Ashtekar's Program: No Escape From Hidden Infinities?

Comments on

Gravity and the Quantum, by A. Ashtekar
gr-qc/0410054 v1, 13 October 2004, 18:12:52 GMT

Experts in quantum gravity may wish to skip this philosophical exercise and read some technical issues here


 

According to my English dictionary WordWeb v3.01, the definition of 'escape' is "an inclination to retreat from unpleasant realities through diversion or fantasy." At the end of this note, I will slightly modify this definition to match the case examined here.

Gary T. Horowitz (see "Spacetime in String Theory", gr-qc/0410049 v2) and Abhay Ashtekar ("Gravity and the Quantum", gr-qc/0410054 v1) have submitted two papers to the January 2005 issue of New Journal of Physics. This is an invited celebratory focus issues related to Einstein's seminal work from 1905, under the title "Spacetime 100 Years Later". The burden of meticulous reading and editing these masterpieces has been shared by their long-time colleagues Richard Price and Jorge Pullin.

I will only try to draw a general picture of Ashtekar's program, offering my critical comments and questions. The question mark in the title of my comments refers to an old, and still unsettled, debate I have with Prof. Karel Kuchar. Back in 1993, Karel Kuchar stated that Ashtekar's quantization program is (quote) self-contradictory (unquote). See K. Kuchar's "Canonical Quantum Gravity", gr-qc/9304012, p. 25: "Secondly, even when one disregards this difficulty, one should notice that the proposal as it stands is self-contradictory." Here, on p. 25 from the pdf file from gr-qc/9304012, K. Kuchar refers to Ashtekar's quantization program from 1991: A. Ashtekar, "Lectures on Non-Perturbative Canonical Gravity", World Scientific, Singapore, 1991, Chapter 10.

I have exchanged email with Gary Horowitz (see below), and now will quote from A. Ashtekar's paper. All numbers in square brackets are references in his gr-qc/0410054 v1. For brevity, I've highlighted some key issues in red.

From the abstract: "The emphasis is on underlying ideas, conceptual issues and overall status of the program rather than mathematical details and associated technical subtleties."

These "technical subtleties" are the most important, if not crucial, part of the Ashtekar's program. I regret that they were not explained in plain English. In the past three years (see gr-qc/0112038, footnote 1), very often the following text has been appearing in his papers:

"To achieve this goal in 3+1 dimensions, one needs a much better understanding of the theory of (intersecting) knots in 3dimensions."

This same text appeared again in his latest paper gr-qc/0410054 v1 (see footnote 9), and I was again unable to understand exactly how Abhay Ashtekar would use "the theory of (intersecting) knots in 3dimensions" to elucidate the problem of hidden infinities. More on this hidden stuff later, suffice it to say that its immediate impact is on (i) the inner-product problem (Hilbert space problem), and (ii) the recovery of the asymptotically flat 3-D space, as will be explained later.

To begin with, let's see footnote 1: "Since this article is addressed to non-experts, except in the discussion of very recent developments, I will generally refer to books and review articles which summarize the state of the art at various stages of development of quantum gravity."

Strangely enough, the seminal papers on quantum gravity by Karel Kuchar, published since 1981, have been omitted. The only reference to Karel Kuchar's research is ref. [15], which is "Canonical methods of quantization" published in 1981.

Another strange omission (shared, accidentally or not, by Carlo Rovelli; see ref. [39]) is Claus Kiefer's monograph Quantum Gravity (Oxford University Press, 2004, 320 pages; ISBN: 0-19-850687-2). Abby Ashtekar is aware of forthcoming publications by Oxford University Press (see ref. [52]), but does not mention Claus Kiefer's contribution to the quest for quantum gravity, nor any other publication by C. Kiefer. (Prof. C. Kiefer has been chairman of the Gravity Section of the German Physical Society since 2000; list of publications here.)

Next quote: "It is because there is no background geometry, for example, that it is so difficult to analyze singularities of the theory and to define the energy and momentum carried by gravitational waves. Since there is no a priori space-time, to introduce notions as basic as causality, time, and evolution, one must first solve the dynamical equations and construct a space-time."

Since this is a general review of quantum gravity addressed to non-experts, I believe it would be fair to tell them more about some difficulties known since the inception of Einstein's GR and its "linearized" approximation that would supposedly contact some fictional "flat spacetime". Such a crude approximation (something like 'a spherical cow') is possible only in 2+1-dimensions (see Steve Carlip below).

"Because the situation is conceptually so novel and because there are no direct experiments to guide us, reliable results require a high degree of mathematical precision to ensure that there are no hidden infinities. Achieving this precision has been a priority in the program."

Here A. Ashtekar hit the nail on the head: no hidden infinities.

"A key open problem in loop quantum gravity is to show that the scalar/Hamiltonian constraint -- either Thiemann’s or an alternative such as the one of Gambini and Pullin -- admits a ‘sufficient number’ of semi-classical states. Progress on this problem has been slow because the general issue of semi-classical limits is itself difficult in any background independent approach.[footnote 12] However, a systematic understanding has now begun to emerge and is providing the ‘infra-structure’ needed to analyze the key problem mentioned above [38,52]. But, while there are promising ideas to complete step iii), substantial further work is necessary to solve this problem.
--
[footnote 12]: "In the dynamical triangulation [30,41] and causal set [37] approaches, for example, a great deal of care is required to ensure that even the dimension of a typical space-time is 4."

Let me enumerate the major problems, as a member of the target audience of non-experts.

1. How do we evaluate quantitatively the vague notion of  "a ‘sufficient number’ of semi-classical states"? They have to be infinitely many, as we know from the elementary school formula of deriving the circumference of a circle.

2. What is "step iii)"? To (quote) "introduce an inner-product and interesting observables, and develop approximation schemes, truncations, etc to explore physical consequences." Well, the inner product problem is severe, to say the least. See Claus Kiefer's "Conceptual issues in quantum cosmology, gr-qc/9906100.

3. Let's not forget that Ashtekar's loop quantum gravity has to recover "the dimension of a typical space-time" that is exactly  4 . The 'proof of the pudding' that the dimension of the spacetime is exactly4  is very simple: we should be able to look around, and see as far as we can, say, 13 billion light years.

Next quote: "As is almost always the case with constrained systems, there are many more solutions and the ‘spurious ones’ have to be eliminated by the requirement that the physical norm be finite."

4. Sure. Otherwise Ashtekar's quantization program will be self-contradictory, meaning it would either eliminate the semi-classical states needed for its "corresponding principle" or would be self-referential due to some closed-loop "dynamics" resulting from its "Hilbert space for quantum kinematics of background independent theories of connections", as explained in ref. [38] below. Perhaps Karel Kuchar or some other expert in quantum dynamics of constrained systems would like to shed light on this tantalizing issue. (If the operators are indeed Hermitian, it shouldn't be terribly difficult to trace the resulting "dynamics", I suppose.)

First and foremost, how do we identify and separate the 'spurious solutions' from the "authentic" solutions that will show up in calculating the circumference of a circle at the scale of tables and chairs?

Next quote: "In 2+1 gravity, the connection formulation used here naturally leads to a complete set of Dirac observables and the inner-product can be essentially fixed by the requirement that they be self-adjoint. In 3+1 gravity, by contrast, we do not have this luxury and the problem of constructing the physical inner-product is therefore much more difficult."

5. What is the exact origin of these difficulties ensuing from 3+1  gravity? A hint from Steve Carlip's web site (URL here) might help for understanding the qualitative difference between 2+1 and 3+1 gravity: "In four spacetime dimensions, this gives the four phase space degrees of freedom of ordinary general relativity, two gravitational wave polarizations and their time derivatives. If n=3, on the other hand, there are no field degrees of freedom: up to a finite number of possible global degrees of freedom, the geometry is completely determined by the constraints. (...) In particular, if there is no matter, the Einstein field equations in 2+1 dimensions imply that spacetime is flat."

Next quote: "However there are many ambiguities [38] and none of the candidate operators has been shown to lead to a ‘sufficient number of’ semi-classical states in 3+1 dimensions."

A quick look at ref. [38]: Ashtekar A and Lewandowski L 2004 Background independent quantum gravity: A status report, Class. Quant. Grav. 21 R53-R152. Available also as gr-qc/0404018 v2. See p. 35 from the pdf file: "This is our Hilbert space for quantum kinematics of background independent theories of connections."

That's it, "the natural Hermitian inner product" resulting from the idea to "introduce an inner product on the space of all cylindrical functions via (4.42)".  See also Eq. (7.15) on p. 71. Only it doesn't work for 3+1 dimensions, as acknowledged by A. Ashtekar himself.

But maybe there is light in the tunnel: "As our knowledge of invariants of intersecting knots deepens, this approach is likely to provide increasingly significant insights. In particular, it has the potential of leading to a formulation of quantum gravity which does not refer even to a background manifold (see footnote 9)."

6. What does footnote 9 say? "To achieve this goal in 3+1 dimensions, one needs a much better understanding of the theory of (intersecting) knots in 3dimensions." Nothing has been said, even remotely, about the Hilbert space problem. How would "the natural Hermitian inner product", which supposedly encodes the probability current, be conserved in time  if  that same "time" is being generated by that same unitary evolution? These are indeed fascinating issues, as acknowledged by A. Ashtekar in July 1993, gr-qc/9302024.

To sum up, my conclusion about A. Ashtekar's "Gravity and the Quantum", dedicated to the non-experts, will be made by modifying slightly the definition of 'escape', as promised in the beginning of this exercise. Now the meaning of 'escape' would be the following:

"An inclination to retreat from the unpleasant realities of the inevitable hidden infinities in loop quantum gravity through diversion to exotic math, 'spherical cow' approximations, and total ignorance of the expectations of the non-experts."

We, the non-experts, know from experts such as John Baez that, in order to have dynamics -- an intrinsic time interval associated to any infinitesimal timelike displacement -- there must be some background, something which can continuously affect the dynamics of the system while remaining unaffected by it. This could only be the Aristotelian First Cause, the ultimate Unmoved Mover, also known as Kuchar's Perennials. And the only entity that can 'remain unaffected while affecting' is the 'the universe as ONE', the last Holon whose "physical" manifestation has vanished completely -- a bare mathematical "point". It is UNspeakable and governs the dynamics of all physical systems "from outside as an unmoved mover", as explained by Karel Kuchar. Read a simple explanation of the non-linear and non-local dynamics of 'relational reality' here, and recall that A. Ashtekar employs the so-called linearized approximation of Einstein's GR. It can work only for 2+1 dimensional spacetime, as explained by Steve Carlip. The notion of 'asymptotically flat 3+1 dimensional spacetime', as tacitly implied in each and every calculation performed within the "linearized approximation" of Einstein's GR, is misleading and wrong. It is, in fact, self-contradictory, since it is valid only and exclusively only for 2+1 dimensional spacetime. Also, we made a huge mistake by introducing double standards for space and time in geometrodynamics, assuming that the intrinsic properties of 3-D space can be excluded from the basic principle of Diff(M)-invariance. To reach quantum gravity, we need to upgrade Einstein's GR, as I tried to argue here, here, and here, following the path of Chris Isham and Karel Kuchar (I'm just a psychologist).

Going back to Ashtekar's quantization program, it is inevitably self-contradictory, as noted by Karel Kuchar in 1993. It contains hidden infinities and hence cannot, even in principle, recover the asymptotically flat 4-D spacetime. You can't recover something (4-D spacetime) that has been deleted from the outset (3-D spacetime), plain and simple.

We must always keep the 4-D spacetime in order to recover it at the scale of tables and chairs. See the two email messages below.
 

D. Chakalov
October 14, 2004
Last update: October 16, 2004

P.S. I just received one complain. In a nutshell, it was claimed that mathematical physicists don't do philosophy, and hence it is not reasonable to expect that criticism expressed on philosophical grounds will be (i) respected and (ii) answered.

Regarding the point (i), let me quote from two mathematical physicists, Hermann Weyl and David Hilbert.

H. Weyl wrote: "The introduction of numbers as coordinates is an act of violence whose only practical vindication is the special calculatory manageability of the ordinary number continuum with its four basic operations." (Hermann Weyl, Philosophy of Mathematics and Natural Science, Princeton University Press, 1949).

And David Hilert wrote (Grundsätzliche Fragen der modernen Physik, Lecture I, Hamburg, July 26, 1923): "A sentence about nature, expressed in coordinates, is only then a proposition about the objects in nature, if the sentence has a content which is independent of the coordinates. ("Ein in Koordinaten ausgedrüuckter Satz über die Natur ist nur dann eine Aussage über die Gegenstände in der Natur wenn er von den Koordinaten unabhängig einen Inhalt hat.")

Since 1999, I've been arguing in my email sent to A. Ashtekar and his colleagues that this quantization program needs a new reference object that will be omnipresent and "outside" 4-D spacetime. To meet the criteria set by H. Weyl, this new reference object (called 'global mode of spacetime') should be written in explicitly coordinate-free form. It is everywhere, and its "proper time", if read by an inanimate physical clock, would be zero, like riding a photon. It shows up in the Hamiltonian constraint in the Wheeler-DeWitt equation, for example. It is a special atemporal luxonic stuff. Pure light (and cognition, maybe). This 'pure light' is the ultimate "background", as explained above. It creates the 'context' of each and every physical stuff that we treat with the principle of general covariance, as stressed by D. Hilbert: we can make a proposition about the objects in nature, if the sentence has a content which is "independent of the coordinates."

Hence the very fact that we use successfully the principle of general covariance presupposes the existence of this special atemporal luxonic "background", which has to be written in explicitly coordinate-free form, after H. Weyl. Should you are interested, please write me back.

As to point (ii), I've lost hope to hear from A. Ashtekar long time ago. I'm ready to acknowledge that my criticism was too harsh, but since I won't hear from him anyway, I decided to be very frank. Perhaps his colleagues, friends, and students cannot afford such luxury.

I always get very frank when I read papers dedicated to Albert Einstein's Annus Mirabilis. How about you?
 

D.C.
Saturday, October 16, 2004, 17:18:12 GMT
 

================
 

Subject: Spacetime 100 Years Later
Date: Wed, 13 Oct 2004 05:32:19 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: gary@physics.ucsb.edu
CC: pullin@rouge.phys.lsu.edu

Hi Gary,

In gr-qc/0410049 v2, you wrote: 

"But the problem has always been: If space and time are not fundamental,
what replaces them? Here the answer is that there is an auxiliary spacetime metric which is fixed by the boundary conditions at infinity."

I wonder, would this "auxiliary spacetime metric which is fixed by the boundary conditions at infinity" help you recover an asymptotically flat 3-D space in which it is possible to look around, and see as far as we can,

http://God-does-not-play-dice.net/Kiefer.html#3D

Sincerely,

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

======
Subject: Re: Spacetime 100 Years Later
Date: Wed, 13 Oct 2004 19:57:12 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Gary Horowitz <gary@physics.ucsb.edu>
CC: pullin@rouge.phys.lsu.edu
References: 1

Dear Gary,

>    At the moment, we do not know how to describe an
> asymptotically flat space in a way similar to the asymptotically
> anti de Sitter space.

I'm afraid the problem is not "at the moment" but is generic to the very nature of 'relational reality',

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

The issue is known since 1944; see the seminal paper by Hermann Weyl at

http://God-does-not-play-dice.net/Jacobson.html#Weyl

I will be happy to elaborate, should you and/or Jorge are interested.

Dimi Chakalov
 
 


Subject: An essential germ of truth
Date: Tue, 18 Dec 2001 12:33:22 +0200
From: "Dimiter G. Chakalov" <dchakalov@surfeu.at>
To: Abhay Ashtekar <ashtekar@gravity.phys.psu.edu>
CC: baez@math.ucr.edu, carlo@rovelli.org, l.smolin@ic.ac.uk
BCC: [snip]

Dear Professor Ashtekar,

I believe it is not an exaggeration to claim that loop quantum gravity has captured "an essential germ of truth", as you claim in your recent gr-qc/0112038 [Ref. 1].

To achieve the goal of loop quantum gravity in 3+1 dimensions, one needs to somehow proceed from kinematics to dynamics. You wrote:

"To summarize, the crux of dynamics in the Hamiltonian approach lies in quantum constraints. While the quantum Gauss and co-vector/diffeomorphism constraints have been solved, it is not clear if any of the proposed strategies to solve the scalar/Hamiltonian constraint incorporates the familiar low energy physics."

May I ask a simple question, following John Baez' dictum "Think deeply of simple things": Why is the issue of quantum dynamics still *not* clear, after fifteen years of research efforts?

Isn't it because quantum dynamics needs a brand new "background" to recover/reach the case of 3+1 dimensions? If so, this new "background" could be the global mode of time, as outlined at my web site. This new "background" is hidden due to the speed of light, and it may be essential to your theory, simply because you need something entirely different from 3+1 dimensions to define quantum dynamics.

Let's think deeply on simple things. In order to understand and explain an apple, you need something that is 'non-apple', such that an apple can be compared to it. In your case, you too need something (the global mode of time maybe) w.r.t.w. you can define quantum dynamics in a meaningful fashion. Otherwise your loop quantum gravity may encounter the paradox of Baron Münhausen: you too will have to 'lift' yourself by pulling up your hair.

You simply need a new reference object, don't you? I believe the global mode of time can obey the requirement of diffeomorphism invariance, the very requirement which leads to the paradoxical (if not pathological) claim that there is no time (the notorious 'problem of time') *nor* space. If you eliminate the time (as advocated by Rovelli, regrettably), you lose space as well due to the full reparametrization invariance of general relativity. Hence to recover space and time, you need something ontologically different than spacetime, some new reference object w.r.t.w. you can understand, define, and explain spacetime.

That's how the brain works, yours included.

I hope that all my email messages sent in the past three years have been safely received.

Regards,

Dimiter G. Chakalov
http://members.aon.at/chakalov
http://members.aon.at/chakalov/dimi.html
--
I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case I would rather be a cobbler, or even an employee in a gaming-house, than a physicist.

A. Einstein, Born-Einstein Letters, 29 April 1924

---
[Ref. 1] Abhay Ashtekar. Quantum Geometry and Gravity: Recent Advances. Mon, 17 Dec 2001 16:55:16 GMT,
http://xxx.lanl.gov/abs/gr-qc/0112038

Report of the plenary talk at the 16th International Conference on General Relativity and Gravitation, held at Durban, S. Africa in July 2001

From the abstract: "These developments spring from a detailed quantum theory of geometry that was systematically developed in the mid-nineties and have added a great deal of optimism and intellectual excitement to the field."

"There *is* a manifold but no metric, or indeed any other physical fields, in the background.

[Footnote 1: In 2+1 dimensions, although one begins in a completely analogous fashion, in the final picture one can get rid of the background manifold as well. Thus, the fundamental theory can be formulated combinatorially  {loop,books} . To achieve this goal in 3+1 dimensions, one needs a much better understanding of the theory of (intersecting) knots in 3dimensions.]
...
"Although there is no natural unification of dynamics of all interactions in loop quantum gravity, it does provide a kinematical unification. (...) A key difference, however, is that while a background space-time metric is available and crucially used in gauge theories, there are no background fields whatsoever now. This absence is forced on us by the requirement of diffeomorphism invariance (or 'general covariance'). (...) In our case, the situation is much more drastic: there is no background metric what so ever!
...
"To summarize, the crux of dynamics in the Hamiltonian approach lies in quantum constraints. While the quantum Gauss and co-vector/diffeomorphism constraints have been solved, it is not clear if any of the proposed strategies to solve the scalar/Hamiltonian constraint incorporates the familiar low energy physics.
...
"Repeated occurrence of such 'unreasonable' simplifications suggests that the ideas underlying loop quantum gravity may have captured an essential germ of truth."
 
 

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Subject: Re: An essential germ of truth
Date: Sat, 25 Jan 2003 23:16:15 +0200
From: Dimi Chakalov <dchakalov@surfeu.at>
To: Abhay Ashtekar <ashtekar@gravity.phys.psu.edu>
CC: carlo@rovelli.org, kuchar@physics.utah.edu
 

Dear Professor Ashtekar,

Regarding my email of Tue, 18 Dec 2001 12:33:22 +0200,

http://members.aon.at/chakalov/Ashtekar.html

I'm wondering if you have replied to the critical remarks by Professor Karel Kuchar in "Canonical Quantum Gravity" (13th Conference on General Relativity and Gravitation (GR-13), Cordoba, Argentina, 29 June - 4 July 1992). He stated that your quantization program is self-contradictory,

http://members.aon.at/chakalov/intro.html#Kuchar93

I haven't detected in the past twelve years some dramatic change in your quantization program. Please advise.

As to my email mentioned above, I hope you have noticed that the so-called global mode of time is very close to Perennials of Prof. Kuchar. It seems to me that the problems of your quantization program have a well-known origin, I believe it was Lao-tze who first noticed that if all things change, there is nothing you will try to hold onto.

I will be happy to learn whether you have managed to solve the problems of your quantization program, as known since the time of Lao-tze and identified by your distinguished colleague in 1992.

Sincerely yours,

Dimiter G. Chakalov
http://members.aon.at/chakalov
--
Dead matter makes quantum jumps; the living-and-quantum matter is smarter.

===========

Message-ID: <360EB192.FE877DB8@omega.bg>
Date: Sun, 27 Sep 1998 16:43:46 -0500
From: "Dimiter G. Chakalov" <maver@omega.bg>
X-Mailer: Mozilla 4.06 [en] (Win98; I)
To: ashtekar@phys.psu.edu
CC: c.isham@ic.ac.uk, john.moffat@utoronto.ca
Subject: Third type measurement?
X-Priority: 2 (High)
 

Dear Professor Ashtekar,

I read your paper on color space [Ref. 1] with great interest. Would like to suggest some speculations perhaps relevant to the metric of color space [Ref. 1, pp. 7 -12], as being defined by some sort of "non-pertubative" or simply "third type measurement" made by the mind-brain system. Will conjecture that it is the human cognition & the human brain that define the metric of color space, not the physical interactions in the retina nor the visual cortex.

First the general idea. Will denote my speculations by S1 through S4.

Cognitive concepts (e.g., an apple) are rooted on Platonic ideas (e.g., an apple 'per se'); they both emerge in our mind along with the perceptions (e.g., visual images of an apple) of the physical reality. The former cannot be detached from the latter. These two constitute 'res cogitans' which too cannot be detached from the brain, 'res extensa'. The brain is what it is because of 'res cogitans'. (The opposite relation is another subject, which I will skip here.)

S1. It is the brain (*not* the mind) that can act on matter.

S2. If we think of an apple as quantum system, I speculate that our brain is coupled to some "intact quantum reality" (corresponding to the Platonic idea of an apple 'per se'), thanks to which we can perform some hypothetical 'third type measurement' -- we can "scan" ALL superposed states a la first type measurement (corresponding to all possible explications of *concrete* apples), AND can select one and only one "projection" of that "apple per se" (corresponding to the act of cognition "I see *this* apple"). We do not decohere/collapse the intact quantum reality that corresponds to the Platonic idea of an apple 'per se', however.

Is there a room for such intact quantum reality in QTM? I do not think so. Here comes S3.

S3. I believe the reason why we cannot observe/register intact quantum reality by inanimate (dead) measuring devices is "encoded" in the linear superposition principle of QM and the requirement for unitarity. The moment we write it down, by preparing the quantum system for measurement, we have *already* stripped the quantum system to some *finite* number of potential states that are inevitably in linear superposition, to define the probabilities for outcomes constrained by the chosen boundary conditions. This is a liner cut-off, and we cannot in principle observe intact quantum reality by inanimate (dead) measuring devices.

S4. If so, perhaps your research on the geometrical formulation of QM [Ref. 2] could shed some light on that intact quantum reality -- if any.

Will be happy to learn if you would be interested in these efforts of mine. Needless to say, there could be many practical applications,

http://www.tcom.co.uk/hpnet/h.htm ,

if we learn how to alter that intact quantum reality -- if any, of course.

Hope you may not feel offended by my 'spherical cow' ideas.

With kind regards,

Dimiter G. Chakalov
http://welcome.to/maver

References

1. Abhay Ashtekar, Alejandro Corichi, and Monica Pierri. Geometry in Color Perception. Erwin Schroedinger International Institute for Mathematical Physics, Vienna, ESI 517, December 22, 1997;

ftp://ftp.esi.ac.at:/pub/Preprints/esi517.ps

Abhay Ashtekar et al.: "In mathematical physics, one generally begins by constructing simplified models in which the actual physical complications of the system under consideration are stripped down to a bare minimum -- the oft-quoted examples is that of a 'spherical cow'" (p. 12).

"Nonetheless, it is important to remember that color perception is in fact a 'psycho-physical' phenomenon while the model treats it simply as a physical phenomenon. Thus, it is quite incomplete. What does it miss?" (p. 13).

"Such re-interpretations are essential for our survival which requires that we should be capable of recognizing the 'sameness' of the object even when it is illuminated quite differently. There are dozens of day to day examples in which the brain interferes non-trivially to make the model completely inadequate" (p. 13).
 

2. Abhay Ashtekar and Troy A. Schilling. Geometrical Formulation of Quantum Mechanics, Mon, 23 Jun 1997 10:55:02 -0500,
http://xxx.lanl.gov/abs/gr-qc/9706069

Ashtekar & Schilling: "The geometric formulation shows that the linear structure which is at the forefront in text-book treatments of quantum mechanics is, primarily, only a technical convenience and the essential ingredients -- the manifold of states, the symplectic structure and the Riemannian metric -- do not share this linearity."

"Also, even in the finite-dimensional case, we do not know if there exist *any* Kahler manifolds other than projective Hilbert spaces for which a satisfactory measurement theory can be developed. Even isolated examples of such manifolds would be very illuminating."

"(D)eeper reflection shows that quantum mechanics is in fact not as linear as it is advertised to be. For, the space of physical states is *not* the Hilbert space H but the space of rays in it, i.e., the *projective* Hilbert space P. And P is a genuine, non-linear manifold. Furthermore, it turns out that the Hermitian inner-product of H naturally endows P with the structure of a Kahler manifold. Thus, in particular, like the classical state space Gamma, the correct space of quantum states, P, is a symplectic manifold!"
 

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Subject: Re: An essential germ of truth
Date: Wed, 29 Jan 2003 02:30:36 +0200
From: Dimi Chakalov <dchakalov@surfeu.at>
To: Karel Kuchar <kuchar@physics.utah.edu>
CC: Abhay Ashtekar <ashtekar@gravity.phys.psu.edu>, carlo@rovelli.org
 

Dear Professor Kuchar,

On Tue, 28 Jan 2003 13:09:32 -0700 (MST), you wrote:

> The sentence which you quote "the PROPOSAL AS IT STANDS
> is self-contradictory" refers to one specific preceding statement
> on the same page marked by a bullet. It is followed by a suggestion
> how `the proposal as it stands' can possibly be amended to remove
> the contradiction and what are the difficulties involved.

I take this as an introduction to your explanation of how the proposal referred to you as [39] can possibly be amended to remove the contradiction, and what are the difficulties involved.

NB: Please send me your rigorous mathematical proof that this task can indeed be achieved.

> What you claim in your letter to Professor Ashtekar is that I
> "stated that YOUR QUANTIZATION PROGRAM is
> self-contradictory".

Isn't it? You didn't put it in plain flat English -- true. But people are not stupid, they know some math and can read your paper,

http://members.aon.at/chakalov/intro.html#Ashtekar_program

If you believe I'm wrong, please see NB above.

> Different words mean different things

But of course! Use math then! You can do it. Please go ahead and save your distinguished colleague. With math. Again, the questions from my preceding email of Tue, 28 Jan 2003 07:07:51 +0200 are:

1. Does Ashtekar's quantization program depend on perennials? 

1.1. If not, do you believe that it can, at least in principle, solve the inner product problem?

1.2. If yes, do you believe that Ashtekar's quantization program makes sense?

> Please, remove my name from your mailing list.

I will, upon receiving your proof that Ashtekar's quantization program can indeed solve the inner product problem, Q1.1 above. Please send your paper ASAP.

Alternatively, please allow me to post this email on my web site, below the discussion of Ashtekar's quantization program.

Please choose one of the two options, and you will never hear from me anymore. Whatever you choose -- thank you.

Once I print my CD ROM,

http://members.aon.at/chakalov/CD.html

you may hear from many kids on many issues, Ashtekar's quantization program included. Mind you, kids are smart and not biased. More on the inner product problem at

http://members.aon.at/chakalov/Margon.html

Ceterum censeo Carthaginem delenda esse.

Sincerely,

Dimiter G. Chakalov
http://members.aon.at/chakalov

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Explanatory note: I'm anticipating the paper by Prof. Karel Kuchar any time soon. I expect a brief and rigorous essay on the inner product problem ("the problem of fixing the inner product in the Hilbert space of physical states by requiring that it is invariant under Diff(M)" , I. Raptis), which will certainly cast new light on the problem of time ("the problem of requiring that the dynamics is encoded in the action of Diff(M) on the space of states", I. Raptis), and most importantly on a well-known but ignored problem: the disappearance of space due to the full reparametrization invariance of general relativity (R. Parentani). Hence I hope to find an answer to the question of the nature of 3-D space: is it a perennial? See the definition of perennials in Prof. Kuchar's fundamental article "Canonical Quantum Gravity", from which this discussion started.

My guess is as follows: The empirical fact that we observe 3-D space is indissolubly linked to some global perennial time. The latter is physically unobservable, according to Prof. Kuchar. But the physical existence of 3-D space is taken for granted by Prof. Kuchar himself. Why? Or maybe I'm guessing wrong? Hope will find out soon.

After receiving Prof. Kuchar's essay, I will remove my email of Wed, 29 Jan 2003 02:30:36 +0200 printed above, as well as this explanatory note. The issue has been raised in November 1999, but nobody replied. It's about time. 

Ceterum censeo Carthaginem delenda esse.
 

Dimiter G. Chakalov
Thursday, January 29, 2003
 


Karel V. Kuchar, Canonical Quantum Gravity, gr-qc/9304012 v1, 8 April 1993.

Sec. 4, Space of solutions, pp. 17-18:

"All physics is to be done on the space F0 of states which solve Eq. (30). (...) To summarize, without the Hilbert space structure on  F  and without boundary conditions or some auxiliary conditions on the states, we are bound to end up with a solution space  F0  that contains many unphysical states. For this reason, I am reluctant to call  F0  ‘the physical space’, and prefer to stick to a more neutral name, the space of solutions."
 

pp. 21-22: Definition of Perennials

Pay attention to p. 22: "The third alternative is to say that because perennials are constants of motion, it does not matter when they are observed. (...) This does not make me too happy either. If all time [tau] is eternally present, all time is irredeemable."
...

"So far I argued that some observables are not perennial. I must now defend my other point, namely, that perennials are often difficult to observe. In this part of the discussion, I take the attitude of physical common sense, that at any instant one can directly observe the position Q of the particle, its momentum P, and the time T on an ideal Newtonian clock, but not the position Q' which the particle had at time T = 0.

p. 23: "The initial position Q', which does not change with T and is a perennial, is inferred from the observed data Q, P, and T by using Eq. (45). For a free particle, such an inference is easy because we know how to integrate equations of motion."

Conclusion on p. 24: "Perennials in canonical gravity may have the same ontological status as unicorns -- a priori , these are possible animals, but a posteriori, they are not roaming on the Earth. According to bestiaries, the unicorn is a beast of fabulous swiftness, strength, and beauty, but, alas, it can be captured only by a virgin [38]. Corrupt as we are, we better stop hunting mythical beasts."
...

p. 25: The proposals on how to find the inner product depend on what position one takes on observables. Let me first discuss the proposal [39], which relies on identifying observables with perennials (...).
--
[39] See [12], Chapter 10: The quantization program

[12] Ashtekar A 1991 Lectures on Non-Perturbative Canonical Gravity (Singapore: World Scientific)

p. 25: "The third problem with the proposal is that the solution space F0 is probably larger than the space of physical states. We have seen that it may contain ‘improper elements’, ‘unbounded states’, and ‘states with negative norms’. (...) In brief, it seems impossible to follow step by step the ‘quantization program’: firstly, to find the space of solutions without having the inner product to determine which states are physical, secondly, on that space of solutions to define the perennials, and thirdly, to find the inner product on F0 which makes all such perennials self-adjoint.
...
p. 26: "To proceed, one should

 * abandon the space of solutions and work instead
in the space of instantaneous states.
"To talk about instantaneous states requires a decision about what is an instant. An instant in a relativistic spacetime is a spacelike hypersurface. However, spacelike hypersurfaces are not elements of the gravitational phase space. (...) These things are more easily said than done. The internal time proposal meets as many difficulties as the approach based on the concept of perennials. I discussed the problems of time in a recent review [40] which complements my present treatment of observables.

"It is sometimes maintained that the approach based on perennials somehow avoids the problems of time. It would be great if it did, but I fear it does not. A closer look reveals that the problems of time and the problem of perennials are rather closely related. A Czech saying has it that the devil thrown out of the door returns through a window."
--
[40] Kuchar K V 1992 Time and interpretations of quantum gravity. Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics ed G Kunstatter et al. (Singapore: World Scientific)
A complementary review of the problem of time is
Isham C J 1992 Canonical quantum gravity and the problem of time, Lectures presented at the NATO Advanced Summer Institute on ‘Recent Problems in Mathematical Physics’, Salamanca, June 15-27, 1992 Imperial College Preprint Imperial TP/91-92/25
....

p. 30: "The connection representation has been linked to the loop representation [42]. One should note that the latter investigation has been successful only for real connections.

"Because the regularization and consistency problems for the constraints have not been satisfactorily resolved, all attempts to find the states which solve the quantum constraints (30) are to a large extent formal. It is notable that connection dynamics actually exhibited a large number (indeed, infinitely many) such solutions. Most of these were obtained in the loop representation and lie outside the scope of this report [16].
...

"The problem of what quantities can be observed (and how they can be observed) is one of the most intriguing and important questions in quantum gravity. A widely held view (which I dispute) is that one can observe only perennials. No true perennials, classical or quantum, have so far been found, and even if they exist, finding them is difficult. I feel we should instead concentrate on formulating and proving (non?)existence theorems about perennials.

p. 31: "Another outstanding problem of canonical quantum gravity is the construction of the inner product. Quantum geometrodynamics has been unsuccessful in this task [22, 36], and connection dynamics has hardly done more than formulate broad guidelines on how one might try to proceed. These guidelines crucially depend on the existence of perennials.
...

"One can never be sure of passing a door before all have been passed. The entries are so interconnected that they cannot be made separately: What is a solution of the quantum constraints depends on the choice of fundamental variables and the form of the constraints. What solutions are physical depends on the inner product. What is an inner product depends on what quantities are observable. What quantities are observable may depend on what solutions are physical. More often than not we are caught in a vicious circle which calls for entering all the doors at once."
 

The only comments I can make are about the problem of "what quantities can be observed (and how they can be observed)", from p. 30 above. Karel Kuchar argues that Perennials are not observable, while I suggest that the correct answer is Jain. Perennials are UNspeakable, they act as 'context', they form a timeless shell/container of all physical observables ("jackets"), and everyone can verify this with her/his brain. This is well-known since Plato and Lucretius. See the John's jacket metaphor here, and recall the excerpt on p. 22 above: If all time [tau] is eternally present, we are dealing with the global mode of spacetime, as in the case of 'relational reality' here. The kind of causality is also very well known -- the Aristotelian final cause. There is nothing here that Karel Kuchar doesn't know.

I guess Karel Kuchar doesn't like these widely known ideas, I haven't heard from him since 28 January 2003 (see above). I heard some rumors that for the last few years he has been working on the consistent histories approach to quantum gravity, but I just can't believe this. Why would Karel Kuchar waste his time with searching for the inner product in consistent histories approach to quantum gravity? We have to enter "all the doors at once", as he wrote on April 8, 1993.

I can't believe Karel Kuchar is doing "consistent histories". We really have to enter "all the doors at once". How? By discovering the intrinsic dynamics of our 'relational reality'. It needs all the "unphysical states" (cf. p. 17 above). All of them have their unique place in the jigsaw puzzle of quantum gravity. Nothing is redundant in Nature, we just have to describe Kuchar's Perennials in a coordinate-free form, after Hermann Weyl.

Unlike A. Ashtekar, I'm open to suggestions and critical comments.
 

D. Chakalov
Sunday, October 24, 2004
--
Man is only wise during the time that he searches for wisdom; when he imagines he has completely attained it, he is a fool.

Solomon Ibn Gabirol, "A Choice of Pearls"