Paul Davies

ZEH TO SMOLIN:

Dear Lee,

I found your clarifying remarks on quantum gravity in your "corrections" very helpful. In fact, I have some further questions which may be of general interest. Let me first address them to you, and then we may find out whether they may be relevant or are simply based on trivial misunderstandings.

I tried to study one of your introductory papers a long time ago. If I remember correctly, the essential point of loop theory is to eliminate gauge degrees of freedom and/or diffeomorphisms (please explain if there is any difference -- I never really understood the distinction between active and passive transformations in this connection). This is done before quantization. First question: may we HOPE (or even expect) that this procedure is equivalent in principle, and in the end, to solving the momentum constraints AFTER we have first quantized in the field representation, for example, (please disregard all "technical" or purely mathematical problems)?

Some people seem to believe that the knot structure remaining after eliminating diffeomorphisms, classified by discrete numbers n, is a quantum aspect. I would believe that these numbers, which appear on the classical level, are analogous to the discrete multipoles of a classical em field. While you can even classically superpose multipoles, quantum theory requires a wave function(al) for their amplitudes, Psi({c_n}). If there is no reparametrization invariance, one can derive a time dependent Schroedinger equation for Psi({c_n},t) instead of the classical time depencence c_n(t), but in gravity we have the Hamiltonian constraint H Psi = 0, and time dependence is lost. (The physical meaning of this formal result should crucially depend on the question whether and in what sense the wave function describes physical states or is "just a tool", as Bill Unruh claimed.)

On a Friedmann sphere, one may also eliminate gauge degrees by restriction to tensor modes only. However, this "field" representation appears much easier to interpret -- for example in terms of "mini superspaces". So here is my second major question: is there anything in loop theory that is equivalent to the hyperbolic WDWE in terms of ln a, the log of the expansion parameter. Claus Kiefer, using ideas of Banks and of Halliwell and Hawking, was able to recover an effective time dependent Schroedinger equation using this approach. This appears impossible under scale invariance; so is there more than usual gauge invariance in loop theory? Or in other words: is there still a wave functional of three-geometries Psi{3_G}, or is there a new and reduced superspace? I am asking this in particular, since the hyperbolic form would also allow us to pose an "intitial" value problem at a=0 in spite of the absence of time. So what is the general "quantum state" of loop gravity, and what happens to the expansion parameter in this formulation, or to Friedmann models in general for that matter?

Even though this discussion becomes a little technical, it seems to be conceptually fundamental in order to understand the universe and its arrow of time.

Best regardsDieter Zeh

____________________________________________________

SMOLIN REPLIES:

Dear Dieter,

Thanks very much for your comments. I'll try to answer the questions raised clearly, please come back to me if something I say is not clear. Also, there is a large literature in loop quantum gravity aimed at the questions you raise, which are of course the key ones, here I mention only a few.

> If
> I remember correctly, the essential point of loop theory is to
> eliminate gauge degrees of freedom and/or diffeomorphisms (please
> explain if there is any difference -- I never really understood the
> distinction between active and passive transformations in this
> connection). This is done before quantization. First question: may we
> HOPE (or even expect) that this procedure is equivalent in principle,
> and in the end, to solving the momentum constraints AFTER we have
> first quantized in the field representation, for example, (please
> disregard all "technical" or purely mathematical problems)?

First, local gauge transformations (The local lorentz transformations, or a subgroup of them, as in the use of Chiral SU(2) in 3+1) are treated differently from spatial diffeomorphisms.

The local gauge transformations are solved BEFORE quantization.

This is done by constructing a classical observables algebra, called the loop algebra, which is locally gauge invariant. It is composed of Wilson loops and corresponding momentum observables. One then makes a choice of representation of this algebra on a Hilbert space. In quantum gravity, as opposed to an ordinary gauge theory, one choices a representation which does not depend on any background metric. This is called the kinematical Hilbert space.

One then defines the action of the spatial diffeomorphisms on this kinematical hilbert space, which is found to be anomolie free. One can then find the subspace of states which are invariant under all spatial diffeo's. Thus, one solves the spatial diffeo constraints ON THE KINEMATICAL HILBERT SPACE, that is AFTER quantization.

Finally, by a rather delicate procedure, one defines the hamiltonian constraint first as a regulated operator on the kinematical hilbert space and then one shows there is a certain limit which results in an operator that is well defined on the kernel of the diffeo constranits.

These days the evolution operators and projectors onto physical states are also defined in terms of spin foam sums over histories.

You might ask if these proceedures could be done in an other order. With respect to local gauge tranformations yes, one can start with a representation of an algebra extended to include gauge dependent observables, and then impose the local gauge constraints quantum mechanically. For technical reasons this is done in some of the mathematically rigorous literature, in papers by Thiemann, Ashtekar, Lewandowski, Baez and collaborators. The result is the same.

I don't think we know how to impose the spatial diffeos classically and then quantize. The closest anyone has gotten is a paper by Newmann and Rovelli-they did succeed in constructing and coordinatizing the phase space mod local gauge transformations and spatial diffeos, but it was not possible so far to see how to represent the hamiltonian constaint on the remaining space. Perhaps this deserves more study.

> Some people seem to believe that the knot structure remaining after
> eliminating diffeomorphisms, classified by discrete numbers n, is a
> quantum aspect.

This is correct. Knot and link and graph classes label the quotient of states in the kinematical hilbert space by the action of all the spatial diffeomorphisms. They are quantum numbers in a quantum mechanial sense.

Certain aspects of the spin network states are understood to have a physical interpretation because they provide a basis for the diagonalization of observables that measure spatial volume and areas. Apart from this, it is not known what observables knotting measures- in the case of vanishing cosmological constant. In the case of non-zero cosmological constant we understand some knot invariants as observables of the quantum gravity theory.

I would believe that these numbers, which appear on
> the classical level, are analogous to the discrete multipoles of a
> classical em field.

No, to my knowledge, the knotting, linking, graph invariants have so far been defined only in the quantum theory. If one wanted to give them a purely classical iterpretation, one could start with the work I mentioned of Newmann and Rovelli.

The spin network states are analogous to the quantized magnetic flux in a superconductor. This analogy has been made precise. The representation in which the spatial diffeo's are well defined is one in which electric flux (of the spacetime connection)-which is the same thing as area- is quantized.

>While you can even classically superpose
> multipoles, quantum theory requires a wave function(al) for their
> amplitudes, Psi({c_n}).

Both the kinematical hilbert space and its quotient by spatial diffeos are genuine hilbert spaces. There is a superposition principle at both levels.

>If there is no reparametrization invariance,
> one can derive a time dependent Schroedinger equation for
> Psi({c_n},t) instead of the classical time depencence c_n(t), but in
> gravity we have the Hamiltonian constraint H Psi = 0, and time
> dependence is lost. (The physical meaning of this formal result
> should crucially depend on the question whether and in what sense the
> wave function describes physical states or is "just a tool", as Bill
> Unruh claimed.)
>

I agree. And the good news is that all the various points of view about the meaning of the hamiltonian constraint and its solutions can be tested in loop quantum gravity. We have explicit solutions (and infinite numbers of them) to all the constraints. In the histories, spin foam formalism one has explicit constructions of both projectors onto the kernel of all the constraints and of genuine evolution operators defined with respect to different notions of physical time (such as the spacetime volume in a history interperlating between an initial and a final state.) So all the different ideas about this deep problem can be studied in detail now in the real theory, and not just in models.

I have my own proposal as to the resolution of the "time in quantum cosmology" problem (see gr-qc/0104097), as do some other practitioners- Rovelli, Crane, Markopoulou...and while we do not yet completely agree, our ideas on it are certainly conditioned by what is and is not possible in a full version of a quantum gravity theory. For this reason I think many of us (certainly those I mentioned) favor some version of a relational quantum theory. This seems favored when one confronts the possibilities available in a real background independent theory with an infinite number of degrees of freedom, and no background symmetries to confuse the discussion.

> On a Friedmann sphere, one may also eliminate gauge degrees by
> restriction to tensor modes only. However, this "field"
> representation appears much easier to interpret -- for example in
> terms of "mini superspaces". So here is my second major question: is
> there anything in loop theory that is equivalent to the hyperbolic
> WDWE in terms of ln a, the log of the expansion parameter.

Yes, one can discuss something like this in detail, see a paper by Chopin Soo, gr-qc/0109046, which extended results in gr-qc/9405015, related results were also given by Szabados in gr-qc/0110106. The time coordinate on the full configuration space of GR that seems most suited for the study of the full theory is shown there to be the imaginary part of the Chern-Simons invariant of the Sen-Ashtekar connection.

But even though one can go some distance towards this idea in the full theory, and hence recover a time variable from a certain set of explicit and exact solutions to all the constraints, as in Soo's work, I think the coordinate only has all the properties one would like a time coordiante to have in the semiclassical limit. So my own view is that this is not the full solution to the issue of time (see the above paper and my previous post on this issue.) But if someone wants to push that point of view in the full theory, this is a place to start.

>Claus
> Kiefer, using ideas of Banks and of Halliwell and Hawking, was able
> to recover an effective time dependent Schroedinger equation using
> this approach. This appears impossible under scale invariance; so is
> there more than usual gauge invariance in loop theory? Or in other
> words: is there still a wave functional of three-geometries Psi{3_G},
> or is there a new and reduced superspace? I am asking this in
> particular, since the hyperbolic form would also allow us to pose an
> "intitial" value problem at a=0 in spite of the absence of time. So
> what is the general "quantum state" of loop gravity, and what happens
> to the expansion parameter in this formulation, or to Friedmann
> models in general for that matter?

When the cosmological constant is positive, there is an exact state whose semiclassical limit is DeSitter, this is the exponential of the Chern-Simons invariant of the Sen-Ashtekar connection, as discovered by Kodama. It is discussed in the references by Soo and Soo and myself, and many others. So I refer you to those papers where these questions are worked out.

Besides this, there is a very interesting series of papers by Bojowald, in which FRW quantum cosmology is recovered in loop quantum gravity (with vanishing cosmological constant.). For a summary and references see gr-qc/0202077. In particular Bojowald shows that there is no initial singularity in the theory.

Finally, I am grateful for your comments because I think there needs to be more interchange between people like yourself who have thought deeply about these hard issues and the people who are working with the full theory. We can provide tools to investigage the key questions in a real theory and not just in models, but I'm sure that we miss key points and make mistakes for not having thought deeply enough.

Thanks, Lee

____________________________________________________

ZEH NOW RESPONDS TO SMOLIN'S ANSWERS:

Dear Lee,

Thank you very much for your informative answers. Let me once more emphasize that I agree with your criticism of string theory approaches. Quantum theory must be fully quantized for consistency (even it is treated as an "effective theory" -- such as QED). I am not quite sure though that we entirely agree about the meaning of quantization (and this question would establish a relation of our discussion to some former contributions on this forum).

Your citation of specific literature will be very helpful for me, but perhaps not so much for this forum in general. (And often the physical relevance of the deep mathematical theorems, which is here at issue, is insuffiently discussed.) So we should try to express the essential results in plain words. In order to please Paul, I will use only one level of > (which appears as a side bar in my e-mail program). I will also omit most of your "technical" remarks in this e-mail, but you may copy them back into a possible reply if required.

My questions aimed primarily at the appropriateness of a Wheeler-DeWitt wave function. Its "stage" (the classical configuration space after hypothetically solving the momentum constraints) is assumed to be the space of coordinate-free three-geometries. Quantization in my sense then means to admit all superpositions of them as physical states. This is the quantum theory that has been confirmed by many experiments. For example in QED, experimentalists were able to construct "mesoscopic Schroedinger cats", and even to observe their decoherence into the quasi-classical "field" states that we usually observe. Erich Joos pointed out long ago how superpositions of different three-geometries would similarly decohere through entanglement with matter (which CAN be formulated in terms of a WDW wave function).

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).

>The local gauge transformations are solved BEFORE quantization.

Fine, but is this the same as reducing the three-metric (h_kl) to an abstract three-geometry?

>This is done by constructing a classical observables algebra, called the
>loop algebra, which is locally gauge invariant. It is composed of
>Wilson loops and corresponding momentum observables.

"Observables" are usually regarded as a quantum concept (quantities which do not necessarily "possess" values). In a classical theory, everything can be described by "beables" (to use John Bell's term). This may be a matter of words, but could be misunderstood. Presumably you mean something like the classical em FIELDS in contrast to their potentials? Then we also have the loop integrals of A, which are nontrivial on not simply connected spaces. Our (macroscopic) space seems to be always simply connected, unless you "exclude" the sources. This may be different on the Planck scale, where we may hope, conversely, to explain sources by topology. Is this what you have in mind when you are discussing (only?) vacuum solutions? This is very fascinating, but as yet somewhat hypothetical (in spite of all mathematical progress).

> One then makes
>a choice of representation of this algebra on a Hilbert space.

Do you have a Hilbert space before quantization, or are you just POSTULATING it in this way? If you say that you "really" want to admit all superpositions, then we seem to agree on quantization.

>In quantum gravity, as opposed to an ordinary gauge theory, one choices a
>representation which does not depend on any background metric. This is
>called the kinematical Hilbert space.

Seems to be the same for the WDWE (with three-geometries used as the basis).

>One then defines the action of the spatial diffeomorphisms on this
>kinematical hilbert space, which is found to be anomolie free.

What does that mean?

> One can then find the subspace of states which are invariant under all
>spatial diffeo's. Thus, one solves the spatial diffeo constraints
>ON THE KINEMATICAL HILBERT SPACE, that is AFTER quantization.
>You might ask if these proceedures could be done in an other order.
>With respect to local gauge tranformations: yes ... The result is the same.
>I don't think we know how to impose the spatial diffeos classically and
>then quantize.

I am satisfied with this answer. You don't seem to imply that it is impossible for some reason. But then we WOULD have wave functionals on classical objects (a restricted class of three-geometries?). Would you still expect the total Hilbert space to be the tensor product of this one and another HS for "matter", or is matter completely gone? (As you know, my main interest is entanglement.) Or would entanglement already arise when eliminating diffeomorphisms? Then this elimination would be genuinely quantum.

I had asked:
> Some people seem to believe that the knot structure remaining after
> eliminating diffeomorphisms, classified by discrete numbers n, is a
> quantum aspect.
>
>This is correct. Knot and link and graph classes label the quotient
>of states in the kinematical hilbert space by the action of all
>the spatial diffeomorphisms. They are quantum numbers in a quantum
>mechanical sense.

That's crucial. What would conceivably happen to them IF you can eliminate diffeomorphisms on the classical level? Usually, quantum numbers appear after one has given boundary conditions to the wave function. You never mentioned them, and I am particularly interested in the boundary condition at a = 0. However, what does the expansion parameter a even mean in a scale invariant theory? Have all exact Friedmann universes to be identified, regardless of their size? (Sorry for insisting on "stupid" questions!) Or can we hope that a becomes meaningful in an "effective" theory of gravity?

>Both the kinematical hilbert space and its quotient by spatial
>diffeos are genuine hilbert spaces. There is a superposition
>principle at both levels.

So you are saying that n is an index of Psi, not of its arguments, but what are these arguments of Psi if they are not certain three-geometries? We know from quantum mechanics that we may start from a continuous representation (|q>), but then replace it by a discrete one that does not have a classical counterpart. If this is analogous, your new basis would consist of genuine superpositions of three-geometries (which later decohere into Gaussians around three-geometries again).

Do these Psi_n individually obey the Hamiltonian constraint, or does this require their (further) linear combination?

I wrote:
> ... we have the Hamiltonian constraint H Psi = 0, and time
> dependence is lost. (The physical meaning of this formal result
> should crucially depend on the question whether and in what sense the
> wave function describes physical states or is "just a tool", as Bill
> Unruh claimed.)
>
>I agree. And the good news is that all the various points of view about
>the meaning of the hamiltonian constraint and its solutions can be tested
>in loop quantum gravity. We have explicit solutions (and infinite numbers
>of them) to all the constraints.

Do they mean different "possible" quantum universes, or is there a further boundary condition for THE universe? How can you "test" them? Is there any chance of interpreting them?

>In the histories, spin foam formalism one
>has explicit constructions of both projectors onto the kernel of all the
>constraints and of genuine evolution operators defined with respect to
>different notions of physical time (such as the spacetime volume in a
>history interperlating between an initial and a final state.)

Does volume have meaning if diffeomorphisms are identities?

> So all the
>different ideas about this deep problem can be studied in detail now in
>the real theory, and not just in models.

Are you referring to mini superspace models? (And may I replace "real theory" by "full theory"?)

>For this reason I think
>many of us (certainly those I mentioned) favor some version of
>a relational quantum theory.

Does "relational" mean diffeo invariance?

I had asked:
> On a Friedmann sphere, one may also eliminate gauge degrees by
> restriction to tensor modes only. However, this "field"
> representation appears much easier to interpret -- for example in
> terms of "mini superspaces". So here is my second major question: is
> there anything in loop theory that is equivalent to the hyperbolic
> WDWE in terms of ln a, the log of the expansion parameter.
>
>Yes, one can discuss something like this in detail, ...
>But even though one can go some distance towards this idea in the full
>theory, and hence recover a time variable from a certain set of
>explicit and exact solutions to all the constraints, as in Soo's
>work, I think the coordinate only ...

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.

>... has all the properties one would
>like a time coordiante to have in the semiclassical limit. So my
>own view is that this is not the full solution to the issue of time
>(see the above paper and my previous post on this issue.) But if someone
>wants to push that point of view in the full theory, this is a place
>to start.

I wrote:
> Claus Kiefer, using ideas of Banks and of Halliwell and Hawking, was able
> to recover an effective time dependent Schroedinger equation using
> this approach [perturbed Friedmann models] ...
>
>When the cosmological constant is positive, there is an exact
>state whose semiclassical limit is DeSitter,

Is the cosmological constant necessary because you have only vacuum solutions in loop theory? That would seem to be a severe limitation when compared with "effective" models.

Trying to come to a conclusion, would you agree that the WDWE, acting on a functional of three-geometries, may well be an excellent effective theory -- appropriate for quantum cosmology except very close to the singularity? All questions of interpretation of quantum theory seem to remain untouched.

Sorry to bother you so much, but some stuff may be of interest for the Wheeler forum.

Best regards, Dieter

__________________________________________________--

SMOLIN REPLIES:

Dear Dieter,

Thanks for your very thoughtful response. I'll do my best to reply:

On Wed, 27 Feb 2002, H.Dieter Zeh wrote:

> Dear Lee,
>
> thank you very much for your informative answers. Let me once more
> emphasize that I agree with your criticism of string theory
> approaches. Quantum theory must be fully quantized for consistency
> (even it is treated as an "effective theory" -- such as QED). I am
> not quite sure though that we entirely agree about the meaning of
> quantization (and this question would establish a relation of our
> discussion to some former contributions on this forum).
>

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.

> My questions aimed primarily at the appropriateness of a
> Wheeler-DeWitt wave function. Its "stage" (the classical
> configuration space after hypothetically solving the momentum
> constraints) is assumed to be the space of coordinate-free
> three-geometries. Quantization in my sense then means to admit all
> superpositions of them as physical states. This is the quantum theory
> that has been confirmed by many experiments. For example in QED,
> experimentalists were able to construct "mesoscopic Schroedinger
> cats", and even to observe their decoherence into the quasi-classical
> "field" states that we usually observe. Erich Joos pointed out long
> ago how superpositions of different three-geometries would similarly
> decohere through entanglement with matter (which CAN be formulated in
> terms of a WDW wave function).

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

> 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.

>
> >The local gauge transformations are solved BEFORE quantization.
>
> Fine, but is this the same as reducing the three-metric (h_kl) to an
> abstract three-geometry?
>

It is analalgous to that, but it is not the same. The reason is that we have a hold of a complete algebra of coordinates on the solution space to the constraints.

> >This is done by constructing a classical observables algebra, called the
> >loop algebra, which is locally gauge invariant. It is composed of
> >Wilson loops and corresponding momentum observables.
>
> "Observables" are usually regarded as a quantum concept (quantities
> which do not necessarily "possess" values). In a classical theory,
> everything can be described by "beables" (to use John Bell's term).
> This may be a matter of words, but could be misunderstood. Presumably
> you mean something like the classical em FIELDS in contrast to their
> potentials? Then we also have the loop integrals of A, which are
> nontrivial on not simply connected spaces. Our (macroscopic) space
> seems to be always simply connected, unless you "exclude" the
> sources. This may be different on the Planck scale, where we may
> hope, conversely, to explain sources by topology. Is this what you
> have in mind when you are discussing (only?) vacuum solutions? This
> is very fascinating, but as yet somewhat hypothetical (in spite of
> all mathematical progress).

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

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. Thus, the queston arises, how to describe and coordinatize the space of gauge fields, extended to include those that are not differentiable or smooth. This can be worked out in detail and is part of the answer to how we construct a space of states on gauge fields that is invariant under the action of diffeomorphisms. The point is that because the measure in the theory is dominated by non smooth configurations, one cannot assume that loop integrals are well approximated for small loops by the local fields and their derivatives. So one must coordinatize the space of gauge fields by loop integrals (in the non-abelian case Wilson loops) in order to construct the action of the diffeos. States of interest are then constant under the action of the diffeos including those that change the size of a loop-so they are exactly not valued on field configurations that are smooth enough for small loop integrals to be approximated by fields.

So, to put the technicalities aside YES, it is exactly as if space is "full of holes" at the Planck scale. The result is that the fields can behave as if the short distance excitations see something with many fewer points than a manifold, and indeed, in a certain sense the hausdorf or scaling dimension is less than 3 below the planck scale- it is closer to 1. This is how the theory manages to be finite and how the theory avoids the problem of PERTURBATIVE non-renormalizablity. The problem with the naive perturbation theory is it assumes that the states are dominated by smooth configurations at all scales. But the explicit detailed construction of the state space shows that this is NOT TRUE. What I've just sketched is the key idea behind why this happens.

> > One then makes
> >a choice of representation of this algebra on a Hilbert space.
>
> Do you have a Hilbert space before quantization, or are you just
> POSTULATING it in this way? If you say that you "really" want to
> admit all superpositions, then we seem to agree on quantization.
>

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

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.

> >In quantum gravity, as opposed to an ordinary gauge theory, one choices a
> >representation which does not depend on any background metric. This is
> >called the kinematical Hilbert space.
>
> Seems to be the same for the WDWE (with three-geometries used as the basis).
> Again, that is a FORMAL construction. 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.

But in loop quantum gravity, the construction of the state space both before and after applications of the diffeos is not formal, it is precisely understood. The result is that it is possible to construct detailed regularization procedures that result in finite operators that represent quantities with real geometrical and physical meaning. Thus, one can get real physical predictions, such as the spectra of the area and volume operators.

So there is not, so far as I understand it, a choice here. If one wants to work with the real diffeomorphism invariant quantum field theory, one has so far at least only loop quantum gravity. The older stuff (3 metrics mod diffeos is a formal construction only, its detailed realization is loop quantum gravity.

> >One then defines the action of the spatial diffeomorphisms on this
> >kinematical hilbert space, which is found to be anomolie free.
>
> What does that mean?

It means one has a precise unitary representation of the full group of diffeomorphisms of the 3 manifold one strated with on this kinematical hilbert space.

>
> > One can then find the subspace of states which are invariant under all
> >spatial diffeo's. Thus, one solves the spatial diffeo constraints
> >ON THE KINEMATICAL HILBERT SPACE, that is AFTER quantization.
> >You might ask if these proceedures could be done in an other order.
> >With respect to local gauge tranformations: yes ... The result is the same.
> >I don't think we know how to impose the spatial diffeos classically and
> >then quantize.
>
> I am satisfied with this answer. You don't seem to imply that it is
> impossible for some reason. But then we WOULD have wave functionals
> on classical objects (a restricted class of three-geometries?). Would
> you still expect the total Hilbert space to be the tensor product of
> this one and another HS for "matter", or is matter completely gone?
> (As you know, my main interest is entanglement.) Or would
> entanglement already arise when eliminating diffeomorphisms? Then
> this elimination would be genuinely quantum.

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.

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

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.

> I had asked:
> > Some people seem to believe that the knot structure remaining after
> > eliminating diffeomorphisms, classified by discrete numbers n, is a
> > quantum aspect.
> >
> >This is correct. Knot and link and graph classes label the quotient
> >of states in the kinematical hilbert space by the action of all
> >the spatial diffeomorphisms. They are quantum numbers in a quantum
> >mechanical sense.
>
> That's crucial. What would conceivably happen to them IF you can
> eliminate diffeomorphisms on the classical level? Usually, quantum
> numbers appear after one has given boundary conditions to the wave
> function. You never mentioned them, and I am particularly interested
> in the boundary condition at a = 0. However, what does the expansion
> parameter a even mean in a scale invariant theory? Have all exact
> Friedmann universes to be identified, regardless of their size?
> (Sorry for insisting on "stupid" questions!) Or can we hope that a
> becomes meaningful in an "effective" theory of gravity?
>

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.

There are other theories which are scale invariant, such as topological quantum field theories.

In the spatially compact case, the volume of the 3 geometry, V, is a completely well defined diffeomrphism invariant operator. So there is a complete counterpart to a. V has a discrete spectrum. The smallest non zero eigenvalue is a number of order one times the Planck volume. The spectrum is complicated, but computed and tabulated for small volumes.

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.

The regularity conditions necessary to get the area and volume quantization are not at volume =0. Instead they are the conditions on the measure necessary to get a well defined unitary represetantation of the 3 diffeomorphisms on the kinematical hilbert space. So they seem imposed by the physical requirement that quotient of the kinematical hilbert space by the diffeos by well defined.

I do not know what would happen if we could mod out by the diffeos classically and then quantize, but if this could be done it would have to include non-smooth fields, for the same reason that path integrals have support on non differentiable fields.

> >Both the kinematical hilbert space and its quotient by spatial
> >diffeos are genuine hilbert spaces. There is a superposition
> >principle at both levels.
>
> So you are saying that n is an index of Psi, not of its arguments,
> but what are these arguments of Psi if they are not certain
> three-geometries? We know from quantum mechanics that we may start
> from a continuous representation (|q>), but then replace it by a
> discrete one that does not have a classical counterpart. If this is
> analogous, your new basis would consist of genuine superpositions of
> three-geometries (which later decohere into Gaussians around
> three-geometries again).

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. But there are no well defined non-vanishing diffeo invariant operators that measure 3 geometry at scales smaller than the planck scale. Again, let me emphasize these are results, they come out of the construction of the theory and they seem forced by the combination of realizing the commutation relations for the loop observables and having a well defined action of the spatial diffeos by unitary operators, so one can mod out by them.

> Do these Psi_n individually obey the Hamiltonian constraint, or does
> this require their (further) linear combination?
> An infinite dimensional subspace is spin net by spin net in the kernal of the hamiltonian constraint, but this seems to be physically trivial-it contains spin nets which are also in the kernal of the volume of the universe. Besides these, one has to take infinite linear combinations to get solutions, but infinite dimensional families of these have been constructed in detail.

> I wrote:
> > ... we have the Hamiltonian constraint H Psi = 0, and time
> > dependence is lost. (The physical meaning of this formal result
> > should crucially depend on the question whether and in what sense the
> > wave function describes physical states or is "just a tool", as Bill
> > Unruh claimed.)
> >
> >I agree. And the good news is that all the various points of view about
> >the meaning of the hamiltonian constraint and its solutions can be tested
> >in loop quantum gravity. We have explicit solutions (and infinite numbers
> >of them) to all the constraints.
>
> Do they mean different "possible" quantum universes, or is there a
> further boundary condition for THE universe? How can you "test" them?
> Is there any chance of interpreting them?

There is no evidence in the spatially compact case for a unque 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?

>
> >In the histories, spin foam formalism one
> >has explicit constructions of both projectors onto the kernel of all the
> >constraints and of genuine evolution operators defined with respect to
> >different notions of physical time (such as the spacetime volume in a
> >history interperlating between an initial and a final state.)
>
> Does volume have meaning if diffeomorphisms are identities?
>

Yes, here I mean spacetime volume and that is an invariant of both spacetime diffeomorphisms and, in the case that the spacetime has boundaries which are spatial manifolds, the diffemorphisms of the boundaries.

> > So all the
> >different ideas about this deep problem can be studied in detail now in
> >the real theory, and not just in models.
>
> Are you referring to mini superspace models? (And may I replace "real
> theory" by "full theory"?)

Yes the first question. By full theory I mean a full quantum field theory, with all degrees of freedom and all gauge invariances and diffeomorphisms taken into account. I would reserve the word "real theory" for the theory found to correspond to experiment. But you could mean the "real" full quantization of GR, in which case yes.

> >For this reason I think
> >many of us (certainly those I mentioned) favor some version of
> >a relational quantum theory.
>
> Does "relational" mean diffeo invariance?

No, relational here refers to a set of ideas invented so far as I know by Crane, Rovelli and myself, and developed further by Markopoulou in which there is not one hilbert space for the universe as a whole, but a family of hilbert spaces, each describing information that an observer living in one part of the universe could have about other parts of the universe. See papers by Rovelli on "relational quantum mechanics" and Markopoulou on "quantum causal histories."

> I had asked:
> > On a Friedmann sphere, one may also eliminate gauge degrees by
> > restriction to tensor modes only. However, this "field"
> > representation appears much easier to interpret -- for example in
> > terms of "mini superspaces". So here is my second major question: is
> > there anything in loop theory that is equivalent to the hyperbolic
> > WDWE in terms of ln a, the log of the expansion parameter.
> >
> >Yes, one can discuss something like this in detail, ...
> >But even though one can go some distance towards this idea in the full
> >theory, and hence recover a time variable from a certain set of
> >explicit and exact solutions to all the constraints, as in Soo's
> >work, I think the coordinate only ...
>
> 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 (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.) Now the version of time you are interested in related to time as observed within a particular spacetime.

The first kind of time is what you must think of formally in a full quantum theory of gravity, because there are no single spacetimes, only quantum amplitudes for paticular histories to be realized. So one wants a time coordinate on the phase space, not on individual histories.

At the same time, what we observes seem to observe, at least in the classical limit, has to do with the past of particular events in a particular history.

So it is very important and interesting to find out how the two kinds of time are related to each other. If they cannot be then there is a fundamental issue with how time is represented in the quantum theory of gravity.

> >... has all the properties one would
> >like a time coordiante to have in the semiclassical limit. So my
> >own view is that this is not the full solution to the issue of time
> >(see the above paper and my previous post on this issue.) But if someone
> >wants to push that point of view in the full theory, this is a place
> >to start.
>
>
> I wrote:
> > Claus Kiefer, using ideas of Banks and of Halliwell and Hawking, was able
> > to recover an effective time dependent Schroedinger equation using
> > this approach [perturbed Friedmann models] ...
> >
> >When the cosmological constant is positive, there is an exact
> >state whose semiclassical limit is DeSitter,
>
> Is the cosmological constant necessary because you have only vacuum
> solutions in loop theory? That would seem to be a severe limitation
> when compared with "effective" models.
>

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

A non zero cosmological constant may not be necessary in general, but it is necessary to get that particular set of results. Given that it dominates the theory at low energies, it is not surprising that a non-zero cosmological constant can have a strong effect on the behavior of the quantum state at all scales. I suspect that it does need to be included to get a good low energy limit, but that is a conjecture at this point.

> Trying to come to a conclusion, would you agree that the WDWE, acting
> on a functional of three-geometries, may well be an excellent
> effective theory -- appropriate for quantum cosmology except very
> close to the singularity? All questions of interpretation of quantum
> theory seem to remain untouched.

The first is a detailed question for research. I believe that the papers of Soo and Bojowald provide a good starting point for the questions you are raising.

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

My own view is that relational quantum theory-and more particularly Markopoulou's quantum causal histories, QCH, give us a good basis for attaching and resolving these problems. QCH are compatible both with the details of loop quantum gravity (in the causal spin foams framework) and with a certain form of the holographic principle, which we call the weak holographic principle (see hep-th/9910146, hep-th/9904009, gr-qc/9811053, hep-th/9912137.

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.

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.

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.

Thanks again so much for your comments and thoughts,

Best wishes, Lee

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Published   2002.02.28