Subject: The atom of geometry
Date: Thu, 28 Apr 2005 14:08:56 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Golam Mortuza Hossain <golam@imsc.res.in>
CC: Ghanashyam Date <shyam@imsc.res.in>,
     Romesh Kaul <kaul@imsc.res.in>,
     Martin Bojowald <bojowald@gravity.psu.edu>

Dear Dr. Hossain,

It is a pleasure to read your latest gr-qc/0504125 v1 [Ref. 1]. You stressed that it is "important to emphasize the meaning of volume in this context. In particular, the volume V = [XXX] of the space is infinite, as it is non-compact. To avoid this trivial divergence in loop quantum cosmology, one considers the volume of a finite cell of universe (see Fig 1.) and studies its evolution."

To avoid this generic divergence in loop quantum gravity, which inevitably leads to hidden infinities,

http://www.God-does-not-play-dice.net/Kuchar.html#Martin

http://www.God-does-not-play-dice.net/Dittrich.html#3

it seems to me that one has to elucidate the *atom of geometry*,

http://www.God-does-not-play-dice.net/Einstein.html#addendum

You acknowledged that the most important issues "related with physical observables, external time evolution, physical Hilbert space are still in nascent stage" [Ref. 1], but I see no reason for optimism whatsoever.

I'm afraid the 'infinite and non-compact triad' [Ref. 1, Fig. 1] is an oxymoron.

Perhaps loop quantum gravity made a false start with its "atom of geometry": it does *not* encode the dynamics of the world at fundamental level. Subsequently, you have to introduce the dynamics 'by hand', by some "effective Hamiltonian using WKB techniques" [Ref. 1]. I'm afraid this is not going to work,

http://www.God-does-not-play-dice.net/Nicolai.html#C3

http://www.God-does-not-play-dice.net/Kuchar.html#March_31

As to the mythical graviton, perhaps you may wish to see [Ref. 2].

Kindest regards,

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

References

[Ref. 1] Golam Mortuza Hossain, Large volume quantum correction in loop quantum cosmology: Graviton illusion? gr-qc/0504125 v1, 25 April 2005,
http://xxx.lanl.gov/abs/gr-qc/0504125

"However, the issues related with physical observables, external time
evolution, physical Hilbert space are still in nascent stage [4, 10]."
--
[4] M. Bojowald, Class. Quant. Grav. 17, 1489 (2000)
[arXiv:gr-qc/9910103]. M. Bojowald, Class. Quant. Grav. 17, 1509 (2000)
[arXiv:gr-qc/9910104]. M. Bojowald, Class. Quant. Grav. 18, 1055 (2001)
[arXiv:gr-qc/0008052]. M. Bojowald, Class. Quant. Grav. 18, 1071 (2001)
[arXiv:gr-qc/0008053]. M. Bojowald, Class. Quant. Grav. 19, 2717 (2002)
[arXiv:gr-qc/0202077]. M. Bojowald, Class. Quant. Grav. 18, L109 (2001)
[arXiv:gr-qc/0105113]. M. Bojowald and H. A. Morales-Tecotl, Lect. Notes
Phys. 646, 421 (2004) [arXiv:gr-qc/0306008]. M. Bojowald,
arXiv:gr-qc/0503020.
 

[Ref. 2] Mario Everaldo de Souza, Gravity cannot be quantized,
gr-qc/0208085 v1,
http://xxx.lanl.gov/abs/gr-qc/0208085
 
 

Note 1: The 'infinite and non-compact triad' is an oxymoron in the following sense. There is a well-known metaphysical dictum that 'the atom' cannot be described by anything pertaining to space and time. If the atom of geometry builds up the spacetime, it cannot be presented with any spacetime concept whatsoever. Can you describe Wilson loops and their LQG cousins without any geometrical concepts? If you can't, you're trying to reduce geometry to something that is geometry. Like trying to explain heat with some tiny little hot particles.

Some of the philosophers advocating spin networks hold different opinion, however. Lee Smolin, for example, claims that if you take 1099 "atoms of volume", these bubbles/loops of spacetime can build up one cubic centimeter of 3-D space without any empty gaps between them. As to the whole universe, it would be "a gargantuan spin network of unimaginable complexity, with approximately 10184 nodes", only you shouldn't be able to see much further the end of your nose: "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" (L. Smolin, Three Roads to Quantum Gravity, p. 205).

Yet all these people are incredibly optimistic, like Abby Ashtekar. Why? Well, maybe because "the underlying (internal time) dynamics is described by a difference equation" [Ref. 1]. That's the big difference: the amazing difference equations.

Surely loop quantum gravity is still in nascent stage [Ref. 1]. Nobody has noted the lessons from Lucretius, Aristotle, and St. Augustine. Why? Well, maybe because these old guys didn't use difference equations.
 

D. Chakalov
April 28, 2005
 

Note 2: Let me stress two crucial conceptual issues of all approaches to quantum gravity. I'll again refer to NPZ paper [Ref. 3].

First, we have to check out whether any of the known approaches to quantum gravity would enable us in principle to recover a smooth classical limit. The so-called 'semi-classical states' are discussed here. The crux of the matter is "whether the familiar Hilbert space formalism of standard quantum mechanics is at all the correct framework for quantum gravity" [Ref. 3].

To elucidate this utterly important issue, we must study the very transition from quantum to classical regime, and back. If we cannot explain this transition, we must seek new quantum theory, after Chris Isham.

Hence the first off task is studying the embedding of a "quantum" event into Minkowski spacetime, as explained here and here. More can be read (in the context of "quantum computing") here.

Secondly, we have to check out whether we've discovered the dynamics of GR. This is also a crucial prerequisite. It seems to me that the dynamics of GR is not known. See the problems of the mythical gravitational waves here, and be aware of the "dark energy". I'm afraid we have not captured the genuine dynamics of GR, and we must seek new gravitational theory, after Chris Isham.

It should be agonizingly clear by now that the combination of (i) the separation of 3-D space from time -- the misfortunate "Hamiltonian formulation" proposed by Dirac and ADM --  and (ii) the full reparametrization invariance of general relativity makes Einstein's GR entirely and exclusively a kinematical theory. As explained by Seth Major, "the equations of motion are a list of expressions which must vanish. At each "instant," the degrees of freedom are the coordinate independent part of the metric known as the "3-geometry." This observation effectively solves all but one constraint -- the Hamiltonian constraint. In the quantum theory each constraint is promoted to an operator. The physical states are those states on which the constraints vanish. The final piece of the quantization is finding solutions of the quantum Hamiltonian constraint equation, the "Wheeler-DeWitt equation."

Only this "final piece" cannot be reached in principle. We made a false start with the Dirac-Bergmann-ADM "Hamiltonian formulation", and can never solve the Hamiltonian constraint problem, because we cannot move from one "instant" (cf. S. Major above) to the nearest one. We can't move over the gaps "between" two successive instants, because we're frozen from the outset. In classical GR, we can, of course, use a static frozen 3-D snapshot, and try to bypass its generic problems (singularities, geodesic incompleteness, and Closed Timelike Curves). Some people, like A. Ashtekar, do acknowledge that we're "doing grave injustice to space-time covariance that underlies general relativity", and yet believe that canonical quantum gravity can be build on these grounds. No way. See the inner product problem, as of September 1991, in: Gary T. Horowitz, "Ashtekar’s Approach to Quantum Gravity", hep-th/9109002 v1, p. 7. We surely must seek new gravitational theory, after Chris Isham.

As to loop quantum gravity (LQG), the quantization program on purely kinematical level fails miserably [Ref. 3]. I'm expecting to hear from Karel Kuchar on this case-specific issue. It's important because LQG is currently in the focus of quantum gravity research, along with the string hypotheses.

Last but not least, let me stress again that the quest for quantum gravity is by no means some purely academic exercise: billions of dollars and euro are scheduled for chasing "gravitational waves" and "the God particle".

These are real money earned by real people with hard work. We don't have to wait for the complete theory of quantum gravity to see whether all these money will be wasted. Not at all. We can check out these two crucial prerequisites of quantum gravity right now. It's all about standard QM and GR. It's all at the tip of your fingers, too.

I plan to talk on these issues at EPS13, provided I will be given a chance to talk. Philosophically speaking, the 'atom of geometry' should be a pre-geometrical entity, and it should succeed where geometry fails. Namely, it should fulfill two seemingly incompatible requirements: provide a discrete structure of the space of states (global mode of spacetime) and a perfectly continual (local mode of spacetime) 'graceful exit' at all scales, from the scale of tables and chairs down to the Planck scale, and back. This is the 'back bone' of quantum gravity. The so-called 'semi-classical states' don't work. If we want to "recover" the nature of continuum, as known since Leibnitz' diff calculus and Archimedes, we must keep it, along with the discrete structure of spacetime, at all scales. It's a package.

Math is available upon request. Please act promptly, time is running out.
 

D. Chakalov
May 2, 2005
Last update: May 3, 2005
 

[Ref. 3] Hermann Nicolai, Kasper Peeters, Marija Zamaklar, Loop quantum gravity: an outside view. AEI-2004-129, hep-th/0501114 v2, 29 April 2005. 55 pages, 11 figures; v2: substantially revised, references added, resubmitted to Class. Quant. Grav.

p. 4: "As a consequence, it becomes non-trivial to see how semi-classical ‘coherent’ states can be constructed, and how a smooth classical spacetime might emerge. In simple toy examples, such as the harmonic oscillator, it has been shown that the LQG quantisation method indeed leads to quantum states whose properties are close to those of the usual Fock space coherent states [21]. In full (3+1)-dimensional LQG, the classical limit is, however, far from understood (so far only kinematical coherent states are known [22, 23, 24, 25, 26, 27]). In particular, we do not know how to describe or approximate classical spacetimes in this framework that ‘look’ like, say, Minkowski space, or how to properly derive the classical Einstein equations and their quantum corrections.
...

p. 6: "Conceptual issues

"Last but not least: although we will have nothing new to say here on the grand conceptual issues of quantum gravity and quantum cosmology (see e.g. [18, 19, 33, 16, 34], and references therein), we wish to remind readers that these problems, which have been around from the very beginning, will ultimately have to be addressed and resolved by all approaches to quantum gravity. This comment concerns not only difficult interpretational problems such as, for instance, the meaning and interpretation of the ‘wave function of the universe’, but also more technical issues. Among the latter we would like to mention the question of whether we have any right to expect the ‘wave function of the universe’ to be normalisable, or whether the familiar Hilbert space formalism of standard quantum mechanics is at all the correct framework for quantum gravity.
...

p. 10: "From this point of view, it appears to us that, beyond the technical subtleties, the kinematical constraints are not the real problem of quantum gravity. The core difficulties of canonical quantum gravity are all connected in one way or another to the Hamiltonian constraint -- irrespective of which canonical variables are used.

pp. 16-17: 
Footnote 6: "Strictly speaking, the Stone-von Neumann theorem also requires that the number of degrees of freedom is finite, which is also not true in canonical gravity.

"4 Quantisation: kinematics

"The failure of operators to be weakly continuous can, as we will see, be traced back to the very special choice of the scalar product (4.7) below, which LQG employs to define its kinematical Hilbert space H_kin. This Hilbert space does not admit a countable basis, hence is non-separable, because the set of all spin network graphs in  E  is uncountable, and non-coincident spin networks are orthogonal w.r.t. (4.7).

"Therefore, any operation (such as a diffeomorphism) which moves around graphs continuously corresponds to an uncountable sequence of mutually orthogonal states in H_kin. That is, no matter how ‘small’ the deformation of the graph in  E , the associated elements of H_kin always remain a finite distance apart, and consequently, the continuous motion in ‘real space’ gets mapped to a highly discontinuous one in H_kin. Although unusual, and perhaps counter-intuitive, as they are, these properties constitute a cornerstone for the hopes that LQG can overcome the seemingly unsurmountable problems of conventional geometrodynamics: if the representations used in LQG were equivalent to the ones of geometrodynamics, there would be no reason to expect LQG not to end up in the same quandary."



 

Note 3: For readers interested in exploring the jungle of LQG, I can recommend two papers. The first one is a general-audience paper by C. Rovelli, which shows the entry point into the LQG jungle [Ref. 4], and the second one, by C. Rovelli and T. Thiemann, elaborates on the generic ambiguities in LQG due to the Immirzi parameter  Y  [Ref. 5].

In the early days of LQG, there was a lot of excitement around the peculiar evidence that the so-called "overcounting" problem might be solved [Ref. 4], but after the Immirzi ambiguity surfaced, it was acknowledged that "there is something we do not yet understand in this respect". [Ibid.]

The jungle is huge, perhaps endless. On the one hand, "the Y parameter is intimately linked (note the poetry here - D.C.) with the affine structure of the configuration space and with the choice of the holonomies as basic operators in quantum gravity" [Ref. 5], and on the other -- it is set to imaginary unit in Ashtekar's jungle. [Ibid.]

If you like this jungle, consult Karel Kuchar. If you nevertheless wish to explore it, do it on your own risk, like C. Rovelli and T. Thiemann.

In my understanding, nothing in LQG can possibly "fix the value of the Immirzi parameter" [Ref. 5], because the generic ambiguities from  Y  originate from the absence of genuine dynamics of LQG. It's a purely kinematical theory which suffers from the paradox of Baron von  Münchausen, as I tried to explain on 26 November 1999.

Since Martin Bojowald was mentioned above [Ref. 1], let's see if he has made some progress [Ref. 6]. The most important problem -- the physical inner product problem -- is not solved. Why? Because "we have a relational scheme to understand what the wave function should mean but the probability measure to be used, called the physical inner product, is not known so far."

If you have a relational scheme to understand, try the ideas here and here. Otherwise you may never escape from Ashtekar's jungle and never comprehend "what the wave function should mean". You will only declare 'more research is needed', because the physical inner product "is not known so far", until you retire.
 

D. Chakalov
May 5, 2005
Last update: May 13, 2005

References

[Ref. 4] Carlo Rovelli, Loop quantum gravity, Physics World, November 2003,
http://academics.hamilton.edu/physics/smajor/Papers/rovelli03.pdf

pp. 2-3: "Loop quantum gravity is the mathematical description of the quantum gravitational field in terms of these loops. That is, the loops are quantum excitations of the Faraday lines of force of the gravitational field. In low-energy approximations of the theory, these loops appear as gravitons -- the fundamental particles that carry the gravitational force.

"This is much the same way that phonons appear in solid-state physics. In other words, gravitons are not in the fundamental theory -- as one might expect when trying to formulate a theory of quantum gravity -- but they describe collective behaviour at large scales.
...

"Two loops that are infinitesimally separated are two different loops, and this implies that there are far too many loop variables to describe the degrees of freedom of the field.

"The breakthrough came with the realization that this "overcounting" problem disappears in gravity. The reason why is not hard to understand. In gravity the loops themselves are not in space because there is no space. The loops are space because they are the quantum excitations of the gravitational field, which is the physical space. It therefore makes no sense to think of a loop being displaced by a small amount in space.

"There is only sense in the relative location of a loop with respect to other loops, and the location of a loop with respect to the surrounding space is only determined by the other loops it intersects. 

"A state of space is therefore described by a net of intersecting loops. There is no location of the net, but only location on the net itself; there are no loops on space, only loops on loops. Loops interact with particles in the same way as, say, a photon interacts with an electron, except that the two are not in space like photons and electrons are. This is similar to the interaction of a particle with Newton's background space, which "guides" it in a straight line.
...

p. 4: "The granular structure of space that is implied by spin networks also realizes an old dream in theoretical particle physics -- getting rid of the infinities that plague quantum field theory. These infinities come from integrating Feynman diagrams, which govern the probabilities that certain interactions occur in quantum field theory, over arbitrary small regions of space-time. But in loop gravity there are no arbitrary small regions of space-time. This remains true even if we add all the fields that describe the other forces and particles in nature to loop quantum gravity.
...

p. 5: "So, does this mean that all is well in loop quantum gravity? Not at all. Some aspects of the theory are still unclear. (...) Furthermore, the theory contains an odd parameter called the Immirzi parameter, Y , which is not fixed. The freedom in choosing this parameter was emphasized by Giorgio Immirzi at the University of Perugia in Italy, and at present it is fixed indirectly by requiring the theory to agree with the Bekenstein-Hawking black-hole entropy.

"This is nontrivial, since the same value of Y matches many different kinds of black holes, and there is some indication that the same value could be obtained in other ways as well. But such an indirect way of determining the Immirzi parameter is not satisfactory, and there is something we do not yet understand in this respect.

"Finally, I repeat that for the moment there has not been any direct experimental test of the theory. A theoretical construction must remain humble until its predictions have been directly and unambiguously tested. This is true for strings as well as for loops."
 

[Ref. 5] Carlo Rovelli and Thomas Thiemann, Immirzi parameter in quantum general relativity, Physical Review D, 57(2), 1009-1014 (1998); gr-qc/9705059 v1,
http://xxx.lanl.gov/abs/gr-qc/9705059

"We find that the quantum theory is in fact undetermined by one parameter. This is due to the fact that the holonomy algebra on which this approach is based depends on a free parameter. This fact gives rise to a one-parameter family of inequivalent quantum theories, which are all, up to additional physical inputs, physically viable.

"In a sense, there is a one-parameter family of "vacua" in quantum general relativity, parameterized by a free (real) parameter, which we call "Immirzi parameter", and denote as Y (iota). Equivalently, there is a symmetry in the classical theory which is realized as a canonical transformation but cannot be realized as a unitary transformation in the quantum theory. 

"The existence of this quantization ambiguity is due to the peculiar kind of representation on which nonperturbative quantum gravity is based [5,6]. This representation is characterized by the fact that the holonomy is a well-defined operator in the quantum theory. Conventional perturbative Maxwell and Yang-Mills theories are not defined using this kind of representation and the  Y  parameter does not appear in that context. But physical and mathematical arguments indicate that this representation is relevant at the diffeomorphism-invariant and background-independent level [6,7]. Thus, the Immirzi parameter appears in the general covariant context.

"In this letter, we describe in some detail how this ambiguity is originated and its consequences. In particular, we address a certain number of questions that have been recently posed concerning the  Y  parameter, and we try to rectify a number of proposed incorrect interpretations of the appearance of this free parameter.
...

"The canonical transformation U(Y) was studied by Barbero [4]. The case Y = [sq.r.] - 1 corresponds to Ashtekar's canonical transformation [12].

"Thus, the fact that U(Y) is a canonical transformation is a simple extension of the important discovery due to Ashtekar [9] which gave rise to the connection formulation of general relativity.
--
[9] A Ashtekar, Phys Rev Lett 57 (1986) 2244; Phys Rev D36 (1987) 1587.
...

"Physically, the constant in front of the general relativity action determines the strength of the macroscopic Newtonian interaction. The freedom in the choice of the Immirzi parameter in the quantum theory consists in the fact that the overall scale of the spectra is not determined by low energy physics.

"In other words, we can measure the Newton constant by means of classical gravitational experiments, and measure the Planck constant by means of non-gravitational quantum experiments. From these two quantities we obtain a length, the Planck length l_P = [sq.r.] hG.

"The point of the Immirzi ambiguity is that the ratio of, say, a given eigenvalue of the area to l_P is not determined by the quantization procedure.
...

"This shows how the Y parameter is intimately linked with the affine structure of the configuration space and with the choice of the holonomies as basic operators in quantum gravity.
...

"Sec V. Conclusions

"Similarly, there are two length scales in quantum gravity: the Planck constant l_P [XXX] and the quantum of area A_0 [XXX] . Unless some non yet understood requirement fixes the value of the Immirzi parameter, these two length scales are independent."
 

[Ref. 6]. Martin Bojowald, Elements of Loop Quantum Cosmology, gr-qc/0505057 v1, 12 May 2005,
http://xxx.lanl.gov/abs/gr-qc/0505057

pp. 13-14: "Here, we encounter the main issue in the role of the wave function: we have a relational scheme to understand what the wave function should mean but the probability measure to be used, called the physical inner product, is not known so far. (...) This is called the kinematical inner product which is used for setting up the quantum theory.

"But unlike in quantum mechanics where the kinematical inner product is also used as physical inner product for the probability interpretation of the wave function, in quantum gravity the physical inner product must be expected to be different. This occurs because the quantum evolution equation (11) in internal time is a constraint equation rather than an evolution equation in an external absolute time parameter. Solutions to this constraint in general are not normalizable in the kinematical inner product such that a new physical inner product on the solution space has to be found. There are detailed schemes for a derivation, but despite some progress [41, 42] they are difficult to apply even in isotropic cosmological models and research is still ongoing. An alternative route to extract physical statements will be discussed in Sec. 4.4 together with the main results.
...

p. 17: "4.4 Phenomenology

"The quantum difference equation (11) is rather complicated to study in particular in the presence of matter fields and, as discussed in Sec. 4.2.2, difficult to interpret in a fully quantum regime. It is thus helpful to work with effective equations, comparable conceptually to effective actions in field theories, which are easier to handle and more familiar to interpret but still show important quantum effects."