouldn't be befuddled by the calculation effectiveness of differential and tensor calculus, but should zoom in the fine structure of the infinitesimal displacement, as did Louis H. Kauffman [Ref. 1]. Perhaps the Machian quantum gravity should start with the well-known saying 'you can't stir your coffee without disturbing a star'. See also the Y-shaped zip of Alan Rendall's leather jacket here.

I hope this is not "spam", because I'm not sending these remarks to anyone.

D. Chakalov
April 6, 2005
 

==========

From: Dimi Chakalov <dchakalov@gmail.com>
Reply-To: Dimi Chakalov <dchakalov@gmail.com>
To: pullin@phys.lsu.edu
Date: Apr 5, 2005 5:21 PM
Subject: Re: failure notice

Hi Jorge,

See the message I got from you (see below): "Mail from 212.39.90.101 refused by blackhole site bl.spamcop.net". As if you have reported my email address as spam.

Perhaps you don't like my ideas,

http://www.God-does-not-play-dice.net/Kauffman.html

http://www.God-does-not-play-dice.net/Pullin.html#last

but do you believe they are "spam"?

Dimi
==========
Subject: failure notice
Date: 5 Apr 2005 13:21:56 -0000
From: MAILER-DAEMON@dsl-pw-ha1.btc-net.bg
To: dimi@chakalov.net

Hi. This is the qmail-send program at dsl-pw-ha1.btc-net.bg.
I'm afraid I wasn't able to deliver your message to the following addresses.
This is a permanent error; I've given up. Sorry it didn't work out.

<pullin@phys.lsu.edu>:
Connected to 130.39.182.71 but sender was rejected.
Remote host said: 550 5.7.1 Mail from 212.39.90.101 refused by
blackhole site bl.spamcop.net

--- Below this line is a copy of the message.

Return-Path: <dimi@chakalov.net>
Received: (qmail 24956 invoked from network); 5 Apr 2005 13:21:44 -0000
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Date: Tue, 05 Apr 2005 16:20:39 +0300
From: Dimi Chakalov <dimi@chakalov.net>
[snip