|Subject: Quantum fluctuations, if any
Date: Tue, 14 Dec 2004 05:42:40 +0200
From: Dimi Chakalov <email@example.com>
To: Ntina <firstname.lastname@example.org>
CC: email@example.com, Roland.Omnes@th.u-psud.fr
May I ask a question in private. I just downloaded your latest "General Relativity Histories Theory", gr-qc/0412059 v1, in which you wrote:
"In quantum theory however, such notions as causality
I believe it is safe to say that nobody knows exactly what is the nature of these "quantum fluctuations", and particularly how a spacelike surface would "change" (?) due to its dependence on the "fluctuating" spacetime metric g .
I will be very grateful to you if you could share with
me your ideas on these "quantum fluctuations". I read very carefully your
paper (just once, will read it again later), but couldn't find an answer,
I just cannot understand how would you address the *initial* problem of these quantum fluctuations (if any). You introduce a *metric dependent* foliation E[g], which is "always spacelike with respect to that metric" (gr-qc/0412059 v1, p. 6), but how did you manage to "freeze" the quantum fluctuations to ensure that E[g] will be always spacelike? Looks like a Catch-22-type problem to me, and I can't understand how you would crack it.
I am very much willing to understand your ideas, and am just asking for your help. As Roland Omnes says, "tell me a story". If I have a choice, may I ask you for a *concrete* example of fixing the lapse function N and the shift vector N^i in such a way that the "quantum realized paths need not be characterised by 'frozen' values of their physical parameters" (p. 8). However, if they are not "frozen", the quantum fluctuations will unleash their destructive influence on the metric, and you'll end up with a quantum mesh, not E[g] that is supposed to be "always spacelike". Catch 22, no?
My efforts to understand the issue of these "quantum fluctuations" is at
Note: I cannot post here the reply from Ntina Savvidou, but let me briefly comment on a statement regarding classical GR, in her paper "General Relativity Histories Theory", gr-qc/0412059 v1:
"Furthermore, the equations of general
relativity are covariant with respect to the action of the diffeomorphisms
group Diff(M), of the spacetime manifold M. This does not pose great
difficulties in the classical theory, since once the equations of motion
are solved the Lorentzian
I think this does indeed pose great difficulties in the classical theory, since the principle of general covariance (active diffeomorphism invariance) leads to severe difficulties in determining any localized form of gravitational energy.
Sir Hermann Bondi has argued that a non-localizable form of gravitational energy is inadmissible in the relativity theory, and therefore its localization can and should be found. I haven't read his arguments [Bondi H., Proc. R. Soc. London A427 (1990) 249], but it seems to me that there is a long way to go from 'should be found' and 'has already been found'. Example: gravitational waves.
Adrian Scheidegger has shown that all the computed radiation terms can be eliminated by suitable coordinate transformations, therefore physical system cannot radiate gravitational energy [Ref. 1]. Had the Einstein-Infeld-Hoffmann method been independent of the coordinate system, there would be no problems, but unfortunately this is not the case [Ref. 2].
The confusion is further increased by Leopold Infeld and Jerzy Plebanski [Ref. 3], who argued that "it is hardly possible to connect any physical meaning with the flux of energy and momentum tensor defined with the help of the pseudo-energy-momentum tensor. Indeed, the radiation can be annihilated by a proper choice of the coordinate system. On the other hand, if we use a coordinate system in which the flux of energy may exist, then it can be made whatever we like by the addition of proper harmonic functions (...)."
On the other hand, there are many physicists who deeply believe that the energy of 'the grin of the cat' can be observable, in line with the basic principle of general covariance. Muhammad Sharif is one of the proponents of such ideas, although he does acknowledge that "this problem arises because energy is not well defined in GR" [Ref. 4].
Where is the catch? We use some time parameter to describe the propagation of these "gravitational waves" [Ref. 4], only they move 'within themselves' and 'with respect to themselves'. Hence the nature of this time parameter is totally unclear. See also the papers by Angelo Loinger here, and my efforts here to explain the reason why this paradoxical situation with the dynamics of GR hasn't been duly noticed.
Perhaps the only viable solution could be in the format 'have our cake and eat it', namely, we agree with Hermann Bondi and suggest a new kind of temporal evolution, in which the mysterious time parameter encoding the dynamics of GR would incorporate the observed pattern of GR waves, but will not allow them to be directly observed, in a remote analogy with quark confinement.
How can we do that? By introducing two components of this time parameter: horizontal and vertical. Perhaps this conceptual solution to the dynamics of GR can provide a typical 'have our cake and eat it' resolution of the puzzle of black holes as well [Ref. 5]: they "exist" only in the horizontal component of spacetime, and do not exist in the vertical one.
In a nutshell, if you stick to Peter Bergmann's opinion [Ref. 6] that the genuine observables in GR must be restricted to diffeomorphism-invariant quantities, you will be perfectly right, only you shouldn't consider it 'the complete truth', because these quantities are not physically equivalent or "tautological": we do observe change in the world around us. How does this "happen", and in what "time", however? The opinion advocated by Karel Kuchar is that there is a dynamical change of data from one hypersurface to another, and there is something that carries data from one hypersurface to the "nearest" one, because the collection of the canonical data on the first hypersurface is clearly distinguishable from the collection of already evolved data on the second hypersurface [Ref. 7]. However, this argument, albeit correct, does not imply that there is a genuine dynamics in GR, because physical quantities do not endure by having different properties at different times at two different adjacent hypersurfaces while preserving their identity or "sameness". There is nothing in GR that can carry the identity of an object from one hypersurface to the nearest one, in line with Kurt Lewin's Genidentität thesis, although such phenomenon (Karel Kuchar called it Perennial) should exist, or else we could never observe any change in the world around us.
These problems are known since the inception of Einstein's GR, and perhaps the time has come to accept the view that we are not dealing with alternatives but with two complementary presentations of the dynamics of Einstein's GR. Peter Bergmann is right, since he talks about the vertical component of time, which any physical clock placed in the horizontal component of time will read as being zero, i.e., frozen. And Karel Kuchar is right, since he talks about the horizontal component of time, in which the passage from one hypersurface to the nearest one is "taking place". The confusion stems from the fact that these two components cannot be disentangled by looking in the past where they are already fused, but only by revealing the emergence of time and space, which is the main unresolved task of quantum gravity. Nothing can really "fly" in Einstein's GR, since all "solutions" to the dynamics of GR are purely kinematical: we choose lapse and shift (which are arbitrary, gauge "variables" and merely reflect our diffeomorphism freedom) at one instant only, and then we fix the so-called "evolution equations" at "later" times, along with the constraints equations being strictly preserved under this fixed "evolution". We cannot repeat this procedure for each and every instant from the trajectory, because we cannot fix gauge variables "online", one-point-at-a-time. This trajectory is indeed a perfect continuum of "points", but only in the horizontal component of time. Mother Nature has access to the vertical component, thus she can choose the correct lapse and shift, and the correct evolution and constraint equations "online", one-point-at-a-time. The resulting dynamics may resemble the (holo)movement of a shoal of fish along a coral reef, but since we are confined inside "one fish only", we cannot have a global view of the whole shoal of fish driven by some "dark energy". Hence in the current version of GR we use purely kinematical models valid for one instant only. The spacetime itself does not "move". There is no genuine dynamics in the current version of GR. To explain the utterly obvious change of things around us, we need quantum gravity.
Why would Mother Nature restrict herself to one possibility only? Perhaps she can 'have her cake and eat it'. The price to pay for such ideas is accepting the possibility for a new (to theoretical physics) kind of reality, potential reality. We don't think with 'either/or' but with 'both/and', in line with the principle of complementarity. Perhaps the spacetime itself does indeed "move"-- how else could we talk about the emergence of spacetime? -- but this genuine dynamics and its "gaps" are hidden by the speed of light. Hence by looking into the past we cannot disentangle the two components, which are already been fused into a perfect continuum of events in the horizontal component of time. But since we can see the remnant from the vertical component in our diffeomorphism freedom and Hamiltonian constraints, we try to 'eliminate alternatives', and engage in endless discussions, since the first days of Einstein's GR. Enough.
Just some 'night thoughts' prompted by Ntina Savvidou's gr-qc/0412059, since her construction is also purely kinematical. I chose to be very frank, however.
The ideas here are presented in plain words only; see also my efforts here to explain the reason why the misleading decomposition of Einstein's equations have obscured the problem of 3-D space in Einstein's GR, and my interpretation here of the effectiveness of the linearized version of Einstein's GR, despite its obvious inconsistencies: strictly speaking, 'linearized gravity' is an oxymoron.
[Ref. 1] Scheidegger A.E., Gravitational Motion, Rev. Mod. Phys. 25, 451-468 (1953).
[Ref. 2] Scheidegger A.E., Gravitational Radiation, Phys. Rev. 99, 1883-1885 (1955).
[Ref. 4] Sharif M., Energy and Momentum in General Relativity, Nuovo Cim. B118 (2003) 669-683; gr-qc/0404001 v1.
"The expressions they gave are called
energy-momentum complexes because they can be expressed as a combination
of Tb_a and a pseudotensor, which is interpreted
to represent the energy and momentum of the gravitational field. These
complexes have been heavily criticized because they are non-tensorial,
i.e. they are coordinate dependent.
[Ref. 5] Logunov A.A. et al., Black holes: a prediction of theory or fantasy?, gr-qc/0412058 v1.
To understand why the conservation laws for energy, momentum, and angular momentum are in principle impossible in Einstein's GR, see the introductory part of A. Logunov's "The Theory Of Gravity", gr-qc/0210005. More from Tullio Levi-Civita, Hermann Weyl, and Andre Gsponer. It's a huge 'can of worms', as John Baez put it.
[Ref. 6] Bergmann P.G., Observables in General Relativity, Rev. Mod. Phys. 33, 510-514 (1961).
[Ref. 7] Kuchar K., Canonical Quantum Gravity, in: R. J. Gleiser, C. N. Kozameh, and O. M. Moreschi (eds.), General Relativity and Gravitation 1992, pp. 119-150. Philadelphia, PA: IOP Publishing; gr-qc/9304012.
Subject: A good
"Again: any background-free quantum theory with local degrees of freedom propagating causally would be a good thing!"
Math is available upon request.
Subject: ... as t' gets closer and
closer to t .
Thank you, once more, for sending me your paper
Christopher J. Isham and Konstantina Savvidou, "Time and
Modern Physics", in: Time, ed. by Katinka Ridderbos, Darwin College Lectures
series No. 14, Cambridge University Press, Cambridge, 2002, pp. 6-26,
On 12 December 2001, I acknowledged that "I have borrowed the terminology of two *modes* of time (and their association as the 'time of being' and the 'time of becoming') from the pioneering research of Konstantina Savvidou [cf. Ref. 1 and references therein]",
Let me explain the reason why I borrowed your terminology. I do not want to leave any trace of doubt that I have besmirched your very interesting ideas. On the contrary, my intentions are, and have always been, to *elaborate* on your ideas in such a way that you can discriminate between the two kinds of time, and "at the end of the day" (a famous expression from a famous MP and Cabinet Minister) solve the problem of time in canonical quantum gravity -- your way.
It is my understanding -- please correct me if I'm wrong -- that you have been able to cast only a *purely kinematical* outline of this highly non-trivial task,
I believe the crux of your problem has been explained in "Time and Modern Physics", in which you and Chris Isham wrote (pp. 18-19):
"The dynamical equation of elementary classical physics is Newton's second law of motion, which, as we have seen before, can be written as the coupled differential equations
[XXX] (Eq. 6)
[XXX] (Eq. 7)
"For our purposes it is important to note that the time derivate dp/dt in equation (7) is to be interpreted mathematically as the limit
[XXX] (Eq. 8)
as t' gets closer and closer to t . A similar remark applies to the time derivative dx/dt in equation (6).
"Now, as we emphasized earlier, the 'time of becoming' variable arises mathematically via the theory of differential equations and differential calculus, as in equations (6) and (7). On the other hand, the quantities t' and t in equation (8) are points of the 'time of being' since they label the specific times at which the momentum variable p has a value.
"This is the sense in which the concept of 'becoming' is structurally dependent on the concept of 'being'.
"From a mathematical perspective, the reason for this dependence is that there are no genuine infinitely small numbers (so-called 'infinitesimals') in standard mathematics, and hence the symbol 'dt' has to interpreted in a limiting sense, as in equation (8). However, there is a subject known as 'synthetic differential geometry' in which genuine infinitesimals *do* exist."
And on p. 26: "The standard way of defining differentiation -- via a limiting procedure -- necessarily mixes up the two ideas of time. However, there are other models of real numbers -- in particular, in synthetic differential geometry -- which allow for genuine infinitesimals, and which hence offer a way of separating mathematically the two concepts of time."
Hence "as t' gets closer and closer to t" (ibid., p. 18), the difference between the two kind of times is obliterated. As you put it (ibid., p. 16), "in fact, for any given physical system the two types of time transformation are locked together, and the actual history of the system is a single path, as represented by the spiral in Figure 2."
Now, I believe you will understand my concerns if you compare the arrow of 'Becoming' in your Figure 2 (p. 15) with the axis Z in Fig. 4 at
I zoom on the infinitesimal timelike displacement AB , as A gets closer and closer to B , along the putative 'global mode of spacetime', depicted with the red Z axis.
I believe this is a whole new ball game. I've done my homework here in Sofia regarding Synthetic Differential Geometry (SDG), and I do *not* believe we can unravel the arrow of 'Becoming' with SDG.
In the context of the recent research project of Chris on the operator-based approach to spacetime structure (gr-qc/0303060 v2), I believe it is the axis Z in Fig. 4 (please follow the link above) that can 'connect' one causal set to another. Should you and/or Chris are interested, please see
Addendum: Since I cannot understand Chris Isham's ideas on the operator-based approach to spacetime structure (gr-qc/0303060 v2), I can only quote the opinion of two theoretical physicists, regarding all causal set theories:
Alfonso Garcia-Parrado and Jose M. M. Senovilla, Causal structures and causal boundaries, gr-qc/0501069 v2, 8 March 2005
"Once a causal set is defined one needs to find its dynamics. Such dynamics must reduce to (say) Einstein’s field equations in the continuum limit. In  a method is sketched to recover the Einstein-Hilbert action from a quantum causal set. Other models were tried in the subsequent follow-ups. Causal set theory has been widely investigated and by now there is a vast literature about this subject, see e. g. [3, 23, 37, 81, 95, 96, 158, 160, 159, 161, 162] and references therein. In fact there is no single causal set theory and plenty of variants abiding by the idea of a basic discreteness of the spacetime have been devised.
"Of course we cannot do justice here
to all this work (a field outside our expertise). We would simply like
to remark that hitherto none of these theories accounts for a satisfactory
continuum limit in which a Lorentzian manifold is recovered. This seems
to be the crux of most of the current approaches to "quantum gravity":
mind-boggling theories are put forward but the key issue of General Relativity
as a limit is often quoted as "under current research"."
 Isham C J (2004) "A New Approach
to Quantising Space-Time: II. Quantising on a Category of Sets" Adv. Theor.
Math. Phys. 7 807-829
Chris Isham wrote that "the analogue of ‘dynamics’ is coded into the decoherence functional that is to be constructed from the basic quantum operators in the history Hilbert space" (A New Approach to Quantising Space-Time: I. Quantising on a General Category, gr-qc/0303060 v2, p. 332), but stressed that "the background space and space-time are manifolds, which is not the case if the space-time is a causal set" (ibid., p. 365). But how do we choose a decoherence functional without even a manifold to 'hold onto'? We need a 'backbone' of the physical world, or else we face a task that has been accomplished only by Baron von Münchausen, who managed to lift himself and his horse by pulling himself up by his hair. Chris Isham says: "new physical principles are needed to decide its precise form", and "this is an important topic for future research" (ibid., p. 365). Well, I'm a bit pessimistic.
To cut the long story short, I believe we need a scale-independent "backbone" of the whole physical world, which is cast on a perfect continuum. This can be done only and exclusively only with the two modes of spacetime outlined at
I say 'only and exclusively only' because this is a matter of logical possibilities. We need both the continuum and the discrete structure of spacetime, and the only way to satisfy these seemingly contradictory requirements is with the two modes of spacetime.
Again, I regret that cannot post here the reply from Ntina Savvidou, but if I've missed something, I will immediately correct my statements.