|Subject: The existence of Dirac observables for gravity
Date: Wed, 03 Nov 2004 13:25:44 +0200
From: Dimi Chakalov <firstname.lastname@example.org>
To: "B. Dittrich" <email@example.com>,
CC: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, R.Loll@phys.uu.nl, email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
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. , 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
I believe the *origin* of the problem has been explained by K. Kuchar in 1992; see a very simple explanation at
Your comments and those from your colleagues will be greatly appreciated.
[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 .
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.
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  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.
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
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."
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
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
is in strong analogy"
In gr-qc/0411031 v1, Sec. 2.3 Evolving Constants, you mentioned some "strong analogy",
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.
Shall I elaborate?
In the first paper,
v1, B. Dittrich and T. Thiemann wrote:
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".
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.
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,
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.  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
BTW I believe a similar problem occurs
with the so-called gravitational
I will appreciate your opinion on these matters.
You can read this email also at
[Ref. 2] Suresh K.
Maran, Relating Spin Foams and Canonical Quantum
"The proposal by Thiemann  for a Hamiltonian constraint operator appears to be set back by anomalies . 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
• 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
• 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."