Dear Professor Giachetta,

May I ask you and your colleagues for your opinion on the
*interpretation* of the conflict between the canonical quantization
method and spacetime covariance, as suggested by A. Ashtekar in "Gravity and the Quantum", gr-qc/0410054 v2 [Ref. 1].

I believe there are at least three points that need to be examined with
utmost scrutiny.

Ashtekar has suggested that some "awkwardness arises only in the
intermediate steps" in the classical limit procedure [Ref. 1]:

"What is awkward in the canonical approach is the classical limit
procedure. In the intermediate steps of this procedure, one uses the
canonical phase space based on a 2+1 splitting. But because this phase space description is equivalent to the covariant classical theory, in the final step one again has space-time covariance. To summarize,
space-time covariance does not appear to have a fundamental role in the full quantum theory because there is neither space nor time in the full theory and it is recovered in the classical limit."

To be honest, I smell a rat here. If we start with the canonical phase space based on a  *2+1*  splitting, we're eliminating the very possibility to recover the classical limit at the scale of tables and chairs.

Ashtekar writes [Ref. 1]: "This point is best illustrated in
3-dimensional general relativity which shares all the conceptual
problems with its 4-dimensional analog but which is technically much simpler and can be solved exactly. There, one can begin with a 2+1
splitting and carry out canonical quantization [23]."

Point #1: I believe we must *not* begin with some totally unrealistic 2+1 approximation, just because in 3-dimensional general relativity the problem is "technically much simpler and can be solved exactly".

I wonder if you and your colleagues agree. Back in 1993, Karel Kuchar stressed that Ashtekar's quantization program is self-contradictory, but the crux of the problem is still unclear to me,

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

I hope you and your colleagues can shed light on this fundamental issue.

My hunch is that we should try to unravel additional degrees of
freedom in the structure on the covariant phase space, called
"polarization". That is, two gravitational wave polarizations and their
time derivatives, as explained by Steve Carlip,

http://God-does-not-play-dice.net/Ashtekar.html#Q5

Ashtekar writes [Ref. 1]: "A similar strategy was tried for general
relativity as well, using techniques form geometric quantization. Recall that quantum states are square-integrable functions of only ‘half’ the number of phase space variables -- usually the configuration variables. To single out their analogs, in geometric quantization one has to introduce additional structure on the covariant phase space, called a ‘polarization’. Quantization is easiest if this polarization is suitably ‘compatible’ with the Hamiltonian flow of the theory. Unfortunately, no such polarization has been found on the phase space of general relativity."

Point #2: I believe the reason why "no such polarization has been found on the phase space of general relativity" is that these additional parameters can be unraveled only in the *dynamics* of Ashtekar's
'relational reality', which Ashtekar wasn't able to achieve,

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

Point #3: Finally, Ashtekar acknowledges that loop quantum gravity program accepts the price of "doing grave injustice to space-time covariance that underlies general relativity" [Ref. 1], and the result is a total mess,

http://God-does-not-play-dice.net/Savitt.html#bare

I will appreciate your professional opinion on the tree points above,
and will keep it strictly private and confidential.

Respectfully yours,

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

Reference

[Ref. 1] Abhay Ashtekar, Gravity and the Quantum, gr-qc/0410054 v2, http://xxx.lanl.gov/abs/gr-qc/0410054

p. 12: "The apparent conflict between the canonical quantization method and space-time covariance is discussed in Appendix A.

p. 32: APPENDIX A: CANONICAL APPROACH AND COVARIANCE

"A common criticism of the canonical quantization program pioneered by Dirac and Bergmann is that in the very first step it requires a splitting of space-time into space and time, thereby doing grave injustice to space-time covariance that underlies general relativity. This is a valid concern and it is certainly true that the insistence on using the standard Hamiltonian methods makes the analysis of certain conceptual issues quite awkward. Loop quantum gravity program accepts this price because of two reasons.

"First, the use of Hamiltonian methods makes it possible to have
sufficient mathematical precision in the passage to quantum theory to
resolve the difficult field theoretic problems, ensuring that there are
no hidden infinities.[footnote 16] The second and more important reason is that the mathematically coherent theory that results has led to novel predictions of direct physical interest.
--
Footnote 16: "The only other background independent approach to quantum general relativity which has faced some of these problems successfully is the causal dynamical triangulation program [30,41], which again requires a 3+1 splitting. The spin-foam approach provides a path integral alternative to loop quantum gravity and does not require a 3+1-decomposition of space-time. If it can be completed and shown to lead to interesting physical predictions, it would provide a more pleasing formulation of ideas underlying loop quantum gravity.
....

"A similar strategy was tried for general relativity as well, using
techniques form geometric quantization. Recall that quantum states are square-integrable functions of only ‘half’ the number of phase space variables -- usually the configuration variables. To single out their analogs, in geometric quantization one has to introduce additional structure on the covariant phase space, called a ‘polarization’. Quantization is easiest if this polarization is suitably ‘compatible’ with the Hamiltonian flow of the theory. Unfortunately, no such polarization has been found on the phase space of general relativity.
...

"Let us therefore return to the standard canonical phase space and use it as the point of departure for quantization. In the classical regime, the Hamiltonian theory is, of course, completely equivalent to the space-time description. It does have space-time covariance, but it is not ‘manifest’.

"Is this a deep limitation for quantization? Recall that a classical
space-time is analogous to a full dynamical trajectory of a particle in
non-relativistic quantum mechanics and particle trajectories have no
physical role in the full quantum theory. Indeed, even in a
semi-classical approximation, the trajectories are fuzzy and smeared.
For the same reason, the notion of classical space-times and of
space-time covariance is not likely to have a fundamental role in the
full quantum theory. These notions have to be recovered only in an
appropriate semi-classical regime.

"This point is best illustrated in 3-dimensional general relativity
which shares all the conceptual problems with its 4-dimensional analog but which is technically much simpler and can be solved exactly. There, one can begin with a 2+1 splitting and carry out canonical quantization [23]. One can identify, in the canonical phase space, a complete set of functions which commute with all the constraints. These are therefore ‘Dirac observables’, associated with entire space-times. In quantum theory, they become self-adjoint operators, enabling one to interpret states and extract physical information from quantum calculations, e.g., of transition amplitudes. It turns out that quantum theory -- states, inner-products, observables -- can be expressed purely combinatorially. In this description, in the full quantum theory there is no space, no time, no covariance to speak of.

"These notions emerge only when we restrict ourselves to suitable
semi-classical states. What is awkward in the canonical approach is the classical limit procedure. In the intermediate steps of this procedure, one uses the canonical phase space based on a 2+1 splitting. But because this phase space description is equivalent to the covariant classical theory, in the final step one again has space-time covariance. To summarize, space-time covariance does not appear to have a fundamental role in the full quantum theory because there is neither space nor time in the full theory and it is recovered in the classical limit.

"The awkwardness arises only in the intermediate steps."
--
[23] Ashtekar A Mathematical problems of non-perturbative quantum general relativity in Gravitation and Quantizations: Proceedings of the 1992 Les Houches summer school eds Julia B and Zinn-Justin J (Elsevier, Amsterdam); also available as gr-qc/9302024
...

"We routinely accept this procedure and the role of ‘phase-space
covariance’ in quantization in spite of an awkwardness in an intermediate step of taking the classical limit. The canonical approachadopts a similar viewpoint towards space-time covariance."
 
 

Note: The above-mentioned "awkwardness" that "arises only in the intermediate steps" is a bona fide case of 'sweeping the garbage under the carpet'. Just read this passage from A. Ashtekar [Ref. 1]: "To summarize, space-time covariance does not appear to have a fundamental role in the full quantum theory because there is neither space nor time in the full theory and it is recovered in the classical limit."

Who says that spacetime covariance is being "recovered in the classical limit"? Abby Ashtekar did acknowledge that his loop quantum gravity is "doing grave injustice to space-time covariance that underlies general relativity", and the result is a total mess: see Point #3 above. More here.

To the best of my knowledge, nobody -- A. Ashtekar included -- has so far succeeded in proposing a hypothesis on quantum gravity that would either solve or circumvent the problem of its classical limit.

Recall that the major idiosyncrasy of standard QM is that it contains classical mechanics as a "limiting case", yet at the same time requires this same "limiting case" for its own formulation (Landau L. and Lifshitz E., Quantum Mechanics, Pergamon Press, Oxford, 1974). Only nobody has managed to produce a rigorous derivation of this "limiting case" in the past thirty years, and we still use QM as a set of calculation recipes only. Hence the "interpretation" of standard QM is still 'shut up and calculate'.

Here the case is far more serious: nothing works in present-day hypotheses on quantum gravity, neither with background nor without. We should not 'shut up and calculate' but begin to rethink our initial steps, and find out where we went wrong.

My hunch is that we will need a Virtual Geodesic Path formulation of general relativity, as I mentioned here, but let's see first whether any of the recipients of my email above will agree to show me my errors.

Please see Point #1, Point #2, and Point #3 above.

And please don't take my critical remarks to A. Ashtekar personally. It's not about Abby Ashtekar and his huge network of friends and colleagues. It's all about Einstein and his Annus Mirabilis. Let's be frank and professional.
 

D. Chakalov
October 20, 2004

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

Subject: Re: The standard canonical phase space of general relativity
Date: Wed, 20 Oct 2004 23:25:07 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Gennadi Sardanashvily <gennadi.sardanashvily@unicam.it>
CC: giovanni.giachetta@unicam.it, luigi.mangiarotti@unicam.it

Gena dorogoi,

Thank you for your feedback.

> I have read your remarks with an interest, but
> unfortunately I am not an expert in this subject and can
> not conclude anything.

You've been schooled by Dmitri Ivanenko (God bless him, he was great
physicist) and have earned your Ph.D. in 1980, you are teaching
Geometric Methods in Quantum Theory at Moscow State University, you are Editor of Advanced Series in Mathematical Physics at World Scientific
(Singapore) and have just written, in collaboration with Giovanni
Giachetta and Luigi Mangiarotti, a colossal monograph, "Geometric and
Algebraic Topological Methods in Quantum Mechanics", 703 pp., and now
you are telling me that you cannot comment on the standard canonical
phase space of general relativity.

I respect you very much, Gena. If you do not *want* to elaborate on
these additional parameters mentioned in Point #2 from my preceding
email -- two gravitational wave polarizations and their respective time
derivatives -- please state it in plain English.

I know very well that A. Ashtekar has developed a huge network of
friends and colleagues, but please recall Dmitri Ivanenko and Albert
Einstein.

Again, please be assured that I will keep your *professional* comments
and suggestions strictly private and confidential, as I promised in my
preceding email,

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

Looking forward to hearing from you and your colleagues,

Best regards,

Dimi

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

Subject: Go "beyond" the intrinsic cut-off of QFT
Date: Fri, 22 Oct 2004 15:11:46 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Gennadi Sardanashvily <gennadi.sardanashvily@unicam.it>
CC: volovik@boojum.hut.fi

Dear Gena,

Here's something real.

In estimating the energy density of the quantum vacuum, we use Planckian cut-off, and the summation over the bosonic and fermionic modes leads to a well-known paradox, hence we need new physics: we do not know the physics of the quantum vacuum "beyond" the cut-off.

G.E. Volovik wrote (gr-qc/0405012 v3): "It is quite possible that we simply are not aware of some very simple principles of the trans-Planckian physics from which it immediately follows that the correct summation over all the modes of the quantum vacuum gives zero or almost zero value for the vacuum energy density, i.e. the trans-Planckian degrees of freedom effectively cancel the contribution of the sub-Planckian degrees irrespective of details of trans- and sub-Planckian physics."

You said that will keep in mind my previous remarks, so let me make them as clear as possible.

If we want to go "beyond" the cut-off, I believe we need to represent all modes in a special propensity-state of 'relational reality',

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

The intrinsic "time parameter" of this relational reality should NOT be replaced with ANY time parameter from present-day cosmology, because it is a brand new entity,

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

Otherwise you may never solve the coincidence problem.

Perhaps you can find this "global time" and its relational reality by studying connections on loop spaces in QFT.

Best wishes,

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

Download the whole web site, 4MB, from
http://God-does-not-play-dice.net/PHI_info.zip
 

There is a strong statement above: nothing works in present-day hypotheses on quantum gravity, which is the reason why I tried to suggest some new ideas about 'relational reality' here.

Let me elaborate on my claim that nothing works in present-day hypotheses on quantum gravity. I will add some new considerations for the putative 'global mode of spacetime' in 'relational reality', by referring to one of the best-known papers by Karel Kuchar, entitled: "Time and interpretations of quantum gravity", in: Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, ed. by G. Kunstatter, D. Vincent, and J. Williams, World Scientific, Singapore, 1992, pp. 211-314. See also refs [33], [95], and [96] therein.

I will quote from the online preprint of Kuchar's paper "Time and interpretations of quantum gravity", which can be downloaded from here. If you read the paper version published by World Scientific, I believe you can find the page [b] there and the quoted text with the formula  b = a + 210. For example, if I quote from p. 23 [a=23] from the online preprint, the corresponding page [b] will be p. 233. So, if you're fluent in GR, please look at p. 23 from the online preprint, sec. Linearized Gravity, and note the final comment on p. 25: "Unfortunately, the flat background used in the decomposition (6.7) - (6.8) loses then its geometric and physical significance, and the constraints written in the new variables become unmanageable." I don't think this should be surprising; see the seminal paper by Hermann Weyl of 1944 here.

Thus, I'll drop the issue of this "linearized" misunderstanding, and will proceed to what I believe is the crux of the matter: the nature of time and Karel Kuchar's paper "Time and interpretations of quantum gravity".

First, what do we imply by 'time'? The metaphysics is well-known, as I tried to explain here. Recall also the concept of genidentity (Genidentität), introduced by Kurt Lewin, which I think can explain beautifully the nature of time and Kuchar's Perennials. Kurt Lewin has stressed that there exist two entities with very different nature: one is changing 'in time', the other provides the 'genidentity' of the object that changes 'in time'. This latter entity is very important: it does not change 'in time', just as the human self does not change 'in time'.

This is Kuchar's Perennial, as explained here. Note that we infer the existence of the Perennial, as I tried to explain here. We never observe it directly, and yet it is needed for those ever-changing states 'in time'. Otherwise the object will lose its Genidentität, as stressed by K. Lewin in 1930s.

I believe this little note can shed some light on the origin of 'spacetime problem', as explained by Kuchar on p. 31. Have you heard about 'the spacetime problem'? It's one of the many issues which Kuchar elucidates with incredible precision and clarity. See p. 32: "So far, I do now know a single concrete example of a decomposition of the fundamental canonical variables based on a scalar time function T(x)." Note that this 'spacetime scalar' is supposed to emerge from the canonical formalism by satisfying a very precise requirement: it should not depend "on the embedding from which it was reconstructed" (p. 31). If this were possible, there would be a way to circumvent the problem of 'multifingered time' in GR, and to eliminate both Kurt Lewin's genidentity thesis and Kuchar's Perennials as redundant.

Karel Kuchar rightly argues against any sort of 'Heraclitian time' (p. 40) and 'reference fluid' (p. 32), and demonstrates (pp. 75-76) that it is impossible in principle to have dynamics in the so-called conditional probability interpretation. In this regard, see the 'relational reality' explanation in Bill Unruh's paper "No time and quantum gravity", in: Gravitation: A Banff Summer Institute. Banff Center, Banff, Canada August 12-25, 1990. Ed. by R. Mann and P. Wesson, September 1991, p. 267.

Also, don't miss Sec. 15, "Frozen Time Formalism and Evolving Constants of the Motion", pp. 82-84 and p. 88, since this is a very good example of how people like Carlo Rovelli (mentioned there) have been systematically ignoring Karel Kuchar's research. See also A. Ashtekar, and ponder on the following simple question: if we are to define rigorously an intrinsic time interval associated to any timelike displacement, what would be the reference object with respect to which we can define such intrinsic time interval? If you can somehow avoid Kuchar's Perennials, you would surpass C. Rovelli and A. Ashtekar, because they haven't managed to do that in the past twelve years. If you cannot recover the spacetime at the scale of tables and chairs, better don't waste twelve years of your time and that of your students, like C. Rovelli and A. Ashtekar. Get back to Karel Kuchar instead, and read his papers which everybody inserts in his reference list but doesn't actually read.

Finally, let me quote the last sentence from Sec. 16, Summary (p. 90): "In my opinion, none of us has so far succeeded in proposing an interpretation of quantum gravity that would either solve or circumvent the problems of time. He who made eternity out of years remains beyond our search. His ways remain unscrutable because He not only plays dice with matter but also with time."

It seems to me that Karel Kuchar hasn't yet found his Perennials. Let me try to help. Sec. 16, Summary, begins with the following (pp. 88-89): "In general relativity, dynamics is entirely generated by constraints. The dynamical data do not explicitly (italics added - D.C.) include a time variable". Surely there is no explicit time variable, and cannot be any explicit time variable in GR. We have Perennials instead. However, we can only infer their existence, as explained by Karel Kuchar here and, most recently, here.

We infer the existence of "points" because we can make tangent vectors at those "points" and calculate values of observables in GR. Similarly, we infer the existence of Perennials because we see things that change 'in time'. We see their different 'jackets' that change 'in time', but we cannot directly observe the Perennial that keeps the genidentity of these ever-changing 'jackets'. Briefly, we have to "shrink" the Perennial down to zero in order to obtain "points" at which we calculate the dynamic state of the 'jackets', which does not, in any way, imply that "He not only plays dice with matter but also with time." Why? Because He can easily play with "points" and their tangent vectors while keeping the regions (Chris Isham) of Kuchar's Perennials widely open to cover any finite time interval from our cosmological time. Besides, He doesn't hold double standards for time and space, as we do (well, some of us).

Don't mess with Him, please. Look at your Perennials. What is intrinsic "time interval" associated to an infinitesimal timelike displacement? With respect to what? And what could possibly "move" one event to the "nearest" one? That's what Perennials do.

Panta rei conditio sine qua non est.

D. Chakalov
October 26, 2004
Last update: October 27, 2004