Subject: The existence of Dirac observables for gravity
Date: Wed, 03 Nov 2004 13:25:44 +0200
From: Dimi Chakalov <dimi@chakalov.net>
To: "B. Dittrich" <dittrich@aei.mpg.de>,
     bdittrich@perimeterinstitute.ca
CC: tthiemann@perimeterinstitute.ca, fotini@perimeterinstitute.ca,     dgottesman@perimeterinstitute.ca, hburton@perimeterinstitute.ca,     rmyers@perimeterinstitute.ca, odreyer@perimeterinstitute.ca, elivine@perimeterinstitute.ca, R.Loll@phys.uu.nl, box@peter-ostermann.de, crane@math.ksu.edu, jdc@uwo.ca, kuchar@physics.utah.edu
 

Dear Dr. Dittrich,

In your gr-qc/0411013 v1 [Ref. 1], you acknowledged that there is not much hope to obtain exact expressions for Dirac observables for gravity, but further investigations into this proposal (constructing a Dirac observable with respect to *all* the constraints) will be undertaken in your forthcoming preprint, ref. [16], co-authored with T. Thiemann.

I would like to make a prediction regarding your forthcoming paper written with Thomas Thiemann: you will discover that you need *infinitely many clocks* to model your Dirac observable at the fictitious instant [tau] from your fictitious time T.

I noticed that you quoted Karel Kuchar's "Time and Interpretations of Quantum Gravity" from 1992 only once, in footnote 3 on p. 13 [Ref. 1]. More about Kuchar's paper and my prediction at

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

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

I believe the *origin* of the problem has been explained by K. Kuchar in 1992; see a very simple explanation at

http://God-does-not-play-dice.net/points.html#NB

Your comments and those from your colleagues will be greatly appreciated.

Regards,

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

References

[Ref. 1] B. Dittrich, Partial and Complete Observables for Hamiltonian Constrained Systems, gr-qc/0411013 v1

p. 2: "Assume that the system is totally constrained so that the constraint generates the time evolution (which is then considered as a gauge transformation). Use a phase space function  T , which is not a Dirac observable, as a clock which "measures" the time flow, i.e. the gauge transformation. Consider another phase space function  f  and calculate the value of  f  "at the time" at which  T  assumes the value [tau] . Since the value of  f  at a fixed time [tau] does not change with time, the result will be time independent, i.e. a Dirac observable.
...

"To define a complete observable we will need infinitly many clocks which describe the embedding of the spatial hypersurface into the space-time manifold. A complete observable is then a phase space function evaluated on an embedding which is fixed by prescribing certain values for the infinitly many clock variables.
...

p. 36: "To construct a Dirac observable with respect to all the constraints (...). Further investigations into this proposal will be undertaken in [16].
--
[16] B. Dittrich, T. Thiemann, in preparation"
 
 

Note: On Thursday, November 4th, at 3:30 PM, Dr. Bianca Dittrich will speak on 'Calculating Dirac Observables from Partial Observables' at the University of New Brunswick, Canada,

http://erdos.math.unb.ca/colloq/dittrich.html

A quote from the abstract: "In the case of gravity this will answer the question whether it is possible to calculate Dirac observables starting from 3-diffeomorphism invariant partial observables."

I wish Dr. Bianca Dittrich best of luck with her talk tomorrow.

But what are these 'partial observables'? See the latest paper by Daniele Colosi and Carlo Rovelli, "Global particles, local particles", gr-qc/0409054, in which they speculated about some "globally defined n-particle Fock states", and posed the following question (p. 16): "Can we view QFT, in general, as a theory of particles? Can we think that reality is made by elementary objects - the particles - whose interactions are described by QFT? We think that our results suggest that the answer is partially a yes and partially a no." Which is, in German, Jain.

Given this latest elucidation of Rovelli's "partial observable", his papers published since 1991, and the devastating critics by Karel Kuchar in "Time and Interpretations of Quantum Gravity" (mentioned briefly by B. Dittrich in a footnote only), I thought that it would be nice to cast my prediction about the forthcoming paper by B. Dittrich and T. Thiemann. I will comment on it shortly after it becomes available.
 

D. Chakalov
November 3, 2004

Note added on November 9, 2004: It looks to me that the forthcoming paper by B. Dittrich and T. Thiemann mentioned above will be a whole set of five papers: B. Dittrich, T. Thiemann, Testing the Master Constraint Programme for Loop Quantum Gravity, parts I. - V., to appear. See ref [4] in the latest gr-qc/0411031 v1 by T. Thiemann.

In this latest gr-qc/0411031 v1, T. Thiemann makes a crucial assumption about the so-called weak Dirac observables:

"It is often stated that there are no Dirac observables known for General Relativity, except for the ten Poincare charges at spatial infinity in situations with asymptotically flat boundary conditions. This is inconvenient for any quantization scheme because it is only the gauge invariant quantities, that is, the functions on phase space which have weakly (footnote 1) vanishing Poisson brackets with the constraints, which have physical meaning and can be measured. These are precisely the (weak) Dirac observables of the canonical formalism.
--
"Footnote 1: We say that a relation holds weakly if it is an identity on the constraint surface of the phase space where the constraints are satisfied."

I wonder what Karel Kuchar will say on these weak Dirac observables. See the problems of constructing weak Dirac observables on p. 16 from gr-qc/0411031 v1.

Further, Thomas Thiemann writes:

"2.3 Evolving Constants

"The whole concept of partial observables was invented in order to remove the following conceptual puzzle:

"In a time reparameterization invariant system such as General Relativity the formalism asks us to find the time reparameterization invariant functions on phase space. However, then "nothing happens" in the theory, there is no time evolution, in obvious contradiction to what we observe.

"This puzzle is removed by using the partial observables by taking the relational point of view: The partial observables f, Tj can be measured but not predicted. However, we can predict Ff,T , it has the physical interpretation of giving the value of  f  when the Tj assume the values j . In constrained field theories we thus arrive at the multi fingered time picture, there is no preferred time but there are infinitely many."
...
"In other words, [XXX] is a weak, Abelean, multi-parameter group of automorphisms on the image of each map F0 f,T . This is in strong analogy to the properties of the one parameter group of automorphisms on phase space generated by a true Hamiltonian. Also this observation, in our opinion due to [5], will be used for a new proposal towards quantization."
--
[5] B. Dittrich, Partial and Complete Observables for Hamiltonian Constrained Systems, gr-qc/0411013"
 

Fine, but  when  the Tj assume the values j ? See my prediction above.

Perhaps it would be a good idea if Bianca Dittrich and Thomas Thiemann consult Karel Kuchar before writing their five papers, "Testing the Master Constraint Programme for Loop Quantum Gravity", parts I. - V. See again T. Thiemann's footnote 1 above.

Also, see a new proposal towards quantization below. Should you find some "weak" observables there, please drop me a line.

Finally, see a recent effort at finding a physical inner product in [Ref. 3]. Surely if something can carry the states of a physical system along a trajectory of "points", then the carrier itself must not change; more from Kurt Lewin here. That's what the human brain and the universe do with the global mode of spacetime. More about the inner product problem from Claus Kiefer here, and about the carrier itself (Aristotelian Unmoved Mover) from Karel Kuchar here.
 
 

D. Chakalov
November 9, 2004

======

Subject: "This is in strong analogy"
Date: Sat, 13 Nov 2004 11:50:01 +0000
From: Dimi Chakalov <dimi@chakalov.net>
To: tthiemann@perimeterinstitute.ca
CC: bdittrich@perimeterinstitute.ca, c.isham@imperial.ac.uk,
     unruh@physics.ubc.ca, kuchar@physics.utah.edu,
     kiefer@thp.uni-koeln.de

Dear Thomas,

In gr-qc/0411031 v1, Sec. 2.3 Evolving Constants, you mentioned some "strong analogy",

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

I'm afraid it is just that: a strong analogy. Just like you have a strong analogy b/w the first law of Ohm and a hose running water, only you wouldn't speculate on the fine structure of electricity from the strong analogy with a hose running water.

More at

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

Shall I elaborate?

Regards,

Dimi
--
Dimi Chakalov
35 Sutherland St
London SW1V 4JU, UK
http://God-does-not-play-dice.net


Note added on November 30, 2004: I'll only quote from the first and the fifth papers mentioned above.

In the first paper, gr-qc/0411138 v1, B. Dittrich and T. Thiemann wrote:

"While there has been progress in the formulation of the quantum dynamics [4], there remain problems to be resolved before the proposal can be called satisfactory. These problems have to do with the semiclassical limit of the theory (...).

"The success of the programme of course rests on the question whether we can really quantize 3+1 General Relativity (plus matter) with this method. (...) The results of this series of papers, in our mind, demonstrate that the mathematics of the Master Constraint Programme succeeds in a large class of typical examples to capture the correct physics so that one can be hopeful to be able to do the same in full 3+1 quantum gravity."

It will "capture the correct physics" only in those cases that cannot be verified nor falsified by experiment or observation at the scale of tables and chairs. The Master Constraint Programme hypothesis is still totally immersed in the quantum realm, and has not shown up its head above the murky water so that we can see it: there is still no semiclassical limit. Why? Because this semiclassical limit must be reached on a bi-directional path: go to the classical regime, and get back to "loop quantum gravity".

Fuhgeddaboudit.

And in the final paper, gr-qc/0411142 v1, B. Dittrich and T. Thiemann wrote:

"(N)ow the burden is on us to show that the theory can also successfully deal with the additional symmetries that have entered the stage by coupling matter to the gravitational field. This is the spacetime diffeomorphism symmetry which finds its way into the canonical framework in the form of the spatial diffeomorphism and the Hamiltonian constraint. It is precisely for this reason that the Master Constraint Programme was created. One now has to apply it to all symmetries of General Relativity, solve the full Master Constraint and establish that we have captured a quantum theory of General Relativity rather than a mathematically consistent but physically uninteresting quantum theory of geometry and matter. This is what has to be done in the close future and finally the mathematical techniques are available in order to make progress."

These "additional symmetries" will probably show the limitations of the model: you can't learn much about the nature of electricity from a hose running water. See above.
 

D. Chakalov
November 30, 2004

 

======
Subject: How to get a physical inner product for canonical quantum gravity?
Date: Tue, 09 Nov 2004 12:52:21 +0000
From: Dimi Chakalov <dimi@chakalov.net>
To: Suresh K Maran <skmst18@pitt.edu>
CC: bdittrich@perimeterinstitute.ca,
     tthiemann@perimeterinstitute.ca, kiefer@thp.uni-koeln.de,
     pullin@rouge.phys.lsu.edu
 

Dear Dr. Maran,

I read the latest version of your gr-qc/0311090 [Ref. 2], and got the impression that you are sort of optimistic about the inner product: "By studying the continuum limit of our elementary transition amplitudes one might be able to get a useful physical Hamiltonian operator (physical inner product) for canonical quantum gravity."

Since canonical quantum gravity is formulated on continuum manifolds, I suppose we have to introduce a generic "discreteness" to the manifold from the outset,

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

Otherwise the *continuum limit* may never be recovered.

As to the proposal due to Thomas Thiemann for a Hamiltonian constraint operator, gr-qc/9606088, which was mentioned as ref. [27] in your paper [Ref. 2], it seems to me that this task involves a generic contradiction.

To the best of my knowledge, the only person who claimed to have solved the inner product problem was Baron von Münchausen, who managed to lift himself and his horse by pulling himself up by his hair. I mean, if we require that the inner product, which encodes the probability interpretation, should be conserved *in time*  T , we face the insurmountable problem of finding the elementary increment  T_0  WITHIN the *instants* of this same  T .  But in canonical quantum gravity these *instants* are 3-D hypertsurfaces with their intrinsic multi-fingered time  t_n , and  t_n  cannot -- even in principle -- represent the DISCRETENESS of  T  needed for its elementary increment/step  T_0 . Hence if we want to do better than Baron von Münchausen, we must introduce a generic "discreteness" in both  T  and  t_n  (please see the URL above).

Hence quantum gravity formulated with the chain of instances  t_n  cannot be unitary, and the Hamiltonian constraint *operator* must involve  T  (called 'global mode of spacetime').

More on this generic "discreteness" at

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

BTW I believe a similar problem occurs with the so-called gravitational
waves, since these waves also have to somehow propagate 'within
themselves'

http://God-does-not-play-dice.net/Kopeikin.html#note_2

I will appreciate your opinion on these matters.

You can read this email also at

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

Kindest regards,

Dimi Chakalov
35 Sutherland St
London SW1V 4JU, UK
http://God-does-not-play-dice.net
 

References

[Ref. 2] Suresh K. Maran, Relating Spin Foams and Canonical Quantum
Gravity: A Discrete Step Evolution Formulation of Spin Foams, gr-qc/0311090 v5

"The proposal by Thiemann [27] for a Hamiltonian constraint operator appears to be set back by anomalies [29]. By studying the continuum limit of our elementary transition amplitudes one might be able to get a useful physical Hamiltonian operator (physical inner product) for canonical quantum gravity.

"There are many open questions that need to be addressed, such as:

• What can we learn from this approach about the physics of
  quantum gravity? For example, is quantum gravity unitary?

• What are the potential applications to the physical problems?

• What is the continuum limit?

• How to include the topologies in our theory that were excluded
  by the conditions that were specified in the beginning of
  section three?

• How to include matter?"

=====
 

[Ref. 3] Karim Noui, Alejandro Perez and Kevin Vandersloot, On the Physical Hilbert Space of Loop Quantum Cosmology, gr-qc/0411039 v1

pp. 3-4: "Without the physical inner product, there is no notion of a probability measure which is an essential ingredient for physical prediction in the quantum theory. Any interpretation of the wave function as a probability amplitude is futile without the probability measure provided by the physical inner product.
...

"In addition the physical inner product provides a means for eliminating pathological solutions to the constraints. For instance it can be shown that the space of solutions of LQC is generally non separable, i.e., an uncountable infinity of solutions to the quantum constraint equation exist.
...

"In the absence of the physical inner product, so far spurious solutions are eliminated using the dynamical initial condition coupled with heuristic arguments based on semi-classicality requirements. Although these requirements are physically motivated, unphysical solutions can be easily identified if a notion of physical inner product is provided. More precisely, on the basis of the notion of physical probability physical states are defined by equivalence classes of solutions up to zero norm states. It is hoped that the ill-behaved solutions mentioned above will be factored out by this means once a suitable notion of physical inner product is provided."