Paul Davies

____________________________________________________

Dear Lee,

Thanks once more for your deep explanations. I am very impressed by the great mathemtical progress that you and others have achieved. Of course, one would now also like to better understand the physical meaning of those many formal solutions that you mentioned. It may be too early yet to expect novel and experimentally testable predictions (after all, that's what is required for a "real" physical theory), but one should be able to derive the two pillars of empirically founded physics: (1) classical general relativity, and (2) a general time-dependent Schroedinger equation for matter. Mathematical consistency and beauty by themselves are not sufficient for a physical theory.

This is the reason why I had hoped that you may, as an intermediate step, derive the WDWE in some sense as a low energy "effective" approximation (much lower than string theory if this is indeed another low energy limit). I never meant that this can be a "full" theory, but I still believe that its role is similar to that of nonperturbative QED in a unified theory. The WDWE offers at least some important novel conceptual consequences (such as timelessness and an "intrinsic initial value problem"). One may even be able to exclude certain classes of solutions of classical GR, such as those with CTCs. Quasi-classical solutions require decoherence, hence an arrow of time, which is incompatible with closed time-like curves. Similar arguments would apply to "white holes" (with an inverse no-hair theorem). This quantum arrow requires an initial QUANTUM state in an appropriate sense (such as the absence of initial nonlocal entanglement). This can all be described in terms of the WDWE.

Of course, there may also be another (direct) way from loop theory to those two "pillars" of physics. If none of them works (I don't think so), so much the worse for loop theory. However, neither would such a direct derivation demonstrate that the WDWE is NOT an effective theory, nor that there may not be another "full" theory from which they may also be derived. We could then discriminate between them only by additional empirical evidence. Therefore, at this stage I have more confidence in the WDWE, which is merely a combination of the two fundamental theories. Its two major shortcomings, (1) that it is not unique (because of factor ordering, for example), and (2) that its application usually requires restriction to a finite number of degrees of freedom, are common for all useful effective quantum theories (such as QED, quantum MECHANICS, quantum rotators as effective objects within the latter, etc.).

So I think that BOTH theories should be further investigated, and their relations be discussed. (Remember that many of our colleagues still believe that quantum physics ends with the "click in the counter" -- see Mohrhoff's new contribution!).

I am replying to your comments in detail in my second e-mail ("Re II") of today.

With my best wishes

Dieter __________________________________________________

Dear Lee,

I am enjoying this "private lesson" on loop theory!

This is the detailed reply to your comments, which may require knowledge of the previous exchange. In order to reduce further "blow up" of the mail, I have dropped most of my former remarks and those of your comments which may be regarded as closed. (See my "Re I" of today first!)

>Even if string theory cannot be by itself a full theory of quantum
>gravity, it could be the low energy approximation to such a theory,
>good in a background dependent regime. It is possibly a mistake, made by
>string theorists as well as skeptics of string theory, to take it
>as an all or nothing proposition. I have argued this in detail
>elsewhere, would be glad to expand on it here if you are interested.

Yes, I would, but we should also hear a string physicist on this issue!

>Loop quantum gravity is conventional in these regards, certainly
>superpositions of physical states exist and can be studied.

So you can have decoherence of geometry by matter in order to get quasi-classical geometries and to avoid Schroedinger cat states? If decoherence leads to quasi-classical three-geometries, the latter must exist as quantum states in the HS.

This is related to my following former question:
> Therefore, my first main question was whether loops form just another
> representation (basis) for this Hilbert space of quantum gravity, or
> whether the underlying configuration space ("field representation")

> of three-geometries is further reduced by (classical) diffeomorphism
> invariance (or even entirely different).
>
>To my understanding spin networks form the only well defined
>representation we have for quantum GR. Three-geometries mod spatial
>diffeos are a formal construction, but the space spanned by a basis
>in one to one correspondence with embeddings of spin nets into a
>manifold is not. In this case one can study all questions
>explicitely because one
>has hold of a complete, detailed construction of the hilbert space.

OK then: how come classical general relativity (without postulating it separately, a la Copenhagen)?

>I mean by a classical algebra of observables, a set of functionals
>on the phase space that is closed under poisson brackets.

You are talking about the mathematics, while I meant the interpretation of "observables" (which is responsible for their name). It becomes relevant only when replacing Poissons brackets by commutators. In quantum mechanics (the major empricial basis of quantum theory!), the role of algebra (first discussed by Born, Heisenberg and Jordan) seems to be often overestimated. For the quantum mechanics of mass points it reduces to the cartesian [p,q] commutator, while nontrivial applications (such as a rotator) are "effective" (derivable) approximate descriptions. However, this may be different in field theories (gauge theories, in particular). Anyhow, in my remark, that observables are quantum objects, I meant the specific quantum interpretation of observables as "formal objects which do not possess values". At this stage of your theory (diffeomorphisms) you seem to need only the algebra.

>Here is an important physical lesson, which is a key point in
>how loop quantum gravity works. I apologise if it seems technical,
>the point is actually very physical.
>
>In any field theory quantum states are valued not on smooth paths or
>fields, but on extended spaces of paths or fields that include-and
>are dominated by-ones that are not differentiable anywhere.

Do you mean Feynman paths? They may be problematic in a time-less theory (see Hawking's derivation of an arrow of time), since they were introduced as propagators for wave functionals. But this may not affect your subsequent arguments (so I drop them here). However, do you "smuggle in" quantization at this point when talking about Feynman paths?

>One has to choose a representation of the observables algebra-there
>is no way out of this being part of the quantization process.

(Preceding quantization.) But I am not sure. In quantum mechanics there are alternative ways. A specific quantum theory need not even be derivable from a classical one. For example, you can determine the inner products between empirically known states empirically (think of different neutrinos or K mesons). However, I do not object to your way of quantizing in this case of loop theory.

>However the choice can be narrowed by the requirment that
>the gauge and diffeomorphism transformations act on the
>resulting hilbert space in a well defined manner. What is
>believed to be the case is that this limits the choice of
>representation uniquely to the one used in loop quantum gravity.

It's an excellent hypothesis, but it NEED not lead to the "real" theory.

I said:
> Seems to be the same for the WDWE (with three-geometries used as the basis).
>
>Again, that is a FORMAL construction.

The difference between formal and not formal is not within mathematics. I guess what you mean it is "incompletely defined".

> One does not know how to proceed
>to construct well defined operators on the quotient of 3 metrics
>by 3-diffeos. As a result, one cannot get detailed results about
>things like the spectrum of various observables such as areas and
>volumes, because one does not have detailed control over how the
>quantum states behave at short distances.

This IS an important advantage (although these "observables" still have to become observable).

>
>There is absolutely no problem including the standard matter fields
>in the construction-including gauge fields, fermions, scalar fields,
>spin 3/2 fields, antisymmetric tensor fields etc. So one gets
>entangled states of spin networks plus matter degrees of freedom.

Do you mean that none of the sources can be hoped to be explained in terms of topology? One would expect some kind of unification in a "full and real" theory.

>I don't know what you mean by "wave funcationals on classical
>objects"?

Wave functionals on a classical configuration space (such as Psi[A(x)] in QED, or Psi[3G] in quantum geometrodynamics). My original question was if there is a similar classical configuration space AFTER eliminating diffeos. (Is it analogous to the Coulomb constraint in QED?)

>The hilbert space is NOT a product of a gravity hilbert space
>and a matter hilbert space once we have moded out by diffeos,
>as the diffeos act on both gravity fields and matter fields.

This is important, although it still does not show that there is no classical configuration space (that would have to consist of mixed classical objects, conceivably massive objects together with their "comoving distortion of geometry" -- similar to the Coulomb field in ED).

Note that these "long range dressings" give rise to mass and charge superselection rules.

>
>The space of diffeomoprhism invariant states is NOT scale invariant
>in theories like GR where there is a scale. The Planck scale comes
>into the canonical commutation relations, so the theory has a scale
>built into it for the same reason that the quantum theory of a
>free massive particle has the mass built into it.

Mass is a quantity that exists on the classical level, while the Planck scale and commutation relations are quantum properties. This may be a good opportunity for you to DEFINE classical diffeomorphisms in non-technical words (if possible). (The definition need not be mathematically clean.) Indeed, I did think that scale transformations are part of diffeos. while the new areas and volumes seemed to be fundamentally quite different quantities (from which the conventional scales of the effective theory then have to be recovered). Is that quite wrong?

>Very roughly speaking, for spin networks with small spins labeling
>the edges (1/2, 1 etc) the volume measures the number of
>nodes of valence 4 or larger.

Nodes of what? The difference between nodes (zeros, I presume) of quantum amplitudes Psi (on some configuration space), and those of some classical fields is very important here. (I guess there is a difference between nodes and topological knots -- the word is the same in German!)

>
>A given spin network state does diagonalize operators which are
>functionals of 3 geoemetry such as the volume of the universe
>or the area of any physically defined surface (the boundary if there
>is such or a surface defined by matter degrees of freedom.)
>So these are as close as one can come to eigenstates of
>3 geometry. That is, 3 geometry is defined when one averages
>over regions large in plank units.

This may answer some of my major questions. Is it not a first step towards an effective theory in terms of three-geometries?

>
>There is no evidence in the spatially compact case for a unique
>physical quantum state. There really are an infinite dimensional
>space of known elements of the simultaneous quotient of all
>the constraints.
>
>The arguments for the existence of such a unique
>state were formal and appear to have been wrong.
>
>Here is an important question we DON'T know the answer to:
>in the case with boundaries (compact or asymptotic) so that
>a non vanishing hamiltonian is defined, does that physical
>hamiltonian have a unique ground state?

Is this a theory WITH time? Or what else would superpositions of different eigenstates of H mean?

More attractive to me are those without boundaries (still H = 0 in loop theory?). Is there then any initial value problem left (effectively equivalant to that at a = 0)? I think you answered "yes" to a similar question before, but is there a chance to postulate initial homogeneity, or a Weyl tensor condition, or a totally symmetric initial state?

I had asked:
> What do you mean by "coordinate only"? By physical time I meant
> something like the "carrier of information about all proper times"
> (or "many-fingered time") -- not a coordinate.
>
>Good, this is something that deserves more investigation. The
>"time" Soo and I studied-the Im part of the Chern-Simons invariant
>of the Sen-Ashtekar connection-is a coordinate on the phase
>space

Why do you speak of a phase space in quantum theory? Do you not simply mean configuration space or another BASIS of HS (which can be only "half" of phase space)?

> (mod spatial diffeos). So it is not a coordinate on a
>single spacetime, it is a coordinate in the infinite dimensional
>manifold which is the phase space within which individual
>spacetimes are families of trajectories. (It is by the way,
>approximated to a certain degree by the York time on large slowly
>varying
>geometries.)

There is something similar in quantum geometrodynamics. See page 188 in Chap. 6 of my 4th edition (available at www.time-direction.de).

>
>We do not have only vacuum solutions, as I said everything goes
>through completely when matter is included.

I have not been well informed, evidently.

>
>I do believe that the questions regarding the interpretations
>of quantum theory are quite touched-because of the issues
>involving time and how one makes measurements in a cosmological
>theory which includes all observers, measureing instruments,
>clocks etc.

Yes, but are there any basic differences in this respect compared to the WDWE or Everett in general (except for time)?

> This is notwithstanding the success of loop quantum gravity at
>solving all the short distance qft issues.

Yes, I think these are different questions.

>
>I would like to emphasize that the foundational issues
>that seem to be the most important, as well as the structures
>that appear necessary to solve them, do not appear
>in mini-superspace models. One needs to be able to have
>local degrees of freedom, light cones, dynamical quantum
>geometry, and the ability to divide the universe into
>different local regions, before one can raise let alone
>resolve the key questions.

This is certainly correct. Therefore, we used a multipole expansion on the Friedmann sphere as a first step. A more general approach is Kiefer's Tomonaga-Schwinger equation, based the Born Oppenheimer approximation with respect to the Planck mass.

>The only issue one can address in mini superspace models is
>the reparameterization invariance in time. But this plays
>a very different role and has a different character than the
>many fingered time reparametrization invariance that appears
>in the full theory.

From what you said above I am not quite sure. But see my Re I as well!

>
>I hope I have convinced you at least that enough progress
>has been made on the full theory to have left a fertile
>field for work and thought on the questions that interest
>you.

Complete agreement on this point.

Best wishesDieter

--------------------------------------------------------------
H. Dieter Zeh Phone: (+49)6223 74097 Gaiberger Str. 38 Fax: (+49)6223 74098 D69151 Waldhilsbach e-mail: zeh@urz.uni-heidelberg.de Germany

See also:www.zeh-hd.de(or www.time-direction.de)

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Published   2002.03.04