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
References [Ref. 1] Golam Mortuza
Hossain, Large volume quantum correction in loop quantum cosmology: Graviton
illusion? gr-qc/0504125 v1, 25 April 2005,
"However, the issues related with
physical observables, external time
[Ref. 2] Mario Everaldo
de Souza, Gravity cannot be quantized,
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 Some of the philosophers advocating
spin networks hold different opinion, however.
Lee
Smolin, for example, claims that if you take 10 Yet all these people are incredibly
optimistic, like
Abby Ashtekar. Why?
Well, maybe because "the underlying (internal time) dynamics is described
by a 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 D. Chakalov
Note 2: Let
me stress two crucial First, we have to check out whether
any of the known approaches to quantum gravity would enable us 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 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 It should be agonizingly clear by
now that Only this "final piece" cannot be
reached 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 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 Math is available upon
request. Please act promptly, time is running out.
D. Chakalov
[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 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.
"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 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:
"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 "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 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 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 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
References [Ref. 4] Carlo Rovelli,
Loop quantum gravity, 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 "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 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,
"This is nontrivial, since the same
value of "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, "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 "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 "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 "The canonical
transformation U( "Thus, the fact that U( "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 "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 "This shows
how the "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, 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 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." |