| Subject:
arXiv:0807.3376v1
[quant-ph], "Comments welcome" Date: Wed, 23 Jul 2008 05:33:59 +0300 From: Dimi Chakalov <dchakalov@gmail.com> To: Steven Weinstein <sw@uwaterloo.ca>, sweinstein@perimeterinstitute.ca Steve: I agree that there is no tendency for a preferred basis to emerge, because the so-called preferred basis is red herring. I think the correct formulation of the smooth, bi-directional, and reversible transition b/w the quantum and classical realms requires rethinking of KS Theorem, http://www.god-does-not-play-dice.net/Szabados.html#Hilbert You also wrote: "... the expansion of the universe, and the consequent enlargement of the environment, plays a crucial role in the emergence of quasiclassicality [12] [13]". Refs [12] and [13] are papers by IGUS Jim Hartle, whose poetry is encoded in the adjective "crucial". See the link above. BTW did you get my email from Tue, 31 Jan 2006 17:25:59 +0200, Subject: Is there a problem of 3-D space in canonical quantum gravity? Take care, Dimi ============ Subject: Re:
arXiv:0807.3376v1
[quant-ph], "Comments welcome" We surely need "two times", but if we consider the global mode of time (cf. the link above), then such "timeless time" will not look like a second temporal direction orthogonal to the time parameter in STR. Nope. It will look like an infinite-dimensional "time" attached to every instant from the local mode of time. I'm afraid it is highly unlikely that Steven Weinstein would comment on my 'Quantum Mechanics 101' at the link above, as he hasn't yet replied to my comments on his article in Stanford Encyclopedia of Philosophy below.
D. Chakalov ======================== Subject: Is
there a problem of 3-D space in canonical quantum gravity?
Dear Steve, I read with great interest your essay on quantum gravity in Stanford Encyclopedia of Philosophy [Ref. 1]. May I offer some philosophical, and completely frank, remarks on the canonical, non-perturbative approach to quantum gravity (QG). Please correct me if I got it wrong. I extend this request to your colleagues as well. We have encountered a 'problem of time' but not 'problem of 3-D space'. You wrote [Ref. 1,
Sec. 4.2, Canonical and Loop Quantum Gravity]
"Although advocates of the canonical approach often accuse string theorists of relying too heavily on classical background spacetime, the canonical approach does something which is arguably quite similar, in that one begins with a theory that conceives time-evolution in terms of evolving some data given on a spacelike surface, and then quantizing the theory. The problem is that if spacetime is quantized, this assumption does not make sense in anything but an approximate way. This issue in particular is decidedly neglected in both the physical and philosophical literature (but see Isham (1993)), and there is more that might be said." I think there is definitely more that can and should be said. What I like in 'the problem of time', exhibited in "that damn equation" (as B.S. DeWitt referred to the Wheeler-DeWitt equation), is that it exposes the initial ERROR in the canonical formulation of GR: the 3+1 decomposition of spacetime and subsequently the alleged "dynamics" of classical GR. Red herring! Regrettably, this error enables some philosophers to even talk about 'evolution of geometry in time', http://www.God-does-not-play-dice.net/Smolin.html This initial error in the canonical formulation of GR has obscured the puzzling fact that there is no 'problem of 3-D space' acknowledged in canonical QG. We start with a wrong premise -- data given on a smooth spacelike surface -- and end up with the problem of time and the Hamiltonian constraint problem. Neither the dynamics nor the "semiclassical limit" of QG observables [Ref. 2] have been recovered from that mess [Ref. 3], http://www.God-does-not-play-dice.net/Nicolai.html#C3 Notice also that the alleged "discreteness at Planck scale distances" [Ref. 2] has already been ruled out by observations, http://www.God-does-not-play-dice.net/Lieu.html#email Going back to the subject: What is 'the problem of 3-D space' -- if any -- in canonical quantum gravity? It amounts to the statement that, according to canonical quantum gravity, the 3-D space [Ref. 3] cannot exist at the scale of tables and chairs. NB: We cannot -- even in principle -- recover the smooth 4-D spacetime of classical GR, at any "semiclassical limit" whatsoever. I suspect that you and all your colleagues will disagree with "even in principle" and "whatsoever", so let me try to be more specific. There are two pitfalls which make the recovery of 3-D space impossible in principle: 1. If we start with classical GR, we cannot -- even in principle -- introduce any "discreteness" in what John Stachel called 'entities that are only individuated dynamically' [Ref. 4, footnote 21]. 2. Once we end up with the Hamiltonian constraint problem, we cannot -- even in principle -- recover the smooth manifold of classical GR. Thomas Thiemann says: "the fact that we started with analytic data and ended up with discrete (discontinuous) spectra of operators looks awkward" [Ref. 5]. Perhaps he is optimistic and hopes to make it 'less awkward', but there is nothing to 'hold onto' in a background-independent QG. Briefly, if you end up with (2),
you can't get back to the smooth asymptotically flat 3-D space of Einstein's
GR, and recover (1). Alternatively, if you try to endow (1) with some "discreteness"
from the outset, you can't do that by introducing *the same kind of discreteness*
Deadlock. Am I wrong? My proposal: We need to unravel the "back bone" of the bi-directional transition between the classical and quantum realms, http://www.God-does-not-play-dice.net/Smolin.html#Bub http://www.God-does-not-play-dice.net/Stachel.html#point This "back bone" is missing in both (1) and (2) above. It is disguised in GR as "empty space", http://www.God-does-not-play-dice.net/Wald.html#note I hesitate to elaborate on my proposal, because I could be all wrong. As Chris Isham put it, I "do not know enough theoretical physics to help with any research in that area", http://www.God-does-not-play-dice.net/Kiefer.html#Isham I do hope you and your colleagues will help me find my errors. I will immediately report them at http://www.God-does-not-play-dice.net/Weinstein.html Thank you very much in advance. Kindest regards, Dimi
References [Ref. 1] Steven Weinstein
(2005), Quantum Gravity, Stanford Encyclopedia of Philosophy,
[Ref. 2] H. Sahlmann and T. Thiemann, Towards the QFT on curved spacetime limit of QGR: I. A general scheme. Class. Quantum Grav. 23 (2006) 867-908; gr-qc/0207030 v1. "Although there are no further inputs other than the fundamental principles of four-dimensional, Lorentzian General Relativity and quantum theory, the theory predicts that there is a built in fundamental discreteness at Planck scale distances and therefore an UV cut-off precisely due to its diffeomorphism invariance (background independence). "A topic that has recently attracted
much attention is to explore the regime of QGR where the quantized gravitational
field behaves "almost classical", i.e. approximately like a given classical
solution to the field equations. Only if such a regime exists, one can
really claim that QGR is a viable candidate theory for quantum gravity.
Consequently, efforts have been made to identify so called *semiclassical
states* in the Hilbert space of QGR, states that reproduce a given classical
geometry in terms of their expectation values and in which the quantum
mechanical fluctuations are small [7, 8, 9, 10, 11]."
[Ref. 3] Lee Smolin:
"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 like." (Three Roads to Quantum
Gravity, Phoenix, London, 2000, p. 205).
[Ref. 4] John Stachel, Structure, Individuality and Quantum Gravity, gr-qc/0507078 v2. pp. 9-10: "For theories, such as
general relativity, that are based on fundamental entities that are continuously,
and even differentiably related to each other, so that they form a differentiable
manifold, permutations become diffeomorphisms. For a diffeomorphism of
a manifold is nothing but a continuous and differentiable permutation of
the points of that manifold (footnote 21). So, maximal permutability becomes
invariance under the full diffeomorphism group. Further extensions to an
infinite number of discrete entities or mixed cases of discrete-continuous
entities, if needed, are obviously possible.
"In both the case of non-relativistic
quantum mechanics and of general relativity, it is only through dynamical
considerations that individuation is effected. In the first case, it is
through specification of a possible quantum-mechanical process that the
otherwise indistinguishable particles are individuated (“The electron that
was emitted by this source at 11:00 a.m. and produced a click of that Geiger
counter at 11:01 a.m.”). In the second case, it is through specification
of a particular solution to the gravitational field equations that the
points of the space-time manifold are individuated (“The point at which
the four non-vanishing invariants of the Riemann tensor had the following
values: ...”). So one would expect the
[Ref. 5] T. Thiemann, Lectures on Loop Quantum Gravity, gr-qc/0210094 v1, p. 40. "In fact, as we shall see later, QGR predicts a discrete Planck scale structure and therefore the fact that we started with analytic data and ended up with discrete (discontinuous) spectra of operators looks awkward." ====== Note: One could imagine a superficial reply to my statement above, in the following format: 'the classical limit is not yet understood, since so far only kinematical coherent states are known, but there could be light in the tunnel, so more research is needed'. Fine, but let's see what we already know. For example, see the task in the framework of Loop Quantum Gravity here. It's hopeless. Recall that you have to recover a smooth 4-D spacetime comprised of infinitely many, infinitely small "points", as we know since the time of Archimedes. These so-called "points" are in fact very peculiar dynamical objects: there is no possibility to "freeze" them and catch their "final" physical content (this should be clear to all people familiar with the Thompson lamp paradox). On the other hand, when we probe the Planck scale, we're inevitably trying to "freeze" the world at this "final" scale. It's like trying to follow an infinitesimal "point" way down to the instant at which it will obtain a fixed, and final, numerical value, as we know from diff calculus. Of course, this is impossible. Hence it isn't surprising that we hit a non-separable, infinite-dimensional Hilbert space, and an uncountable infinity of solutions to the quantum constraint equation, as acknowledged here. Can we recover the world of tables and chairs [Ref. 3] from this quantum mesh? It's hopeless, again. As to the problem of time, I believe we have to accept the ontological principle of relational reality (link here), because the theory is background-free. Hence the problem of time can be explained with the Buridan donkey paradox here. As to my "back bone" Anzats mentioned above, note that the task has been identified by Erwin Schrödinger in 1931. See the 'flipping a quantum coin' quiz here.
NO WAY. To quote Seth Major, "as a consequence of general covariance, relativity becomes entirely constrained; 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."' Thus, what your inanimate wristwatch "reads" is a total miracle: the transition of the dead-frozen 3-geometry from one "instant" to the nearest "instant" of dead-frozen 3-geometry. And you haven't noticed this total miracle, because you've introduced a fixed background for these miraculous "timelike displacement" jumps: the 3-D space. How come you acknowledge the absence of some "hidden unmoved mover", after Karel Kuchar, but not some 'problem of 3-D space in classical GR'? Are you comfortable with miracles? You poor watch would be reading the dynamics of GR iff the group of transformations were passive, i.e., inducing coordinate re-descriptions only (like changing the coordinates of the same building displayed in different street maps). With the active Diff(M)-invariance, however, there is no background of 'the same building', as we know from Einstein's Hole Argument. If you wish to keep Einstein's GR and seek an adequate notion of time, which would account for the quasi-local nature of gravitational interactions, see the 'shoal of fish' metaphor, p. 7 in gw.pdf. If presents the main task of reconstructing "a classical spacetime from nonlocal diffeomorphism-invariant observables" (Steve Carlip, gr-qc/0108040). The task is on the table since 1972, if not earlier. Read again Seth Major and Steve Carlip. Last but not least, we talk about Closed Time Curves (e.g., W.B. Bonnor, gr-qc/0211051), but preserve the 3-D space intact, as some fixed Newtonian background that can remain neutral to flipping the course of time. We lose the identity of a 'world point', we lose the notion of absolute time, but treat the 3-D space in GR as some 'absolute space'. Hence there is no 'problem of 3-D space' in canonical quantum gravity, and no progress either. To sum up, let me compare 'quantum
reality' with 'gravitational reality' in Einstein's
GR: what do they have in common? In both cases, the notion of
There is no water "in between" two adjacent molecules of water. Likewise, there is nothing in GR that can "move" canonical data from one frozen 3-D hypersurface to the "nearest" frozen 3-D hypersurface: please see the ladder metaphor here, and recall Karel Kuchar's Perennials and 'unmoved mover'. He has explained the distinction between the observables and Perennials as follows (emphasis added): "[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 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). See also Karel Kuchar: "In general relativity, dynamics is entirely generated by constraints. The dynamical data do not explicitly include a time variable" (pp. 88-89). Yes, we surely observe dynamical evolution, but according to GR textbooks it should be a miracle, because the textbooks do not even mention the 'unmoved mover' and Karel Kuchar's Perennials. Recall the basic postulate in Einstein's GR that "spacetime does not claim existence on its own but only as a structural quality of the gravitational field", hence any prior-geometry plenum or 'unmoved mover' is excluded from GR textbooks. If this were correct, there should be some 'structural quality of the gravitational field' -- you name it -- that would literally move the canonical data from one hypersurface to the "nearest" hypersurface, "over" the gaps of total absence of any background whatsoever between these adjacent hypersurfaces (notice the "logic" implied in this excercise: it is like trying to explain the nature of heat with some tiny little and very "hot" particles, say, heatino and heatinino). If we follow this line of reasoning, this 'structural quality of the gravitational field' would be the perfect clock of the whole universe, since it would be the physical source of the elementary timelike displacement. But it is impossible to have a physical clock that would work perfectly, regardless of what its state is, as explained by John Baez. Thus, we need the 'unmoved mover' and Karel Kuchar's Perennials, although they are not, regrettably, discussed in GR textbooks (nor is the substrate of the "dark" energy that springs from the "empty space" of the "gaps" between adjacent hypersurfaces). Needless to say, the 'unmoved mover' or Aristotelian First Cause is totally invisible (hence "dark") and cannot be directly observed, otherwise it will be just another 'structural quality of the gravitational field'. It is simply the last, and truly fundamental, layer of the physical reality. Only it cannot be physical. It is pure geometry, and is hidden in the gaps of 'non existence' "between" points. To be specific, please go back to the ladder metaphor here, and notice that the first hypersurface and the "nearest" second hypersurface are completely fused into one single continuous "horizontal" step of the ladder (local mode of spacetime): we observe a ball [X] moving along a 'continuous step of the ladder', from the left to the right (I've provided three such hypersurfaces, which are "vertical steps" in the global mode of spacetime). In Einstein's GR, there is no room for the "gaps" of Perennials, in which Mother Nature makes gauge-invariant, observable states relationally, -- one-at-a-time, -- to produce the elementary timelike-displacement "jumps" of the ball. Just as the quantum reality is not "fixed" prior to observations, and depends of the 'context' (Kochen-Specker theorem), the gravitational reality is not fixed before the contextual, and fully relational, emergence of the fleeting content of the spacetime "points", due to the active diffeomorphisms. It shouldn't be surprising that Einstein's GR is marred with intractable pathologies, since its "dynamics" is indeed red herring. It shouldn't be surprising that "the last remnant of physical objectivity" (A. Einstein) was eliminated by the requirement of general covariance either. To reveal the hidden dynamics of GR, read Karel Kuchar, particularly "Time and interpretations of quantum gravity" (reference here). It is sad that some people have completely ignored Kuchar's Perennials (C. Rovelli even flouted by talking about 'evolving constants' and 'partial observables'). The problems of the dynamics of GR were recognized by Einstein and Hilbert: see the 'reference fluid' on p. 32. More from St. Augustine and Kurt Lewin here. But of course I could be all wrong -- see Chris Isham above. Should anyone find errors, I will immediately post them here. One final question has to be cleared: why am I doing this? What makes me think that I will get a professional reply from the established theoretical physics community? Not because they care about the efforts of some guy from Botswana, Borneo, Bulgaria, or whatever. No. The reason is quite different. I believe will hear from Steve Weinstein and his colleagues, because they do respect their field of research. This is not a hobby. If they were collecting bottle labels, and I was suggesting to switch to paper napkins -- well, I may never hear from them. Here the case is entirely different. Or is it? D.
Chakalov
============
Message-Id:
<8c6d2244fd5e6b83d8274f2f9f3e193d@brandeis.edu> Note: Given the 11-hour difference in the time zones (GMT -8 and GMT +3), Professor Stanley Deser was amazingly fast to reply after just 44 minutes, by typing his reply in the subject line: "I'm afraid mine are far more prosaic considerations than those. yes, I do understand GR, but cannot discuss that now.sd". I was never able to understand the logic in his seminal paper [Ref. 2]. I fully agree with the first paragraph (after E. Kretschmann and H. Weyl), but look at the second paragraph: "Thus (?!? - D.C.) it is necessary that the metric field be separated into the parts carrying the true dynamical information and those parts characterizing the coordinate system." Why would Stanley Deser treat the metric field as some separable entity? What are the legitimate reasons to believe that one part from the metric field would carry "the true dynamical information", while the other part can and should be neglected, because it "does not involve any observable changes in the physics, since it merely corresponds to a relabeling under which the theory is invariant" [Ref. 2]? It seems to me that Stanley Deser and his prominent colleagues have made a logical error -- non sequitur -- in their conclusion that "the parts carrying the true dynamical information and those parts characterizing the coordinate system" can be separated.
The way I see it, the metric
field is a dynamical entity, which is the end product from a
bi-directional "talk" in the global time.
Thus, the true dynamical degrees of freedom, encoded in the metric field, are being re-created in each and every "point" from the geodesic. How does Mother Nature create this dynamical geodesic? Relationally, of course, since there is no "background" in the local time, as we know since Einstein's Hole Argument. I wonder if Stanley Deser would agree. He probably wouldn't, but then will have to elaborate on his ADM paper published in 1959. He replied today with "yes, I do understand GR". Well, I was never able to understand the alleged "dynamics" of GR [Ref. 2], since I see no real dynamics in Dirac-ADM layout [Ref. 3]; just a kinematical snapshot at an instant of time, and a number of highly suspicious "constraints" that have no counterparts in classical physics [Ref. 4]. We can derive dynamical laws from the laws ruling the data at a single instant in classical mechanics only, since we operate there with an explicit time variable. And why is it explicit? Because the bi-directional "talk" of matter and geometry yields no observable effects, hence we can use the fixed spacetime grid from STR, and can indeed derive dynamical laws from the laws ruling the data at a single instant. We don't have this luxury in GR: the alleged dynamics is entirely generated by constraints [Ref. 4], which totally obscure the two kinds of time in Einstein's GR, as well as the curious facts that there is no hint to some 'problem of time in classical GR' (see the Buridan donkey paradox here) nor any 'problem of 3-D space' in canonical quantum gravity -- they both were 'swept under the rug' in Dirac-ADM hypothesis from the outset [Ref. 3]. I wonder if Stanley Deser would agree. That is going to be very interesting. Three years ago, he reiterated "the silent points" of Dirac's metaphysics [Ref. 3], but failed to mention any current spacetime quantization program that could eventually be deemed promising. No quantum gravity program works with the Hamiltonian formulation of GR, loop quantum gravity notwithstanding. This persistent negative result shouldn't be surprising, given the fact that "the silent points" of Dirac's metaphysics -- a 3+1 decomposition of the gravitational field variables, rather than maintaining manifest 4-covariance [Ref. 3] -- is marred with grave injustice (A. Ashtekar) to the founding fathers of the theory of relativity, and total ignorance of the basic principles of fundamental dynamical evolution, as known since the time of Aristotle. To cut the long story short, if you take just one snapshot from the bi-directional "talk", you may speculate that its 'parts' can be separated [Ref. 2], but all you can do is to make Einstein spin in his grave like a helicopter. As I mentioned in my email above, I'm afraid Stanley Deser will deeply hate my viewpoint on Dirac-ADM hypothesis. I hope he will nevertheless reply professionally. Back in 1984, he had assured Paul Dirac that there exists a finite quantum field theory (N=4 supersymmetric Yang-Mills), but Dirac didn't believe him [Ref. 3]. Dirac was right; see L. Susskind here. Stanley Deser now claims "yes, I do understand GR". Do you, really?
[Ref. 3] S. Deser, Some Remarks on Dirac's Contributions to General Relativity, gr-qc/0301097 v1. "As it happens, in our (ADM)
contemporaneous and independent development of Einstein theory [12],
understanding that the Einstein-Hilbert (or any other diffeo-invariant) action
was necessarily an "already
parametrized" system a la (3b) was an essential, beautiful, confirmation
of our canonical formulation...
Before going into the iconoclastic
ideas above, it is worth trying to look "passively"
at the active diffeomorphisms, as suggested by Peter
Bergmann and Arthur Komar [The Coordinate Group Symmetries of General Relativity,
I wish Luca
Lusanna, Massimo Pauri, and Michele Vallisneri
best of luck with their approach to classical GR. If do hope they can choose
the lapse and shift dynamically, as functions of the evolving
geometry, and keep adjusting the lapse and shift dynamically,
to avoid the perplexing pathologies of classical GR, such as 'closed time
curves' (CTCs), the Cauchy
Problem, and "singularity",
as shielded by some mythical "event
horizon".
D.
Chakalov
--
p. 22: "Instead, we are now able to claim that any coordinatization of the manifold can be seen as embodying the physical individuation of points, because it can be implemented [again, at least locally] as the Komar-Bergmann intrinsic coordinates after we choose the correct Z[A] and we select the correct gauge. The byproduct of the gauge fixing is the identification of the form of the physical degrees of freedom as nonlocal functionals of the metric and curvature. "Summarizing, each of the point-events of space-time is endowed with its own physical individuation (the right metrical fingerprint!) as the value, as it were, of the four canonical coordinates (just four!), or Dirac observables which describe the dynamical degrees of freedom of the gravitational field. "However, these degrees of freedom are unresolveably entangled with the structure of the metric manifold in a way that is strongly gauge dependent. "As a final consideration, let us
point out that Eq. (24 - the Bergmann-Komar intrinsic coordinates - D.C.)
is a numerical identity that has an inbuilt noncommutative structure,
deriving from the Dirac-Poisson structure on its right-hand side. The meaning
of this structure is not clear at the classical level, but we believe that
it could be relevant to the quantization of general relativity.
p. 27: "In addition, we have established
that within the Hamiltonian framework we can use a gauge-fixing procedure
based on the Bergmann-Komar intrinsic coordinates to turn the primary mathematical
individuation of manifold points into a physical individuation of point-events
that is directly associated with the value of the gravitational degrees
of freedom (Dirac observables). The price to pay is the breaking
of general covariance.
"Indeed, would it be possible to
build a fundamental theory that is grounded in the reduced phase space
parametrized by the Dirac observables? This would be an abstract and highly
nonlocal theory of gravitation that would admit an infinity
of gauge-related, spatio-temporally local realizations."
Subject: The 'proof
of the pudding' for loop quantum gravity
Dear Drs. Singh and Pawlowski, I am trying to understand your latest paper [Ref. 1] and LQG in general. May I offer you the 'proof of the pudding'. To endow your "space" with the structure of a Hilbert space, you used the so-called group averaging method [Ref. 1, p. 3), and wrote: "Since we have the explicit Hilbert space and a complete set of Dirac observables, we can now construct states which are semi-classical at late times -- e.g., now -- and evolve them numerically ‘backward in time’." The states obtained in [Ref. 1] are not classical but *semi-classical*, since are just "sharply peaked" around the alleged classical trajectory. To construct the 'proof of the pudding', let's make the following Gedankenexperiments: 1. Pick up an object around you, say, your PC placed on your desk, and evolve it numerically ‘backward in time’ at some instant t_n , then evolve it numerically ‘forward in time’ at the instant 'now', denoted with t_3 . 2. Repeat the step (1) above: evolve your PC numerically ‘backward in time’ at the same instant t_n , then evolve it numerically ‘forward in time’ at the next instant 'now', denoted with t_2 . 3. Repeat the step (2) above: evolve your PC numerically ‘backward in time’ at the same instant t_n , then evolve it numerically ‘forward in time’ at the next instant 'now', denoted with t_1 . You will obtain the history of your PC, which can be viewed ‘backward in time’ (the deflation time) as states t_1, t_2, and t_3 . As you put it [Ref. 1, p. 4]: "We only asked that the quantum state be semi-classical at late times. This is an observational fact rather than a new theoretical input or a philosophical preference." Here comes the questions of the 'proof of the pudding': Q1: Where would you place t_n in the history of the universe, to obtain the *sameness* of your semi-classical PC in the three consecutive instants t_1, t_2, and t_3 ? Q2: Given a particular value of t_n , what will be the compilation of errors of the semi-classical state of your PC during the steps (1)-(3) above? If these questions are too hard, may I ask a simple one: Q3: Do you know the dynamics of GR? If you say 'yes', please see http://www.god-does-not-play-dice.net/Weinstein.html#shoal Your comments will be greatly appreciated. I will be happy to learn the opinion of your colleagues as well. Looking forward to hearing from you, Regards, Dimi Chakalov
"... so far the physical Hilbert
space, Dirac observables and semi-classical states have not been specified.
Consequently, physical ramifications of the singularity resolution could
not be worked out. The present investigation overcomes these limitations.
We find that the emergent quantum space-time is significantly different
from that in the early papers.
"Since we have the explicit Hilbert
space and a complete set of Dirac observables, we can now construct states
which are semi-classical at late times -- e.g., now -- and evolve them
numerically ‘backward in time’.
"However, if it is evolved via (2),
the situation becomes qualitatively different. The state remains sharply
peaked at the classical trajectory...
p. 4: "We only asked that the quantum state be semi-classical at late times. This is an observational fact rather than a new theoretical input or a philosophical preference." ======== Note: Last year, in his Public Lecture "Space and Time: From Antiquity to Einstein and Beyond", Abby Ashtekar stated (slide 42): "Quantum theory of Geometry developed primarily at the Center for Gravitational Physics and Geometry at PSU. Now used by research groups world-wide." Coincidently or not, this Public Lecture was reserved for April 1st. Once I hear from the proponents of LQG, I will remind them of the question posed by Claus Kiefer regarding the Hilbert space problem (gr-qc/9906100): "What is the appropriate inner product that encodes the probability interpretation and that is conserved in time?" For if you use some blurred "semi-classical" states that can only be "sharply peaked" around the alleged classical trajectory, you don't have some fixed and unique classical path that can be used to solve the Hilbert-space problem. You have to play with a set of (uncountable?) possible classical paths, and it is not at all clear how you would discover the "appropriate inner product that encodes the probability interpretation and that is conserved" in such utterly blurred multi-fingered "time", if any. The explanation of the Hilbert space problem was given by Karel Kuchar fifteen years ago, in May 1991, in "Time and interpretations of quantum gravity", pp. 7-8; see also the internal many-fingered time (hypertime variables) XA(x) in the functional Schrödinger equation (2.7), as well as p. 22, pp. 31-32 (the spacetime problem and the 'reference fluid', "a tenuous material system whose back reaction on the geometry can be neglected"), p. 40 (the 'Heraclitian time' [tau] of Unruh & Wald), p. 83 (the "frozen time"), and ref. [129] therein (link here). Fifteen years ago, there was no such thing as 'loop quantum gravity' (LQG). As of today, the proponents of LQG can only ignore the fundamental research by Karel Kuchar. The problems of GR dynamics were recognized even by the founding fathers of GR: see the 'reference fluid' story on p. 32 above. Anyway. As to the Q3 above,
follow the link from the text "dynamics of GR",
which will bring you to Steve Carlip's web site,
"One way to understand the simplification (lower dimensional models - D.C.) is the following. In n dimensions, the phase space of general relativity -- the space of generalized positions and momenta, or equivalently the space of initial data -- is characterized by a spatial metric on a constant-time hypersurface, which has n(n-1)/2 independent components, and its time derivative (or conjugate momentum), which adds another n(n-1)/2 degrees of freedom per spacetime point. It is a standard result of general relativity, however, that n of the Einstein field equations are constraints on initial conditions rather than dynamical equations. These constraints eliminate n degrees of freedom per point. Another n degrees of freedom per point can be removed by using the freedom to choose n coordinates. We are thus left with n(n-1)-2n = n(n-3) physical degrees of freedom per spacetime point. "In four spacetime dimensions, this gives the four phase space degrees of freedom of ordinary general relativity, two gravitational wave polarizations and their time derivatives. If n=3, on the other hand, there are no field degrees of freedom: up to a finite number of possible global degrees of freedom, the geometry is completely determined by the constraints." Can you unravel these "two gravitational wave polarizations and their time derivatives" in the "ordinary" GR? A. Ashtekar did acknowledge in gr-qc/0410054 that, "unfortunately, no such polarization has been found on the phase space of general relativity". I wonder why. Here's a hint from A. Ashtekar: "In mathematical physics, one generally begins by constructing simplified models in which the actual physical complications of the system under consideration are stripped down to a bare minimum -- the oft-quoted examples is that of a 'spherical cow'." So, what are the 'physical degrees of freedom per spacetime point'? No 'spherical cows' [Ref. 2], please. To sum up, let me explain what I like in loop quantum gravity: it seems to me that it does contain 'an essential germ of truth' -- although hugely disguised -- about the cosmological "singularity". M. Bojowald elaborated as follows: "This can be visualized with an ideally spherical balloon which looses air. It remains an empty balloon such that all its parts clash together -- as in a singularity. Now one has to imagine that instead of clashing the parts can freely move through each other and simply move forward. The balloon then again expands with the former inside pointing outward, and vice versa". I really like that story, it reminds me of the old joke about how a mathematician can catch a lion in Sahara. First, he will make sure that there is at least one lion available, then he will build a cage, go inside it, and will invert the space w.r.t. the cage surface, in such a way that all points from inside will be teleported outside, and the other way around. Then the mathematician will wind up outside the cage, and the poor lion will be trapped inside! Now, forget about the lion, and just try to imagine two simultaneous and CPT-invariant "copies" of the mathematician: one outside the cage, and another inside, with their opposite time directions and inverted (left-right) arms. If we allow these simultaneous presentations of the mathematician to be performed in the putative global mode of spacetime, perhaps the transitions will be perfectly smooth. See another exercise here. Perhaps we may solve the puzzle
of 3-D space by November 2015. As of today,
we're stuck with Einstein's confession of 1920: "In the first place, we
entirely shun the vague word "space," of which, we must honestly acknowledge,
we cannot form the slightest conception."
D.
Chakalov [Ref. 2] Abhay Ashtekar,
Gravity and the Quantum, gr-qc/0410054 v2,
"A common criticism of the canonical quantization program pioneered by Dirac and Bergmann is that in the very first step it requires a splitting of space-time into space and time, thereby doing grave injustice to space-time covariance that underlies general relativity. This is a valid concern and it is certainly true that the insistence on using the standard Hamiltonian methods makes the analysis of certain conceptual issues quite awkward. Loop quantum gravity program accepts this price because of two reasons."
I read the basic ideas at this web page, and
they didn't come out brief and clear. Sorry. I will try now to be very
brief and clear, and will focus on the reasons why both the
"dark" energy and the reference fluid have been excluded from GR from
the outset. 1. I stated above that the dynamics of GR, as presented in many GR textbooks, is 'red herring'. The proper way to put it would be 'essentially incomplete' (recall Karel Kuchar: "In general relativity, dynamics is entirely generated by constraints. The dynamical data do not explicitly include a time variable", pp. 88-89). My argument stems from one of the basic postulates in Einstein's GR, which is that "spacetime does not claim existence on its own but only as a structural quality of the gravitational field". Hence the long sought 'reference fluid' (see #2 below) is being inevitably 'swept under the carpet', because it cannot fit this postulate. The true dynamics of GR should be determined by both 'physical stuff' and the Aristotelian First Cause. The latter is nothing but the 'reference fluid' and Kuchar's 'unmoved mover'; see p. 32 from his paper quoted above, and his 1999 article The Problem of Time In Quantum Geometrodynamics (emphasis added): "The profound message of general relativity is that spacetime does not have any fixed structure which is not dynamical but governs dynamics from outside as an unmoved mover." I have suggested the so-called potential point(s) as an 'unmoved mover' that governs the dynamics 'from outside'. Thus, the dynamics of GR should be determined by both physical stuff and the Aristotelian First Cause. To explain this proposition, let's see what will happen if we drop the Aristotelian First Cause, and switch to the Marxist-Leninist interpretation of GR. Consider, for example, the line of reasoning of Bob Geroch: Suppose we fix a smooth, four-dimensional manifold M , which we take to be connected, paracompact, and Hausdorff. The points of M will be interpreted as the events of spacetime, and, thus, M itself will be interpreted as the spacetime manifold. We do not, as yet, have a metric, or any other geometrical structure, on M . Fine. But can we explain the 'principle of locality' -- the nature of the phenomenon that carries physical interactions from one "point" to the nearest "point" with a speed that cannot exceed the speed of light in vacuum? Nope. So, what do we do next? We introduce a Lorentzian metric and add Levi-Civita connection, for proper parallel displacement of vectors. But again: can we explain the 'principle of locality' by introducing Lorentzian signature and Levi-Civita connection over that cute spacetime manifold M ? Of course not. We just describe the effect from the phenomenon implied in the principle of locality, but not the phenomenon itself. Notice that the nature of spacetime continuum -- from one point to the "nearest" point -- is intermingled with the so-called speed of light. It's a bundle, and is all produced by the underlying 'reference fluid'. How do we approach this whole puzzle? If we stick to the Marxist-Leninist interpretation of GR, we must derive the complete underlying structure of spacetime only and exclusively only from 'the structural quality of the gravitational field'. We know from Leibnitz' differential calculus that we can assign a tangent vector to a dimensionless "point" from a trajectory, say. We also know from classical mechanics that we can unambiguously define 'instantaneous velocity' that pertains to a given "point" from a trajectory. Hence we believe that the physical content of any classical dynamical variable can indeed be carried down to the "point" at which it can, and should, be treated as 'geometry', like 'the grin of the cat without the cat', as observed by Alice. Fine. But notice the expression 'only and exclusively only' above. That's the crux of the issue raised by Karel Kuchar. In my not-so-humble opinion, we cannot -- even in principle -- derive the underlying structure of the spacetime manifold M exclusively from the fleeting physical content of the "points" produced by active diffeomorphisms. They can only provide the necessary condition for deriving the underlying structure of spacetime. The sufficient condition is the 'unmoved mover' or the Aristotelian First Cause. It shows up as 'context' or 'potential point(s)', and is completely and utterly "dark" reference fluid. I've repeatedly, and very politely, asked Bob Geroch and Bob Wald to comment on my proposal, but they never replied. I also asked Karel Kuchar, and he did reply by stating that my reading of his papers were "superficial", but didn't explain why. The opinion of Chris Isham can be read below. You'll be the judge. 1.1. One of the reasons why we haven't noticed the absence of the reference fluid in the modern GR is that, in most cases, GR works very well as a calculation tool. If you're dealing with an object of the size of the Solar system, you won't encounter some obvious "dark" effects, and all your calculations will be FAPP correct (unless of course you try to solve the puzzle of Pioneer anomaly). This situations reminds me of the Schrödinger equation, which too works very well in QM (the 'shut up and calculate' interpretation of QM), but we know that it does not take into account the quantum vacuum. Thus, we have to acknowledge that we don't have the equivalent of QFT in present-day GR, which makes the latter applicable to a very limited domain. If this sounds trivial, recall that 395 distinguished scholars from the LIGO Scientific Collaboration (LSC) are playing with the "Schrödinger equation" (the so-called linearized gravity) to catch the gravitational waves "online". The task of GW astronomy is not feasible, because we don't know the dynamics of GR, nor the "vacuum" effects from the so-called dark energy. More in gw.pdf; the basic arguments are spelled out here. 1.2. Another reasons why we haven't noticed the absence of the reference fluid in today's GR is that we have supplied some "obvious" background which has already incorporated the "gaps" of the reference fluid: the 3-D space. To cut the long story short, there is no 'problem of 3-D space' in canonical quantum gravity, and no progress either. Also, since there is no fixed background in classical GR and all the physical stuff is being individuated dynamically and relationally, we should have acknowledged a 'problem of time' in it, as a precursor to the problem of time from the Wheeler-DeWitt equation. See the Buridan donkey paradox here, and the proposed 'ontological principle of relational reality' here. 1.3. If somebody claims that has managed to develop some version of quantum gravity that can recover Einstein's GR at some semi-classical or low-energy limit, don't buy it. We simply don't know the true dynamics of GR. Currently, we are tacitly invoking the reference fluid in GR, in the sense that the intrinsic time interval associated to any timelike displacement, as produced by the reference fluid, is a total "miracle". Any time we use "smooth, nonsingular" transport (cf. Bob Wald here), we invoke the same "miracle" as well, because the reference fluid which binds the "points" into a "smooth manifold equipped with Lorentzian signature" is totally ignored in today's GR textbooks, by virtue of the basic postulate mentioned at #1 above. 1.4. Any time we use the phrase 'dynamical dark energy', we talk like parapsychologists, because only parapsychologists are not obliged to explain 'energy of what ?'. I suggest that the 'dynamical dark energy' pertains exclusively to the reference fluid in GR. Let me try to explain.
On the other hand, recall that the dynamical "dark" energy is a perfect fluid, which "provides an all-pervading energy density and negative pressure that are the same to all observers, at all places, and at all times in the history of any universe model, even the expanding ones" (B. Schutz, GRAVITY from the Ground Up: An Introductory Guide to Gravity and General Relativity, Cambridge University Press, Cambridge, 2003, p. 257). This "fluid has zero inertial mass! It can be accelerated with no cost, no effort" (ibid., p. 255). NB: Hence we arrive at the 'dynamical "dark" energy of the reference fluid'. It requires two ontologically different kinds of time (explained here and here), which produce the Heraclitian time of the whole universe, driven by the holistic (hence "dark") energy of the reference fluid. The way I see it, the first off task would be to understand and solve the puzzle of the gravitational contribution from the quantum vacuum, as implied from the original idea of Einstein's cosmological "constant". It is pointless to introduce new exotic scalar fields 'by hand' and try to resolve their "dynamics": see T. Padmanabhan, Advanced Topics in Cosmology: A Pedagogical Introduction, astro-ph/0602117 v1, p. 32. Again, I hesitate to
elaborate on my proposal above, because I could be all wrong.
As Chris Isham put it, I "do not know enough
theoretical physics to help with any research in that area." I will always repeat this bold, and unsupported so far by any evidence, statement by Prof. C. Isham, until he or anyone else finds at least one error in my ideas and proposals. Please do not hesitate to prove me wrong. You can do that by (i) showing some error in my ideas and proposals, or (ii) suggesting some alternative ideas, or (iii) both. I am also fully open to new ideas, like a 'potential point'. To sum up, 'the grin of the cat without the cat', as observed by Alice, is just the necessary condition for deriving the underlying structure of spacetime, while the sufficient condition is the reference fluid that binds the "points" of spacetime manifold and carries physical influences from one point to the "nearest" one, with speed that cannot exceed the "speed" of of light. This sufficient condition is missing in GR textbooks, regrettably. More from Graham Nerlich. I will regret if I've produced the impression of offering some purely academic exercise in the metaphysics of quantum gravity. There are some very important implications from 'the dynamical "dark" energy of the reference fluid', which I believe can be verified experimentally, but -- see again Chris Isham above. Panta rei conditio sine qua non est.
Ignore it at your peril. D. Chakalov
Happy Birthday, Albert! The fun part is just
around the corner!
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I
also wrote above that the so-called dynamics of classical GR is '
