Subject: Quantum gravity, gr-qc/0508120 v1
Date: Tue, 30 Aug 2005 15:15:01 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Claus Kiefer <kiefer@thp.uni-koeln.de>
CC: Chris Isham <c.isham@imperial.ac.uk>,
     Hermann Nicolai <Hermann.Nicolai@aei.mpg.de>,
     Friedrich Hehl <hehl@thp.uni-koeln.de>,
     Christian Heinicke <chh@thp.uni-koeln.de>,
     Ulrich Eckern <ulrich.eckern@physik.uni-augsburg.de>,
     Fay Dowker <f.dowker@imperial.ac.uk>,
     Ariel Caticha <ariel@albany.edu>,
     Johan Noldus <johan.noldus@gmail.com>,
     Paul Tod <tod@maths.ox.ac.uk>,
     Dorje C Brody <d.brody@imperial.ac.uk>,
     Lane P Hughston <lane.hughston@kcl.ac.uk>,
     Szabados Laszlo <lbszab@rmki.kfki.hu>,
     Robert M Wald <rmwa@midway.uchicago.edu>,
     Norbert Straumann <norbert.straumann@freesurf.ch>,
     Domenico Giulini <giulini@physik.uni-freiburg.de>,
     Jorge Pullin <pullin@phys.lsu.edu>,
     Karel Kuchar <kuchar@physics.utah.edu>

Dear Claus,

I believe you used the question mark 40 times in your recent paper. Perhaps it won't hurt if I remind you of my unpublished work, which would eventually add just another question mark.

Time is external in ordinary quantum theory (the parameter  t  in the Schrödinger equation), and in general relativity is dynamical because it is part of spacetime described by Einstein's equations. Your conclusion reads: "Both concepts cannot be true fundamentally."

I believe you've read my email sent in the past four years, so I'll be very brief. It all boils down to your question: "At which scale(s) would one expect that effects of quantum gravity necessarily occur?" At the scale we can obtain a *perfect continuum*. That is, at ALL scales.

Thus, we need two modes of spacetime, to 'have our cake and eat it': a perfect continuum in the local mode of spacetime (which holds at all scales) and a discrete *potential reality* in the global mode of spacetime. The latter can be revealed in the dynamics of embedding a quantum event into Minkowski spacetime,

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

If you try to trace back the "history" of the relativistic collapse, you'll end up with the "point" at which the quantum event "had entered" the Minkowski spacetime, but you won't be able to go further, much like in the case of the so-called geodesic incompleteness.

The story goes back to the famous tripod of Ernst Specker,

http://www.God-does-not-play-dice.net/Landsman.html#note_last

It's actually a very old story,

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

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

Not surprisingly, it showed up in the so-called dark energy puzzle as well; please see p. 12 in

http://www.God-does-not-play-dice.net/paper.doc

Consider this. The Sun is an 'objective reality' in the sense that, if you look at the Sun and record the event of your observation with  t_1 , you can say that (i) there *was* an objective state of the Sun 8 min prior to t_1, which you see at t_1, and (ii) there *is* an objective state of the Sun at t_1 as well, only it will be observable after 8 min.

In this context, the 'dark energy of X' cannot be considered an objective reality, because any considerations of its effect on the spacetime metric would immediately lead to 'reductio ad absurdum': the effect of 'dark energy of X' at  t_1  applies simultaneously to the distances between all spacetime "points". Had the 'dark energy of X' been an objective reality like the Sun, you would have had not one but two concurrent spacetime hypersurfaces at  t_1 : one at t_1 and another one at (t_1 + 8) min: reductio ad absurdum. Hence the 'dark energy of X' is also 'potential reality'. Surely it has its own dynamics and history (the coincidence problem), but we cannot, and should not try to find the dynamic dark energy INSIDE the local mode of spacetime. It's not there. Recall that the 'dark energy of X' is roughly 18 times more than the world described in GR textbooks (73 % vs. 4 %), so it's like trying to find an elephant in your office, which is 18 times bigger than your desk.:-)

I'll be happy to elaborate, of course. Should you or some of your colleagues have questions, please don't hesitate.

Kindest regards,

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

====

Subject: Re: GW argument
Date: Tue, 27 Sep 2005 01:29:14 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Brian Josephson <bdj10@cam.ac.uk>
CC: Chris Isham <c.isham@imperial.ac.uk>
References: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10
 

> The first paragraph is OK.  The problem is in the next one:

> > In this context, the 'dark energy of X' cannot be considered an
> > objective reality, because any considerations of its effect on the
> > spacetime metric would immediately lead to 'reductio ad absurdum':
> > the effect of 'dark energy of X' at  t_1  applies simultaneously to
> > the distances between all spacetime "points".

> This is one of these typical unclear arguments that I've complained
> about. Where is the argument for this assertion?

The context here is 'objective reality out there', as explained in the first paragraph with the two states of the Sun,

http://www.God-does-not-play-dice.net/Kiefer.html#Sun

To be specific:

1. At the instant t_1, as recorded with your wristwatch on Earth, you
observe the state #1 of the Sun, which was the actual state of the Sun at (t_1 - 8) min.

2. At the same instant t_1, there IS an actual state #2 of the Sun, which will inevitably become observable to you 8 min later, at (t_1 + 8) min.

Now, let's apply this line of reasoning to the state of X, which denotes the state of the dynamic dark energy of X. Notice that X evolves in time as well, as we know from the coincidence problem.

NB: At the instant t_1, as recorded with your wristwatch on Earth, you observe THE state of X, which has produced a snapshot of THE WHOLE UNIVERSE.

Namely, the dynamic dark energy applies the expansion of the distance between all points in a fully "democratic" fashion, without any preferences to their locations viewed from Earth, say. Also, the dynamic dark energy of  X  is a PERFECTLY smooth force which does not leave any irregularities or "clumps" of residual  X  in any place in the universe. Had there been any such "clumps" of  X , they would have lead to tremendous catastrophes due to gross violations of energy conservation laws in these places. (I'm trying here to explain Matt Visser's gr-qc/0309109 v4: "Exotic matter is powerful stuff: Apart from possibly destroying the universe in a future "big rip" singularity [7], if the exotic matter clumps to any extent there is real risk of even more seriously bizarre behaviour -- everything from violations of the positive mass condition (that is, objects with negative asymptotic
mass), through traversable wormholes, to time warps [4, 14, 15, 16, 17].")

Thus, at the instant t_1 from the cosmological time of the 'expanding
balloon',  X  has produced a snapshot of THE WHOLE UNIVERSE (balloon).

The reason why I use capital letter is to stress the SIMULTANEOUS action
of X on ALL points from the balloon surface. There is no preferred
location of the "source" of X in 3-D space. X acts simultaneously on all
points and does not leave any irregularities in the expansion of the
balloon: it is a perfectly isotropic and homogeneous expansion covering
all points EN BLOC. X does not propagate in 3-D space from some point to
the next one. X is invisible, directly unobservable, or simply "dark".

Because X acts simultaneously on all spacetime points, had the 'dark energy of X' been an 'objective reality' like the Sun, you would have had not one but two concurrent spacetime hypersurfaces at  t_1, corresponding to state #1 of X and state #2 of X (please see above).

That is, if the relativistic status of X were like that of the Sun, you would have one 'snapshot of the whole universe' at t_1, with a given snapshot from the evolution of the dynamic dark energy X. That's your state #1 of X.

Unlike state #2 of the Sun, however, you cannot have state #2 of X at
t_1.

You cannot have two concurrent snapshots of THE WHOLE UNIVERSE at t_1. See NB above.

Thus, X is not 'objective reality out there'. Besides, it is 18 times more than everything that lives "inside" 3-D balloon (73 % dark energy vs. 4 % normal stuff, as described in GR textbooks).

Please tell me where my explanation failed to reach your brain. It will be entirely my fault, as I said before.

D.

====

Addendum

For some reason or another, the explanation above was not clear enough. It is perhaps very difficult to consider the possibility that 'the dark energy of X' can act on the whole spacetime en bloc. But it is the same 'dark energy of X' which has produced the inflationary stage. It is not something that obeys the laws of STR. Thus, if 'the dark energy of X' was able to expand the whole 3-D space shortly after The Beginning, it should today have the same faculty as well.

Imagine a global, now-at-a-distance reference frame at  t_1 . Think of it as an instantaneous cross-section of the whole universe, made by some transcendental tachyon. What can you see and what will be "dark" to you?

In such now-at-a-distance reference frame, there is an actual state #2 of the Sun at  t_1 ,  which will inevitably become observable to you 8 min later, at (t_1 + 8) min (cf. above). You cannot see it at  t_1 , however. You can only see the state #1 of the Sun, and that's all the stuff you can find in your spacetime: it's all made of things seen post factum. You cannot see faster-than-light, hence cannot "see" the dynamic dark energy of X "online", as it expands the 3-D space. If you could "see" in the global, now-at-a-distance reference frame at  t_1 , you could be able to "time" the expansion of space, and tell other people whether it is "accelerated" or not. But you can't. Hence it is "dark".

Recall again the famous statement from the Hindu catechism: The One is an unbroken Circle (ring) with no circumference, for the circumference is
nowhere and the center is everywhere. The "circumference" is nothing but the global, now-at-a-distance reference frame. It is completely "dark" as well. 

Had 'the dark energy of X' been a normal physical stuff obeying the laws of STR, it would have exposed a privileged "center" of expansion, hence a privileged reference frame inside your 3-D space. This would certainly ruin STR, so we're enjoying yet another case of 'peaceful co-existence', STR and 'the dark energy of X'. 

Since all points from 3-D space are fully legitimate "centers" of the universe, the 'dark energy of  X'  picks them up en bloc in the global, now-at-a-distance reference frame. I don't know why this is difficulty to understand, given the fact that the inflationary cosmology was introduced in 1981. It certainly doesn't make sense in the framework of STR, but that's not my fault. It doesn't make sense in the framework of GR either: the so-called dark energy is 18 times more than everything we observe in our spacetime post factum, like the state #1 of the Sun (cf. above).

This whole story is really bizarre, yes. Please do send me your comments.  If the explanation above has failed to reach your brain, it will be entirely my fault, as I stated before. More can be read here and here.
 

D. Chakalov
October 9, 2005

===========
 

Note: I read today the recent neo-realist paper by Chris Isham [Ref. 1], in which he elaborated on the nonstandard (in the logical sense [Ref. 2]) "way of saying ‘how things are’ in regard to the quantity A when the state is |psi>" [Ref. 1].

The puzzle is known since the first days of QM, after Erwin Schrödinger: "The rejection of realism has logical consequences. In general, a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has. But then what does it mean?"

It is a potential reality, as argued yesterday (cf. above), and the truth value of the propositions is Jain, as suggested on December 28, 2003.

As to the "long-standing question of whether the standard quantum formalism can be given an interpretation that does not involve measurement as a fundamental category" [Ref. 1], let's take a closer look at Sec. 3.4, The question of normalisation, p. 16 [Ibid.]:

"This is no problem in the conventional formalism since, there, one never gets reduction to an eigenstate for which there is zero probability of finding the associated eigenvalue. Or, more precisely: such zero probability events are swept under the carpet as never happening. However, for our neo-realist view, the normalisation problem is a genuine issue since in the action of the monoid SP(H) on a state |psi>, there will of course be strings Q for which ˆQ |psi> = 0."

The normalisation procedure is a rigorous presentation of the statement 'something will happen'. If we don't wish to include the measurement as a fundamental category, we inevitably introduce counterfactuals: the result of a series of reductions which could have been made, but were not actually performed; see p. 15 and Eq. 3.12 [Ibid.].

Thus, if we are to consider the "reduction to an eigenstate for which there is zero probability of finding the associated eigenvalue", after Chris Isham, but without actually performing the reduction, how can we define the set of (potential) eigenstates for which there is zero probability of finding the associated (potential) eigenvalues? Isn't that an empty set?

Big mess, isn't it? The confusion comes from the combination of 'objective reality' (see Erwin Schrödinger above) with the normalisation procedure 'something will happen'. If we keep the notion of 'objective reality' (see the example with the Sun above), then a neo-realist interpretation will inevitably encounter a set of eigenstates for which there is zero probability of finding the associated eigenvalues.

This is how the 'potential reality' shows up in the distorted mirror of 'objective reality' in the neo-realist interpretation of Chris Isham. These eigenstates simply do not exist in the local mode of spacetime, hence the probability of finding their associated eigenvalues in the local mode is zero. In the context of the Schrödinger cat paradox, the two cat states are potential reality prior to observation/reduction, and it doesn't matter if the latter is attended by observer/hammer or not. The two cat states are in the global mode of spacetime. The situation does not change if we extend the Hilbert state from two to infinite-dimensional.

Going back to the question posed by Erwin Schrödinger in 1935 (reference here), a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has had in the local mode of spacetime, but ascertaining the potential value that it has, and will always have in the global mode of spacetime. These potential values are not killed by the "collapse" in the local mode of spacetime; they are merely projected there from the  Schrödinger cat per se  residing in the global mode of spacetime.

Dead matter makes quantum jumps; the living-and-quantum matter is smarter. I tried to explain this to Chris Isham during our first meeting on November 13, 1998, and in our last meeting on March 9, 2006. He wasn't convinced [Ref. 1], and still isn't. Three years ago, on Wed, 23 Oct 2002 19:24:15 +0100, he delivered his opinion with the following statement: "You do not know enough theoretical physics to help with any research in that area."

Pretty harsh statement, isn't it? But it is a double-edged sword as well. It would be hugely embarrassing for his reputation if it turns out that the alleged "time parameter in the Schrödinger equation" refers to 'potential reality'. It has to be a reality, albeit a potential one, or else we're playing a very dangerous game, as warned by Einstein fifty years ago [Ref. 3]. Let me try to be more specific.

If Chris Isham really wants to explore the "long-standing question of whether the standard quantum formalism can be given an interpretation that does not involve measurement as a fundamental category" [Ref. 1], he should never abuse the quintessential issue of Quantum Mechanics: "a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has" (Erwin Schrödinger). Any time we introduce "that sharp time" of STR to QM, we utterly abuse QM. As it is known since 1931, we contradict the very foundations of QM by applying the value of the variable  t , because it is taken from the 'objective reality out there' of the classical world of STR (for example, the "instantaneous position" of the measurement device at a given instant). The "sharp time" of STR is totally alien to the spirit of QM, because it requires fixed values of time prior to observation. If we want to learn something about the quantum reality without involving "measurement as a fundamental category" [Ref. 1], we must never "filter" the quantum reality through "that sharp time" of STR. It is a grave and unjustified error to approach the quantum reality with the time variable of STR, which pertains to  facts  that exist as an objective reality 'out there'. Chris Isham should be fully aware of this (see the 'flipping a quantum coin' quiz here), and yet he suggests just a new kind of logic: neither true nor false, but "somewhere in between" [Ref. 2]. That is not going to work, because the potential reality exists "outside" the Hilbert space, and the truth value of the propositions is Jain. All my efforts to explain this crucial issue to Chris Isham, from November 13, 1998 until our last meeting on March 9, 2006, have been totally ignored. My proposal from December 16, 2004 was not even acknowledged, as if I was talking about cat food.

Anyway, I'm curious to see how Chris Isham would explain the 'dark energy of X'. It's like trying to find an elephant in his office, which is 18 times bigger than his desk. Or maybe some day he will manage to solve the normalization problem in his recent paper [Ref. 1, Sec. 3.4, p. 16]. The way I see it, the task is impossible (the potential reality is a genuine non-Archimedean reality), but since I "do not know enough theoretical physics to help with any research in that area", I will patiently wait to see how Chris Isham will defend his professional opinion. The sooner, the better.

Again, I don't talk about cat food here, but defend the opinion of Schrödinger and Einstein. Ignore it at your peril.
 

D. Chakalov
August 31, 2005
Last update: March 14, 2006
 

[Ref. 1] C.J. Isham, A Topos Perspective on State-Vector Reduction, quant-ph/0508225 v1, 30 August 2005.
 

[Ref. 2] C.J. Isham, Is it true; or is it false; or somewhere in between? The logic of quantum theory. Contempory Phys., 46(3), 207-219 (2005).
http://taylorandfrancis.metapress.com/link.asp?id=t1446p7w6v84626n
 

[Ref. 3] "Except for Laue, you are the only one who realizes that you cannot avoid accepting reality if you are honest. Most of the others don't even see what kind of a dangerous game they play with reality."

Albert Einstein to Erwin Schrödinger, 1950 (E. Schrödinger, M. Planck, A. Einstein, H.A. Lorentz: Briefe zur Wellenmechanik (hrsg. K. Przibram), Springer Verlag, Wien, 1976.)
 
 

=======

Subject: quant-ph/0507231 v1
Date: Thu, 01 Sep 2005 14:41:47 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Daniel Lehmann <lehmann@cs.huji.ac.il>
CC: Dov Gabbay <dg@dcs.kcl.ac.uk>,
     Kurt Engesser <secretary@dcs.kcl.ac.uk>,
     "Isham, Christopher J" <c.isham@imperial.ac.uk>

Dear Dr. Lehmann,

You wrote: "A proposition holds in some state if and only if this state is a fixpoint for the proposition."

If we consider a counterfactual (potential) state of a quantum system before measurement, then the truth value of *any* proposition will be "Jain",

http://www.God-does-not-play-dice.net/Kiefer.html#Isham

I wonder if the Illegitimate State 0 (quant-ph/0507231 v1, Sec. 3.3) matches the case of a counterfactual (potential) state, as explained at the URL above.

As you put it, "But, here, the assertion of the proposition, i.e., the measurement, changes the state of the system. The assertion holds in the state resulting from the measurement, but did not necessarily hold in the state of the system before the measurement was performed. In fact it held in this previous state if and only if the measurement left the state unchanged."

I believe the counterfactual (potential) state never changes, since it is a genuine non-Archimedean reality,

http://www.God-does-not-play-dice.net/York.html#8

If so, it cannot be normalized in principle.

Kindest regards,

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

=======

Subject: Dead-end in gr-qc/0508104 v2?
Date: Tue, 06 Sep 2005 13:53:53 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: johan noldus <johan.noldus@gmail.com>, J.Noldus@phys.uu.nl
CC: H.vanBeijeren@phys.uu.nl, bdewit@phys.uu.nl,
     G.tHooft@phys.uu.nl, tumulka@mail.mathematik.uni-tuebingen.de

Hi Johan,

On Tue, 6 Sep 2005 10:20:45 +0200, you wrote:
[snip]

> but so far I had only heard the same slogans about QM from you.

I have offered just one "slogan": Dead matter makes quantum jumps; the living-and-quantum matter is smarter. More at

http://www.God-does-not-play-dice.net/Kiefer.html#Isham

> It would be better to diversify a bit :-)

With utmost pleasure!

The first sentence from your abstract (gr-qc/0508104 v2) reads: "This paper can be seen as an exercise in how to adapt quantum mechanics from a strict relativistic perspective while being respectful and critical towards the experimental achievements of the contemporary theory." Specifically, you reexamine quantum mechanics "in a Bohmian spirit from a strict, die hard relativistic point of view and see how far it brings us."

I'm afraid it brings you to a dead-end.

You wrote: "Of course our new equations have the correct classical limit."

That's your dead-end: the reconciliation of your QM interpretation with STR. You don't have any classical limit whatsoever.

The task is to provide a bi-directional transition from the quantum realm and the world of tables and chairs for which STR holds. See the "slogan" of John Bell below, and the description of the task (in the context of Cramer's TI) at

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

NB:All you have to do is to reconcile the "nonlocal guidance mechanism" (gr-qc/0508104 v2, p. 3) with the time parameter in STR, thus explaining the dynamics of embedding a quantum event into Minkowski spacetime.

If you start from the quantum realm, I bet you too will reach the dead-end (see the URL above), and won't be able to recover the classical world of STR. Alternatively, if you start from STR, you won't be able to introduce a "smooth" imaginary unit in the phase of QM waves.

Just try it, either way, and I'll elaborate, with many references.

Lastly, you wrote: "I am entirely convinced of the idea that elementary particles are just ..."

Neither you nor anyone can be "entirely convinced" of anything, because neither you nor any of your colleagues can explain the bi-directional link between the quantum and classical worlds. If you could, I would have heard about your discovery on CNN Breaking News.

BTW your colleague Gerardus 'The Head' had promised me to mention my web site in his fundamental paper on the cosmological "constant",

http://www.God-does-not-play-dice.net/Gerard.html#201

I'll be happy to diversify my "slogan" with some help from his (forthcoming?) paper as well. Can't wait!

Regards,

Dimi
--
So one of my missions in life is to get people to see that if they want to talk about the problems of quantum mechanics -- the real problems of quantum mechanics -- they must be talking about Lorentz invariance.

John S. Bell
http://plato.stanford.edu/entries/qm-bohm/
 
 

Note: Regarding the link between the quantum and classical worlds (cf. above), recall the paper by Erwin Schrödinger "Specielle Relativitätstheorie und Quantenmechanik" [Ref. 1, p. 247]. He wrote (quoted after [Ref. 2]):

"Ein solcher Systemzustand ist wohl physikalisch sinnlos; die Idealuhr, von welcher die Q.M. durch Verwendung der Variablen t Gebrauch macht, ist mit den Grundlagen der Q.M. im Widerspruch."

("Such a state of the system is physically meaningless; the ideal clock, that quantum mechanics uses by the application of the variable t, is in contradiction with the foundations of quantum mechanics." Thanslated by Jan Hilgevoord [Ref. 2].)

The problem is this: "whereas classically the spatial coordinates and the time coordinate are on an equal footing -- the basic idea of the Lorentz transformation -- in quantum mechanics this is not so. Time in quantum mechanics is not an operator but a c-number; its value is supposed to be
exactly known." [Ref. 2, p. 49]

Thus, we know since 1931 that we contradict the very foundations of QM by applying the value of the variable t . The latter is taken from the 'objective reality out there' of the classical world of STR, as I stressed seventy years after Schrödinger's paper, in May 2001.

Seventy years ago, in 1935, Erwin Schrödinger stressed in "Die gegenwärtige Situation in der Quantenmechanik" that "a variable has no definite value before I measure it" (recall Kochen-Specker theorem), yet we use the value of the "variable"  t  in blatant contradiction with the foundations of QM.

QM and STR are not, and cannot be compatible.

Now, we take the "variable"  t  from the cosmological time, don't we?

Dead silence. So far there is no CNN Breaking News today, nor any reply to my email from May 8, 2001. All I suggested was to develop quantum gravity by supplying QM and GR with what they do not have: reality. It's just a potential reality.

Do we have to wait another seventy years to understand the problem of the so-called dark energy? Can you find an elephant in your office, which is 18 times bigger than your desk?

Actually, the situation is even more bizarre: you stay quietly in your car, and notice that it is being accelerated by some perfectly smooth "dark" force, some dark elephant maybe, which is 18 times bigger than your car. All you know is that the dark elephant "behind" your car is not 'objective reality out there', as we know it from STR and GR. Moreover, it has its own dynamics (the coincidence problem), and if you wish to play with the Equation Of State (EOS) of the dark elephant "behind" your car, try to figure it out whether its size increases in cosmological time as well: the faster the car/expansion, the bigger the dark elephant. Where does it come from so that it can grow bigger and bigger? Looks like we're dealing with some utterly non-unitary dark elephant evolution. As John Wheeler put it, "Time is Nature's way to keep things from happening all at once."

Do you really want to embed the dark elephant in your car?
 

D. Chakalov
September 12, 2005
Last update: September 23, 2005
 
 

[Ref. 1] Erwin Schrödinger (1931), Specielle Relativitätstheorie und Quantenmechanik, Sitzungsberichte der Preussischen Akademie der Wissenschaften, phys.-math. Kl., Bd. 12, S. 238-247.

[Ref. 2] Jan Hilgevoord, Time in quantum mechanics: a story of confusion, Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, Volume 36, Issue 1, March 2005, Pages 29-60. ScienceDirect link here, PDF file here.
 
 

=====
 

Subject: Pencils of ideal observers
Date: Thu, 15 Sep 2005 22:45:00 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: José Pereira <jpereira@ift.unesp.br>
CC: Chris Isham <c.isham@imperial.ac.uk>

Dear José,

I believe it is possible to suggest a peaceful way of reconciling the equivalence and the uncertainty principles, contrary to the claim in your recent gr-qc/0509051 [Ref. 1.] All we have to do is to supply QM and GR with what they do not have: the notion of reality. It's just a *potential* reality. This should have been clear since 1931,

http://www.God-does-not-play-dice.net/Kiefer.html#note_1931

Then instead of quantizing gravity, we should seek a quantum theory that yields General Relativity as its classical limits, as suggested by Chris Isham 20 years ago [Ref. 2].

Why invent the wheel?

Best regards,

Dimi
--

[Ref. 1] R. Aldrovandi, J. G. Pereira, K. H. Vu, Gravity and the Quantum: Are they Reconcilable? gr-qc/0509051 v1.
http://arxiv.org/abs/gr-qc/0509051

"Quantum mechanics, on the other hand, is fundamentally based on the
uncertainty principle, which is essentially nonlocal in the sense that a particle does not follow one trajectory, but infinitely many trajectories, each one with a different probability. This difference precludes the existence of a quantum version of the strong equivalence principle, and consequently of a quantum version of general relativity.
...

"To begin with, we note that the strong version of the equivalence principle, which requires the weak one, presupposes an ideal observer [3], represented by a timelike curve which intersects the space-section *at a point*. In each space-section, it applies at that intersecting point.

"The conflict comes, for the strong principle, from that idealization and extends, clearly, also to special relativity. In the equation for a curve, gravitation only appears through the Levi-Civita connection, which can be made to vanish all along. An ideal observer can choose frames whose acceleration exactly compensate the effect of gravitation. A real observer, on the other hand, will be necessarily an object extended in space, consequently intersecting a congruence of curves. Such congruences are described by the deviation equation and, consequently, detect the true covariant object characterizing the gravitational field, the curvature tensor which cannot be made to vanish. Quantum Mechanics requires real observers, pencils of ideal observers.

"The inconsistency with the strong principle, therefore, is a mathematical necessity. It is not possible, as a consequence, to define a quantum version of the strong equivalence principle [4]."
 

[Ref. 2] A. Davies and D. Sutherland, eds. Superstrings and Supergravity. Proceedings of 28th Scottish Universities Summer School in Physics. SUSSP, Oxford, 1986, p. 8.
 
 


 
 

Comments on "Quantum Cosmology and the Arrow of Time", by Claus Kiefer, gr-qc/0502016 v1. 

(I have outlined some portions of the text with bold and red. - D.C.)

In a nutshell: According to a well-known idea due to H.-D. Zeh, the increase of entropy defines the direction of time (gr-qc/0502016 v1, p. 6).

If the increase of entropy defines the direction of time, then the increase of entropy should be related to some fundamental phenomenon which defines the direction of time.

A thermometer reads the temperature, but does not define it. Claus Kiefer believes that the increase of entropy defines the direction of time, while I believe that no physical phenomenon can completely define the direction of time, because the missing effect of the Holon will inevitably make your physical phenomenon look like a jolly good thermometer. The physics of time asymmetry is a perfect case for studying the "dark" effect of the Holon, all we need is to take seriously the atemporal medium of John Cramer and the "explicit (but unmeasurable) time" of Bill Unruh, and search for the Unmoved Mover of Aristotle and Karel Kuchar.

But let's go back by Claus Kiefer's gr-qc/0502016 and look closely at "the important property" that Vi --> 0  for  [alpha] --> - [infinity].

Here [alpha] is the so-called intrinsic (C. Kiefer) cosmological time. H.-D. Zeh has set it to approach asymptotically - [infinity]

Hold on.  Before inflation, the universe is timeless and there is no classical evolution (gr-qc/0502016).

Hence if we instruct [alpha] to approach asymptotically - [infinity], it should hit the "timeless" stage of The Beginning after just 13.7 billion years. If so, it can't be "intrinsic" anymore. Let's drop this possibility.

Then perhaps [alpha] should contact the "inflationary" stage only. Hence [alpha] should include the whole "inflationary" stage that has started "right after" ... WHAT?  No, no way, we should drop this possibility too.

Do you see the catch? C. Kiefer uses only one kind of time. It's the "intrinsic" time of a physical clock, as defined in GR. Once you allow the clock to even touch the "inflationary stage", you can't stop it, and the poor clock will rush toward The Beginning. However, it cannot, even in principle, survive the inflationary expansion of the spacetime metric. This process is totally destructive and irreversible. Hence it should define a new "arrow of time" following the deflation time. This new "arrow of time" goes straight to The Beginning, only the latter cannot be reached by any test particle viewed as a clock (cf. the vacuum cleaner paradox here). This new "arrow of time" is just as good as the "arrow" produced by the so-called decoherent quantum world. I mean, both "arrows" simply don't make sense.

Tough. I mean, total mess. Which reminds me of a very nice letter I received from Yakov Zel'dovich in April 1986, in which he wrote: "Long time ago, there was a short period of time during which there was still no time at all." I suppose he meant the whole Universe as the only strictly closed quantum system. It's a genuine timeless entity, also known as the Unmoved Mover, after Aristotle and K. Kuchar.

More on the nature of time here.

By the way, the late Yakov Borisovich made another joke in his letter: he addressed me with 'Professor Chakalov'. Of course I'm not. C. Kiefer and H.-D. Zeh are professors in theoretical physics, and you should read their papers, not mine. Just don't forget that you too are indissolubly linked to the whole Universe as the only strictly closed quantum system, and there is nothing you can do about it. Unless of course you use the global mode of spacetime.

Not sure? Consider this. Suppose that, just for the sake of the argument, it were possible to discover the fundamental time asymmetry from the local mode of spacetime, along the lines of C. Kiefer and H.-D. Zeh. You would find some physical process or phenomenon that makes the elementary timelike displacement. But the local mode of spacetime is not only T-invariant. It is physically unbounded as well, in the sense that any physical mechanism that you discover will inevitably require for its explanation some new meta-theory and a new, next-level physical mechanism (recall Kurt Gödel's theorems). This kind of infinity is known from the ancient Greeks, they called it apeiron. We call it 'potential infinity'. It never ends in the local mode of spacetime, because any physical explanation there leads to another physical explanation, much like the Thompson's lamp paradox here. To understand the nature of continuum, we need both the local and the global modes of spacetime, as I tried to argue in my email to C. Kiefer below.

You can't beat this simple metaphysics with any advanced math, because it is embedded in your thinking. Surely "no reference to an external measurement agency can be made", as stressed by C. Kiefer. But the "external agency" does not make measurements. It is in the global mode of spacetime, and makes the elementary transitions "between" two "points" from the continuum of the local mode of spacetime. That's all it does. If you believe can catch it from the local mode of spacetime, check out Confucius' cat here.

C. Kiefer quotes in gr-qc/0502016 M. Gell-Mann and J. B. Hartle who claim that "quantum mechanics is best and most fundamentally understood in the framework of quantum cosmology". I fully agree, in the sense that nobody understands quantum mechanics nor quantum cosmology anyway.
 




The line of reasoning in current quantum cosmology is like this: you see a big gorgeous hen, and you say -- okay, this hen has surely evolved from a nice big chicken, which in turns has evolved from a tiny little cute chicken, which has obviously came from an egg, but the egg has evolved from a totally unphysical "inflationary" oven, and no egg can possibly be traced back to that terribly hot oven. On the top of everything, the "oven" has had an incredibly unlikely low entropy, that is, an extremely unprobable initial condition of low gravitational entropy and an incredibly fine adjustment of the boundary conditions for the evolution of the hen and its fundamental "constants". Then you shrug your shoulders and admit that perhaps something very essential is missing.

This shouldn't be surprising at all, since you know that 96 per cent of the hen is of totally unknown nature, some "dark hen" stuff that behaves like mass (23 per cent) and energy (73 per cent). But if somebody suggests that there is no need for any "hot oven" nor "inflationary stage" of the egg, because the evolution of the hen could be non-unitary, and the hen could have evolved from an extremely ordered initial state of 'light and cognition' (also known as [John 1:1]), you say -- no, we don't like such non-unitary, creatio ex nihilo evolution, and we will definitely find the physical origin of all that dark hen stuff, sooner or later.

Meanwhile you teach your undergraduates the machinery of "decoherence" (J.J. Halliwell) and explain the subtleties of 'how the quantum hen became classical'. You introduce the notion of time to these kids, and say that their clocks read the time of hen's evolution, which is why "quantum mechanics is best and most fundamentally understood in the framework of quantum cosmology" (M. Gell-Mann and J. B. Hartle). Then you admit to the kids that we, of course, need quantum gravity to understand quantum cosmology and quantum mechanics, and a full-blown quantum gravity is still out of sight, but the good news is that we at least understand quantum theory and the superposition principle, and can bet all our money that the "time parameter" in the Schrödinger equation can indeed be converted into the time parameter of hen's evolution, which is being read by the clocks of your undergraduates. They shouldn't worry about some "collapse", since it can be avoided in the decoherent history approach to quantum mechanics, which again is "best and most fundamentally understood in the framework of quantum cosmology" (M. Gell-Mann and J. B. Hartle).

If you still have some doubts about 'how the quantum hen became classical', replace the 'hen' with the CPU in your computer, and apply the theory of decoherence to explain the operation of your processor as a decoherent quantum system. For example, it can calculate the circumference of a circle, and will never make errors.

More from C. Kiefer's "Quantum Gravity" below. I can wholeheartedly recommend it, since it reveals most of the problems from using just one kind of time (local mode of spacetime). Read this:

"A third motivation is the problem of time. Quantum theory and GR (in fact, every general covariant theory) contain a drastically different concept of time (and space-time). Strictly speaking, they are incompatible. In quantum theory, time is an external (absolute) element, not described by an operator (in special relativistic quantum field theory, the role of time is played by the external Minkowski space-time). In contrast, in GR, space-time is a dynamical object. It is clear that a unification of quantum theory with GR must lead to modifications of the concept of time."

Join the club! Not sure? Recall Yakov Zel'dovich: "Long time ago, there was a short period of time during which there was still no time at all." He was joking, of course. We always keep this timeless entity (global mode of spacetime) in our brains. It's an effect of the Holon and is very, very "dark" indeed.
 

D. Chakalov
February 4, 2005
Last update: February 14, 2005
 
 

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

C. Kiefer, Quantum Cosmology and the Arrow of Time, gr-qc/0502016 v1.

p. 1: "Apart from microscopic and some mesoscopic systems, it is usually not possible to isolate a system from its environment. Following this chain, the environment is coupled to its environment, and so on, leading ultimately to the whole Universe as the only strictly closed quantum system. Universality of quantum theory thus dictates that the Universe as a whole has to be described by quantum theory -- this is the realm of quantum cosmology. For its interpretation, no reference to an external measurement agency can be made. Since such an interpretational scheme provides insight into quantum theory in general, it was claimed that "quantum mechanics is best and most fundamentally understood in the framework of quantum cosmology" [2].
...

p. 2: "The topic addressed here is the observed irreversibility of the world and its possible justification from quantum cosmology [5]. (...) I shall argue that the formal structure of the equations of canonical quantum gravity by themselves suggests a simple boundary condition from where the arrows of time follow naturally. I shall start with a brief review of the quantum cosmological formalism and the problem of the arrows of time. I then attempt to trace the origin of these arrows to a simple boundary condition in quantum cosmology. Finally I shall briefly discuss possible consequences for black holes."

p. 3: "A most important question is how to address appropriate boundary conditions. Since there is no external time, boundary conditions have to be imposed with respect to intrinsic (dynamical) degrees of freedom. For the Wheeler-DeWitt equation in quantum cosmology, the scale factor presents itself as the appropriate timelike variable."

p. 4: "An important ingredient is decoherence of relevant variables (such as the volume of the universe) by irrelevant variables (such as small density fluctuations). Otherwise one would encounter superpositions of macroscopically different universes. Decoherence 'starts' with the onset of inflation; before inflation, the universe is timeless and there is no classical evolution [15]."

(...) Because of decoherence, one must use a master equation instead
of the Schrödinger equation, valid for the evolution into the forward direction of (semiclassical) time t. If this master equation holds in that direction of t that corresponds to increasing scale factor  a , it cannot be valid across a classical turning point into a recollapsing phase. This indicates that the emergence of an arrow of time for a classically recollapsing universe is subtle, see Section IV."

p. 5: "Our universe is thus characterized by an extremely unprobable initial condition of low gravitational entropy, or, in other words: Why did the universe start so smoothly?"

p. 6: "How can one derive an arrow of time from a framework that does contain no time? I shall discuss in the following section how this can be achieved, at least in principle."

p. 6: "The potential appearing in (4) is asymmetric with respect to 'intrinsic time' [alpha]; one has, in particular, the important property that Vi --> 0 for [alpha] --> - [infinity].

"This allows one to impose a very simple boundary condition in this limit. As Zeh has suggested [5], one can demand that

 [XXX]    (5)

that is, an initial condition where the various degrees of freedom are not
entangled. Solving then the Wheeler-DeWitt equation with this condition,
entanglement automatically increases with increasing [alpha]."

"This increase of entropy then defines the direction of time. All the arrows of time discussed in Section III would then have their common root in this entropy increase. (...) Quantum cosmology thus not only specifies the beginning of the classical evolution (when decoherence sets in at the onset of inflation), but also its end."
 
 

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

From: Dimi Chakalov <dimi@chakalov.net>
To: Claus Kiefer <kiefer@thp.uni-koeln.de>
Cc: [snip]
Subject: Archimedes & Kalina vs Ashtekar & Smolin
Date: Wed, 28 Jan 2004 23:00:05 -0000
 

Dear Claus,

I spotted your name in an article by Mr. Rüdiger Vaas [Ref. 1]. I don't know what you've actually said, but it seems to me that you have given spin networks some sort of 'benefit of the doubt': "the transition of spin networks to classical spacetime is not yet completely understood" [Ibid.].

As if the transition of spin networks to classical spacetime *might* be possible, only it is not yet *completely* understood.

If this is indeed your standpoint, I would like to share with you my disagreement. I will use math suitable for the readers of "Bild der Wissenschaft" and my ten-year old daughter, Kalina. She is fluent in German (she actually speaks German far better than Bulgarian), and I will suggest her to write to "Bild der Wissenschaft" and to request some clarifications. Here's why.

Kalina knows how to derive the circumference of a circle, after Archimedes, who used the limiting process on the area and base of polygons inscribed in circles -- as n  approaches infinity -- to determine [pi] ,

http://www.math.psu.edu/courses/maserick/circle/circleapplet.html

As the number of sides, n , on the inscribed polygon increases, its perimeter approaches the circumference of a circle. Then comes the magic of the infinitesimal: as  n  approaches (Sic!) infinity, we get the well-known formula and the number [pi] . Only we don't know what actually happens with  n  "when" it approaches infinity. So much about quantum geometry,

http://members.aon.at/chakalov/Pestov.html#continuum

Perhaps Kalina will ask two very simple questions (in German, of course):

Q1: Exactly how big is n , the number of sides? Is it, for example, more or less than the "atoms of volume" in a cubic centimeter, 1099,

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

Q2: If we cannot answer Q1, can we at least be certain that  n  can be presented with *any* number? If yes, can spin networks explain the circumference of a circle?

I believe her questions will be forwarded to you for consideration, so please reply to Kalina as well. Her email is: kalina@chakalov.net More of her questions at

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

Smart kid, eh? Some day I'll tell her more about Einstein and the nature of gravity,

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

Kindest regards,

Dimi
--
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

Reference

[Ref. 1] Rüdiger Vaas, Jenseits von Raum und Zeit, Bild der Wissenschaft (2003), No. 12, pp. 50-56 (English translation by Amitabha Sen), Beyond Space And Time, physics/0401128,
http://arxiv.org/abs/physics/0401128

"Were Albert Einstein alive today, it would have given him great pleasure. For the goal is to fulfill the big dream of a unified theory of gravity and the quantum world.
...
"This is how quantum geometry revolutionizes our worldview: Space is quantized like matter!
...
"The main competitor to quantum geometry is string theory. In contrast to this, Ashtekar's formulation contains no unification of all forces - 'only' gravitation is quantized separately," says Claus Kiefer, professor for physics at the University of Cologne and one of the leading experts in the field of quantum gravity in Germany. "Independent of the correctness of the approach, its value lies in pointing out which aspects in a final theory of quantum gravity are to be expected."
...
"Certainly this also has gaps and weaknesses. Like the transition of spin networks to classical spacetime is not yet completely understood, and the calculation of entropy of black holes creates some problems," says Kiefer."

====

Subject: Re: The cosmological constant problem
Date: Sat, 07 Feb 2004 16:44:21 +0200
From: Dimi Chakalov <dimi@chakalov.net>
To: Claus Kiefer <kiefer@thp.uni-koeln.de>

Dear Claus,

In case you're too busy to respond to my daughter,

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

perhaps you could write a paper on the Hilbert space problem (Conceptual issues in quantum cosmology, gr-qc/9906100) and the Barber Paradox,

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

Hope to see you at GR17 in Dublin; the deadline for abstract submission is March 19th.

More at

http://members.aon.at/chakalov/Seevinck.html#note

Best regards,

Dimi

======

Subject: The quantization of GR
Date: Thu, 05 Aug 2004 14:36:20 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: kiefer@thp.uni-koeln.de
CC: unruh@physics.ubc.ca

Dear Claus,

I read very carefully your recent gr-qc/0408010 v1. You wrote:

"There should therefore exist an encompassing theory of quantum gravity in which these concepts are modified. In fact, the straightforward quantization of GR leads to the result that no external time can exist at all at the most fundamental level."

I wonder if you have mentioned Bill Unruh's ideas

Bill Unruh, No time and quantum gravity, in R. Mann and P. Wesson, editors, Gravitation: A Banff Summer Institute. Banff Center, Banff, Canada, August 12-25, 1990. September 1991, pp. 260–275.

in your recent book

C. Kiefer, Quantum gravity (Oxford University Press, Oxford, 2004).

Perhaps the "mysterious time" is what we need. As you put it, "Mass generates curvature which in turn acts back on the mass." It seems to me that some "mysterious time" is needed for the *timing* of this bi-directional talk.

Perhaps you can check it with your brain,

http://God-does-not-play-dice.net/Azbel.html#self

You also wrote: "Since gravity couples to all forms of energy, it is clear that coupling occurs, and we want to stress that it is of utmost importance to investigate this interaction further; (...)."

Please note that I'm available.

Regards,

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

Comments: The quantization of GR is a daunting task; see Angelo Loinger, "Quantum gravity": an oxymoron, physics/0308042. I haven't yet read Claus Kiefer's monograph Quantum Gravity (Oxford University Press, Oxford, 2004, 320 pages; ISBN: 0-19-850687-2), but I'm sure he is well-aware of the cornerstone issues in Einstein's GR, which are explained in length by Prof. Angelo Loinger.

Let me briefly comment on three excerpts from the outline of C. Kiefer's book, which can be downloaded from this URL. The author of this outline, "WHY QUANTUM GRAVITY?", is C. Kiefer.

"A third motivation is the problem of time. Quantum theory and GR (in fact, every general covariant theory) contain a drastically different concept of time (and space-time). Strictly speaking, they are incompatible. In quantum theory, time is an external (absolute) element, not described by an operator (in special relativistic quantum field theory, the role of time is played by the external Minkowski space-time). In contrast, in GR, space-time is a dynamical object. It is clear that a unification of quantum theory with GR must lead to modifications of the concept of time. One might expect that the metric has to be turned into an operator."

One might also expect that, should the metric has to be turned into an operator, we will need the global mode of spacetime. I hope to hear also from Bill Unruh on this issue.

"In a universally valid quantum theory, genuine quantum effects can occur on any scale, while classical properties are an emergent phenomenon only (see Chapter 10). This is a consequence of the superposition principle."

Chapter 10, Quantum gravity and the interpretation of quantum theory, is probably the crucial part of C. Kiefer's book. The challenge is well-known, see C.N. Yang. I cannot agree with the standard speculation about the "emergence" of classical properties, which was recently mentioned by H. Nikolai.

And the last excerpt from C. Kiefer's "WHY QUANTUM GRAVITY?":

"According to the Copenhagen interpretation of quantum theory, all structures related to space-time would probably have to stay classical because they are thought to be necessary ingredients for the measurement process, cf. Chapter 10. For the purpose of quantum gravity, such a viewpoint is, however, insufficient and probably inconsistent."

Here again we're referred to Chapter 10. It seems to me that all structures related to spacetime can and should stay classical in the local mode of spacetime, since the solution to the measurement problem is given by including the global mode of spacetime pertaining to the Holon. Otherwise how can we find the circumference of a circle or any other "point-like" event, after Thompson's lamp paradox?

It's always been a pleasure to read Claus Kiefer. I'll read his monograph Quantum Gravity very soon.
 
 

D. Chakalov
July 5, 2004

===============
Subject: Asymptotically flat 3-D space (private)
Date: Mon, 27 Sep 2004 05:40:19 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: kiefer@thp.uni-koeln.de
CC: cw@thp.uni-koeln.de, c.isham@imperial.ac.uk, r.loll@phys.uu.nl

Hi Claus,

I wonder if you have elaborated in your monograph "Quantum Gravity" on the statement:

"In a universally valid quantum theory, genuine quantum effects can occur on any scale, while classical properties are an emergent phenomenon only (see Chapter 10). This is a consequence of the superposition principle."

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

Regarding the dimension of the 3-D space, Renate Loll et al., hep-th/0404156 v4, got d_h = 3.10 ± 0.15. I wonder how far one could see in a spatial hypersurface with d_h = 3.10 ± 0.15. Lee Smolin, for example, wasn't quite successful: "One of the biggest mysteries is that we live in a world in which it is possible to look around, and see as far as we can" (Three Roads to Quantum Gravity, p. 205).

It seems to me that if we live in an asymptotically flat 3-D space, we need to find out (i) what is carefully keeping it 'asymptotically flat', and (ii) why its dimension is *exactly*  3 . Not d_h = 3.10 ± 0.15,

http://mathworld.wolfram.com/HausdorffDimension.html

Regarding (i), see the story of the twin elephants at

http://God-does-not-play-dice.net/energy.html#elephant

Only the 'dark elephant' lives "outside" 4-D spacetime,

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

http://God-does-not-play-dice.net/energy.html#gravity

http://God-does-not-play-dice.net/Barbara.html#imaginary

A penny for your thoughts! Will keep them private.

Kindest regards,

Dimi
 

Note: Let me elaborate on the word 'carefully' above (highlighted in red). Apart from the incredible "coincidence" that our universe "happened" to be equipped with asymptotically flat 3-D space (here we enter a big can of worms from inflationary cosmology and the initial condition, as well as all sorts of speculations about the astonishing balance between open and closed 3-D topologies, which make our universe truly unique), there is something far more real: the dynamical balance between the two forms of gravity, revealed in its two invisible components, "dark" matter and "dark" energy. They form 96 per cent from the stuff of the universe, so we cannot ignore them. This dynamical balance didn't happen just once, say, 13.7 billion years ago. It is happening now, in each and every instant of time, as read by your wristwatch.

If we talk about dynamical balance, we imply the notion of time. The corresponding 'conservation law' that leads to this incredible balance is not known, for the following reason: we cannot employ the notion of time from Einstein's GR. What physicists call 'time' is the end-product from this "talk" between the two forms of gravity (see the Catch 22 paradox in Einstein's GR here.) All this is intermingled with the coincidence problem. As explained by Paul Frampton (astro-ph/0409166 v3), "we live right in the middle of the spike of the delta function. If the dark energy had appeared earlier it would have interfered with structure formation: if later, we would still be unaware of it". See also Julien Lesgourgues' "An overview of Cosmology", astro-ph/0409426 v1, p. 57 and Sec. 2.4.4.

Then there is another puzzle: the so-called dark energy is smooth, isotropic, and omnipresent, just as the good old 3-D space. We cannot, however, address the issue properly, since there is no preferred or absolute reference frame in Einstein's GR. All we know is that, if there were just one "clump" of this exotic energy, we would have immediately noticed it from the anomalous, if not catastrophic, behavior of matter fields around the "clump". As explained by Matt Visser in gr-qc/0309109 v4, it's a very powerful stuff: "Apart from possibly destroying the universe in a future "big rip" singularity [7], if the exotic matter clumps to any extent there is real risk of even more seriously bizarre behaviour -- everything from violations of the positive mass condition (that is, objects with negative asymptotic mass), through traversable wormholes, to time warps [4, 14, 15, 16, 17]."

Hence in order to address the whole bundle of issues attached to the seemingly simple expression 'asymptotically flat 3-D space', we need to confront yet another puzzle: the positivity mass "theorems". We don't actually prove these "theorems". We postulate them. Tom Roman was very clear on this issue (gr-qc/0409090 v1): "Energy conditions tell us what are "physically reasonable"  distributions of mass-energy, which in turn tell us what are physically reasonable spacetime geometries. They are also crucial in proving many theorems using global techniques in general relativity, such as singularity theorems and the topological censorship theorem [15]. However, the energy conditions are not derivable from GR."

Clearly, we need new physics. And we haven't yet address the second issue above, denoted with (ii): why is the dimension of space exactly  3 ? It can't be some "coarse-grained" value, as usually stated, because we can see billions light years away in the universe. We see "as far as we can" (cf. L. Smolin above). Which brings another issue on board: the hypothetical Planck-scale "fluctuations" of the very metric of spacetime. Because these fluctuations would be extremely small, they would only be evident in light that travels a great distance, correct? To the best of my knowledge, no such effects have been observed. More on this puzzle here.

Since the introduction of geometrodynamics by John Wheeler forty years ago, many physicists talk about some "coarse-grained" stuff. If this philosophy were true, how can we obtain any exact value with differential calculus? How can we have anything exact, point-like in our 3-D world of tables and chairs? Are we surrounded by some "coarse-grained" illusion of a stable 3-D world? Explaining this illusion should be the first-off task of all physicists who seriously care about quantum gravity.

John Baez, for example, specifically warned that, in a full-fledged background-free theory such as quantum gravity, not only the spacetime metric should be dynamical but the dimension of spacetime as well: "We can even try to treat the dimension of spacetime as dynamical - although we don't really know how to do this very well. But how can we do physics without any prior assumption on the nature of spacetime, e.g. that it is a manifold, or a spin foam, or some other kind of gadget? (...) Personally I think one can dig oneself into a hole by trying to do physics without any background structure - it's a bit like trying to paint a painting without any canvas" (John Baez, 5 May 2000, What is a background-free theory?). See 'my kind of gadget' here. For an alternative approach, see Renate Loll here.

As I tried to explain on numerous occasions, the task begins with fixing the first error we made many years ago, by introducing double standards in Einstein's GR: one for the treatment of time, and quite another for the 3-D space. [See R. Sorkin et al., General Covariance and the "Problem of Time" in a Discrete Cosmology, gr-qc/0202097: "In the canonical quantization of gravity, as it is normally understood (and sought) the fundamental object of attention is not spacetime but space alone (corresponding to something like a Cauchy surface in spacetime) and anyone following this approach is sooner or later faced with the problem of recovering time from a frozen formalism (as it's sometimes called) from which time as such is absent. The problem of time in this sense is, I believe, insoluble (...)"].

Time as some background parameter disappears in Einstein's GR; see a very clear explanation here. But not the 3-D space. It is not some Perennial, as introduced by Karel Kuchar, but a serious bug in the linearized version of Einstein's GR.

As explained by Karel Kuchar in Canonical Quantum Gravity, gr-qc/9304012, "if we could observe only constants of motion, we could never observe any change". On this basis he suggested two classes of variables: observables and Perennials. The former are dynamical variables that remain invariant under spatial diffeomorphisms but do not commute with the Hamiltonian constraint, while the latter are some special "observables" that do commute with the Hamiltonian constraint from the Wheeler-DeWitt equation. Kuchar's main idea is that we can only observe dynamical variables that are not Perennials. But how do we observe any change? What moves something from one hypersurface to another, knowing very well how to find this "next" hypersurface? Only Perennials can do this miracle. They can govern the dynamics "from outside as an unmoved mover", says Karel Kuchar (Karel Kuchar, "The Problem of Time In Quantum Geometrodynamics", in The Arguments of Time, ed. by Jeremy Butterfield, Oxford University Press, Oxford, 1999, p. 193).

Now, the distinction between the two classes of variables, observables and Perennials, is a bit technical but very elucidating: "[H] generates the dynamical change of data from one hypersurface to another. The hypersurface itself is not directly observable, just as the points {x} are not directly observables. However, the collection of the canonical data (qab(1), pab(1)) on the first hypersurface is clearly distinguishable from the collection (qab(2), pab(2)) of the (already - D.C.) evolved data on the second hypersurface. If we could not distinguish between those two sets of data, we would never be able to observe dynamical evolution" (K. Kuchar, Canonical Quantum Gravity, gr-qc/9304012). Hence Perennials do exist: we observe dynamical evolution, just because there are Perennials spanned "over" two or more hypersurfaces. And the Atom of Lucretius does exist too, just because we observe dynamical evolution. And the gaps of spacetime do exist too, just because we observe dynamical evolution. Only the gaps of spacetime accommodating Kuchar's Perennials and Lucretius' Atom are dynamic objects tending asymptotically toward zero, as we know since the time of Zeno. All this goes back to the paradox of continuum and its total ignorance in the linearized version of Einstein's GR. "No perennial ever changes along a dynamical trajectory", says Karel Kuchar in Canonical Quantum Gravity, gr-qc/9304012. Sure, but we cannot find these Perennials in the linearized version of Einstein's GR. That's the problem, as told by Hermann Weyl in 1944 and by Lucretius some 2060 years ago.

I like Perennials, and I always carry them in my brain. I have a number of ideas about the pre-geometrical "gaps" of spacetime needed for the phenomenon of emergence of our 4-D spacetime. It's a jigsaw puzzle with many pieces, and we have to keep all of them in our mind, because 'at the end of the day' we should arrive at some nice smooth asymptotically flat 3-D space, and should be able to see as far as we want. The first rule of the puzzle is simple: keep all elephants invisible in the gaps. Second rule: upgrade Einstein's GR with two virtual worlds with "inverted" spacetime basis. Third rule: make a pool of "dark energy" to expand the 4-D spacetime with constant acceleration. Forth rule: make 4-D "points", one-at-a-time, in the local mode of 4-D spacetime. Make sure that the "size" of these 4-D "points" is always running toward zero, but will never actually reach zero. Hence you may hope that will get many blueprints from the "gaps" (global mode of spacetime) cast in the local mode of spacetime, and will eventually solve all the puzzles in Einstein's cosmological constant. There are many such blueprints, but the most important ones are that we have 3-D space per se: we have 'big' and 'small', inside and outside. Perhaps we need to extend the notion of Diff(M)-invariance with a new symmetry group of 'space inversion', as I tried to suggest on April 12, 2003. And here come the ideas about the pre-geometrical "gaps" of spacetime, as mentioned above, and some non-smooth topological transitions of the two virtual worlds in the "gaps", creating 4-D "points" one-at-a-time. It's a bit complicated, but if you are interested, please drop me a line.

Albert Einstein wrote in December 1916: "In the first place, we entirely shun the vague word "space", of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by "motion relative to a practically rigid body of reference." (...) With the aid of this example it is clearly seen that there is no such thing as an independently existing trajectory (...), but only a trajectory relative to a particular body of reference." (A. Einstein, Relativity: The Special and The General Theory, Three Rivers Press, Crown Publ., 1995, translated by Robert W. Lawson, Part III, Space and Time in Classical Mechanics; ISBN: 0517884410. Original title: Über die spezielle und allgemeine Relativitätstheorie, gemeinverständlich, 1917. See an introduction by Soshichi Uchii here.)

Hence we have to expect that the world at the fundamental level should not possess any background whatsoever, correct? Jain! Surely nothing physical can possibly serve as background. The ultimate background should not be physical and should govern the dynamics "from outside as an unmoved mover", as explained by Karel Kuchar above. The "unmoved mover" is, of course, the Aristotelian First Cause: the Universe as ONE.

Do we enjoy 3-D space in which it is possible to look around and see as far as we can? Is it a perfect (not "coarse-grained") continuum? If the answers to these questions is in the affirmative, we need to construct a perfect isotropic and smooth continuum of "points", and to insert the "gaps" in our 4-D spacetime "from outside as an unmoved mover", as explained by Karel Kuchar. Everything we can physically observe is being already cast in our past light cone, and is laid out on a perfect continuum. The gaps are not there, again. We need a new mode of spacetime, a global mode, to accommodate the 'context' that comes from the last Holon: the Universe as ONE. But we cannot find these "gaps" in the linearized version of Einstein's GR, since this "linearized" misunderstanding is totally blind to these gaps. That's the problem, as told by Hermann Weyl in 1944 and by Lucretius some 2060 years ago. All we can say, from the standpoint of the linearized Einstein's GR, is that an infinitesimal piece of spacetime "looks flat" (Michael Weiss and John Baez, Is Energy Conserved in General Relativity? see more here). Why does it look "flat"? Because in order to compute the material content of spacetime, we have to eliminate its 'context' from the Holon (also known as Karel Kuchar's Perennials).

To sum up, were the spacetime created only and exclusively only by the material content of spacetime, it would have inevitably suffered from the pathologies of the same material content, as seen in a fully background-free theory such as quantum gravity. We need not just 'the words in the sentence' but their 'common context' as well. We need to discover the dynamics of prior-geometry, which is responsible for the emergence of time and space, as explained by Chris Isham. The dynamics of prior-geometry cannot be built on "the a priori, Euclidean four-dimensional space, the belief in which amounts to something like a superstition", as stressed by Einstein (Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein, Oxford University Press, 1983, p. 235). We don't want to introduce some non-dynamical, Riemann-flat metric à la Nathan Rosen's bimetric theory either. We need to zoom on the fine structure of the spacetime gaps. The content of the gaps is not some pre-existing and immutable arena of spacetime, however. It changes along the evolution of the universe: subtle is The Lord.

Again, if you are interested, please drop me a line. I can't promise some detailed mathematical revelations, but besser ein Laus im Kraut als gar kein Fleisch. (I don't know how to translate it into English, so perhaps you may wish to contact Claus Kiefer. He is fluent in both English and math.)
 
 

D. Chakalov
September 27, 2004
Last update: October 10, 2004
 

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

Subject: The spectra of lengths, areas and volumes
Date: Sat, 13 Mar 2004 19:17:32 +0200
From: Dimi Chakalov <dimi@chakalov.net>
To: Jerzy Lewandowski <lewand@fuw.edu.pl>
CC: kiefer@thp.uni-koeln.de
 

Dear Dr. Lewandowski,

In your article "Space and Time Beyond Einstein" [Ref. 2], co-authored by Dr. Abhay Ashtekar,

http://cgpg.gravity.psu.edu/~ashtekar/nyt/abhay.html

you and your colleague wrote:

"The rigorous mathematics of quantum geometry predicts that lengths, areas and volumes are 'quantized' in a very specific way and enables one to calculate their 'spectra,' i.e., allowed, discrete values. These results have been used to resolve some long standing puzzles."

I wonder if you have resolved the transition of spin networks to classical spacetime, which is "not yet completely understood", according to Dr. Claus Kiefer,

http://members.aon.at/chakalov/Kiefer.html#spectra

In case there are some vague points in your model of quantum geometry, I trust you will sort them out at GR17. My tentative contribution is outlined at my web site.

Regards,

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

Reference

[Ref. 2] Space and Time Beyond Einstein, by Abhay Ashtekar and Jerzy Lewandowski,
http://cgpg.gravity.psu.edu/people/Ashtekar/articles/
Rzeczpospolita_SaTBE.html
 

Note: Perhaps Jerzy Lewandowski can shed some light on the most basic idea in Ashtekar's program: the so-called parallel transport. It has been explained to the general audience seven years ago (Beyond space and time, by Robert Matthews, New Scientist, 17 May 1997, pp. 38-42):

"With connections, the shortest distance is defined as the route along which every tangent vector is parallel to its neighbour so there is no need for coordinates X and Y" (Ibid., p. 41).

The immediate, and perfectly obvious, question follows: How could any "point" know that its next neighboring "point" might have a tangent vector that is parallel to it, so it can anneal its tangent vector to the one in its immediate future? Something utterly non-local is implied here, on the most fundamental level. I call it 'global mode of spacetime'. Join the club!

Not sure? Recall A. Ashtekar: "(W)e do not have a natural prescription for unifying our description of gravity with that of sub-atomic particles." (Ibid., p. 42) This is 'Die gegenwärtige Situation' seven years later, as stressed by Claus Kiefer above.

Regarding the string hypotheses, Robert Matthews wrote: "Using the Minkowski metric in your attempts to understand gravity smacks of assuming the properties of the very thing you are trying to understand." (Ibid., p. 40)

It's a bit like that: if you are doing analytical chemistry and want to prove that there is NaCl in your test sample, you must not contaminate it with NaCl from the outset.

In the case of string hypotheses, however, you know from school that your idea of NaCl is initially inadequate for searching and identifying 'the right NaCl': your NaCl is not salty. It has a fixed background. And you know from school that there is no such fixed background in GR due to general covariance. Hence you're wasting your time with those strings. Not only yours, but that of your students as well, in case your teach them "semi-classical quantum gravity" and "brane worlds" in 10, 11, or whatever dimensions.

So eventually you decide to explore a 'real salty NaCl', without any fixed background. How? Relationally, of course. You make a very important assumption: the necessary and sufficient condition for something, call it A, to exist at a given instants its relation to not-A. Hence your operational definition of 'reality' is purely and exclusively relational: A is real only in relation to another object that it is interacting with, and only at/during the very instantof interaction. That's how you begin your painting relationally, without any fixed canvas whatsoever. That's your idea of the dynamical description of your relational reality.

Probably misled from your childhood experience with a kaleidoscope, you tacitly assume that every time you shake your "relational reality", it fixes the reality of all colored pieces "relationally". So you examine the chain of instants at which your kaleidoscope has fixed some "relational reality", and claim that there is no need to think of the (global mode) of shaking the kaleidoscope, since all that matters are those frozen instants of fixed correlations of all colored pieces. But you are not a kid, you are fully aware of the fact that the correlation of all colored pieces in the kaleidoscope is not "taking place" in the state space of your already-correlated pieces, where they are at rest. You do know that something inherently non-local is "taking place" "during" the unobservable shaking of your kaleidoscope. It has been cut off from your "dynamical" description from the outset. Just like in string hypotheses, your NaCl is not salty, and you know it bloody well.

What to do, then? The current approach in spin networks is to zoom on the very structure of the state space of fixed, already-correlated colored pieces, and try to unravel the non-local correlation "during" the (global mode of) shaking the kaleidoscope in the structure of those fixed, already-correlated colored pieces. How, exactly? "To achieve this goal in 3+1 dimensions, one needs a much better understanding of the theory of (intersecting) knots in 3 dimensions", sources say. Why are you searching for something that you know is not there?

The hardest thing of all is to find a black cat in a dark room, especially if there is no cat, says Confucius, which gently reminds us of another wisdom, from Lao-tzu: If you realize that all things change, there is nothing you will try to hold onto.

Zoon on the dynamics of the human brain. The Angels are in the details.
 

D. Chakalov
March 15, 2004
Last update: March 17, 2004