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Josh Willis replies to comment on LQG and the diffeomorphism group

  1. Mar 12, 2004 #1

    marcus

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    There has been a lot of internet discussion of a paper by Ashtekar, Fairhurst, and Willis called "Quantum gravity, shadow states, and quantum mechanics" gr-qc/0207106, and one of the authors, Josh Willis, replied on SPR today (in the thread "LQG and diffeomorphism group cocycles")

    Here is a representative exerpt of the longer post:

    --------exerpt from Willis post------

    Well, since Urs keeps referring to this paper I co-authored, I feel I
    ought to respond with what I see.

    So I will start by replying to the remarks about gr-qc/0207106, and
    then work backwards towards some more general remarks about the
    relationship between group averaging and anomalies, as I see it. This means I take the quoted article in reverse order. My apologies if this is confusing, but I think in the long run it will be clearer.

    So, let me start with:

    Urs Schreiber wrote:
    > Sometimes LQG papers like gr-qc/0207106 are mentioned which supposedly show
    > that the LQG-like 'form of quantization' reproduces ordinary quantization in
    > some limit. But having a closer look at this paper shows that this _only_
    > works when the usual quantum corrections are copied to the LQG-formalism
    > (lower half of p.14). This is however not the case in the 'LQG-string' or
    > the LQG treatment of the spatial diffeo constraints of 3+1d gravity.

    I do not understand your assertion that the "quantum
    corrections" are copied into the model system we study, and that this
    is somehow very different from what is done with, for instance, the
    diffeomorphism constraint in LQG.

    So, for people who haven't read page 14 of gr-qc/0207106, let me
    summarize quickly what is going on.

    And let me start by saying that page 14 of that paper is the wrong
    place to be looking to begin with: there we are finding candidate
    semiclassical states, not considering commutation relations. Instead,
    look at pages 7 and 8, to start.

    We are looking at the one-dimensional point particle in quantum
    mechanics. We have not yet considered any particular Hamiltonian, but
    are instead at this point in the paper looking only at the canonical
    commutation relations.

    One normally learns these as:

    {q,p} = 1

    and then seeks in the quantum theory for self-adjoint operators
    \hat{q} and \hat{p} that satisfy the corresponding relations:

    [\hat{q},\hat{p}] = i\hbar

    But these commutation relations among the p's and q's of course imply
    commutation relations among the exponentiated operators, which---if
    \hat{q} and \hat{p} are self-adjoint---will be unitary. This is the
    relation:

    (*) U(l)V(m) = exp{-ilm}V(m)U(l)

    where:

    U(l) = exp{il\hat{q}}
    V(m) = exp{im\hat{p}}

    The relation (*) is that of the Weyl-Heisenberg algebra, and what we
    do in our paper is look at a non-standard unitary representations of
    this algebra, rather than looking at a self-adjoint representation of
    the CCR. The particular representation we look at is motivated by
    analogy with LQG, and the main point of the paper is to look at the
    low-energy limit of this theory, using techniques that it is hoped
    will be relevant for LQG. However, as that does not seem to be what
    the questions are about at the moment, I will not say more about that
    here.

    And this is the key point: the relation (*)---including the factor of
    exp{-ilm}, which is what in other places on the net Urs seems to think
    is put in by hand, i.e. by "knowing" what the quantum theory should
    be---is in fact dictated *classically* by the Poisson algebra of the
    basic observables. Any representation by unitary operators which
    purports to be a canonical quantization *must* have this relationship,
    and in particular the representation we consider does (as of course
    also does the standard Schroedinger representation)...

    --------end quote-------
     
    Last edited: Mar 12, 2004
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  3. Mar 12, 2004 #2

    marcus

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    It is fascinating that this discussion should relate in one way or another the spatial diffeomorphism group in LQG, and how LQG implements the diff-invariance found in General Relativity.
    Especially interesting, I think, since Rovelli's treatment does not use the diffeomorphism group but rather the "almost smooth" group Diff*.
    One result of extending the idea of diff-invariance is "separability"---making the kinetic state space of LQG have a countable basis.
    A new paper of Rovelli and Fairbairn underscores this shift in the way spatial diff-invariance is handled and it seems to have interesting mathematical consequences ("Separable Hilbert space in LQG" gr-qc/0403047).

    This is an area where LQG is in flux and where there are deep differences between Rovelli's and for example Thiemann's approaches, some of which is described in section 4.2 of the Rovelli/Fairbairn paper. It has become difficult to say frankly just what a "LQG-like" handling of the diffeomorphism constraint should be, or if there is one unique way.

    Several people have joined the discussion at SPR, and indeed there has been a lot of talk about this same thing here at PF too, not to mention other internet forums.
     
    Last edited: Mar 12, 2004
  4. Mar 12, 2004 #3

    selfAdjoint

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    I like his characterization of group averaging, which you may remember we all made some heavy weather over. It STARTS with unitary products. We finally got there, and it's nice to know we weren't wrong.
     
  5. Mar 13, 2004 #4

    marcus

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    Urs reply today on SPR (brief sample)

    Unfortunately, since these SPR posts are long, I can only copy representative samples. Later I will try to include links to the Cornell SPR archive. I copied about a third of Josh previous post and will show an even smaller fraction of Urs here, just to give a sample.

    Today Urs replied with noteworthy civility and clarity. It raises hopes in me that this could be a very helpful discussion to listen to.
    Here is a brief sample of today's post by Urs on SPR:

    ----quote---
    ...I do understand why the Weyl algebra (equation III.1) in your paper looksthe way it does, including that factor. I understand that this factor is not put in by hand. My point is that another factor is.

    > How does one see this? By (as Urs has mentioned elsewhere) the
    > Campbell-Baker-Hausdorff theorem, which lets me calculate what (*)
    > should be, entirely in terms of iterated commutators of p's and q's.
    > For the CCR, this is in fact the easiest way to go, because the CBH
    > series stops after the first commutator.

    I understand this and totally agree. But I think for those following this one should emphasize that the BCH theorem applied to exp(p)exp(q) can only serve to motivate certain steps in the LQG construction, because the p's and q's do not exists (not both at least) in your framework.

    > But for more complicated Lie algebras, as for example the
    > diffeomorphism algebra in gravity, the CBH series does not terminate
    > and this is not a terribly enlightening way to view the problem.
    >
    > But it's also not necessary, because a self-adjoint representation of
    > a Lie algebra exponentiates to a unitary representation of the Lie
    > group. So what one should check instead is that one has an honest
    > group representation, that is, that:
    >
    > U(g_1)U(g_2) = U(g_1 * g_2)
    >
    > And this *does* hold for the diffeo group in LQG.

    Yes, as it does for Diff(S^1) x Diff(S^1) in Thomas Thiemann's 'LQG-sting'. By construction. I.e. operators U can be found which do satisfy this relation. I do understand and agree that these operators can be constructed.

    The problem is, that these operators don't capture the usual quantum
    effects. They do not come from what ordinarily is called 'canonical
    quantization'. That's because the ordinary prescription of 'canonical
    quantization' at some point says that we are to promote the classical
    canonical coordinates and momenta to self-adjoint operators on some Hilbert space. This step is explicitly violated in LQG, since half of these objects are not represented as operators at all.

    So with this step violated, something else has to be found to replace it. Among other things, I claim that this replacement introduces a large ambiguity.

    Namely whenever there is now an object in the classical theory in the
    enveloping algebra of p and q, you don't know precisely how to quantize it (even modulo operator ordering issues) since one of p and q is not represented on the Hilbert space.

    I think that this is a problem both in gr-qc/0207106 as well as in the
    'LQG-string'. Here is why:....

    ----end quote, read the rest on SPR---
     
  6. Mar 13, 2004 #5

    marcus

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    It's great to have some of the same topics revisited in a slightly different light. I am hoping for a better understanding this time thru. Must say I do like Willis style, calm, and the care with which he organizes and explains.
     
  7. Mar 13, 2004 #6

    jeff

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    Re: Urs reply today on SPR (brief sample)

    This is nonsense and you'll forgive me if I don't trust your judgement of what a "representative sample" would be. Here's the entire post:

    "Josh Willis" <jwillis@gravity.psu.edu> schrieb im Newsbeitrag
    news:c2slkk$g2k$1@f04n12.cac.psu.edu...
    > Well, since Urs keeps referring to this paper I co-authored, I feel I
    > ought to respond with what I see.

    Many thanks for your answer!

    > I do not understand your assertion that the "quantum
    > corrections" are copied into the model system we study, and that this
    > is somehow very different from what is done with, for instance, the
    > diffeomorphism constraint in LQG.

    I'll try tomake my concern more precise below.

    > (*) U(l)V(m) = e^{-ilm}V(m)U(l)

    [...]

    > The relation (*) is that of the Weyl-Heisenberg algebra, and what we
    > do in our paper is look at a non-standard unitary representations of
    > this algebra, rather than looking at a self-adjoint representation of
    > the CCR.

    Ok.

    > And this is the key point: the relation (*)---including the factor of
    > e^{-ilm}, which is what in other places on the net Urs seems to think
    > is put in by hand,

    No, sorry, that's a misunderstanding of what I was saying. I am grateful
    that you took the time to answer so that I can try to clarify this.

    I do understand why the Weyl algebra (equation III.1) in your paper looks
    the way it does, including that factor. I understand that this factor is not
    put in by hand. My point is that another factor is.

    > How does one see this? By (as Urs has mentioned elsewhere) the
    > Campbell-Baker-Hausdorff theorem, which lets me calculate what (*)
    > should be, entirely in terms of iterated commutators of p's and q's.
    > For the CCR, this is in fact the easiest way to go, because the CBH
    > series stops after the first commutator.

    I understand this and totally agree. But I think for those following this
    one should emphasize that the BCH theorem applied to exp(p)exp(q) can only
    serve to motivate certain steps in the LQG construction, because the p's and
    q's do not exists (not both at least) in your framework.

    > But for more complicated Lie algebras, as for example the
    > diffeomorphism algebra in gravity, the CBH series does not terminate
    > and this is not a terribly enlightening way to view the problem.
    >
    > But it's also not necessary, because a self-adjoint representation of
    > a Lie algebra exponentiates to a unitary representation of the Lie
    > group. So what one should check instead is that one has an honest
    > group representation, that is, that:
    >
    > U(g_1)U(g_2) = U(g_1 * g_2)
    >
    > And this *does* hold for the diffeo group in LQG.

    Yes, as it does for Diff(S^1) x Diff(S^1) in Thomas Thiemann's 'LQG-sting'.
    By construction. I.e. operators U can be found which do satisfy this
    relation. I do understand and agree that these operators can be constructed.

    The problem is, that these operators don't capture the usual quantum
    effects. They do not come from what ordinarily is called 'canonical
    quantization'. That's because the ordinary prescription of 'canonical
    quantization' at some point says that we are to promote the classical
    canonical coordinates and momenta to self-adjoint operators on some Hilbert
    space. This step is explicitly violated in LQG, since half of these objects
    are not represented as operators at all.

    So with this step violated, something else has to be found to replace it.
    Among other things, I claim that this replacement introduces a large
    ambiguity.

    Namely whenever there is now an object in the classical theory in the
    enveloping algebra of p and q, you don't know precisely how to quantize it
    (even modulo operator ordering issues) since one of p and q is not
    represented on the Hilbert space.

    I think that this is a problem both in gr-qc/0207106 as well as in the
    'LQG-string'. Here is why:

    In the text leading to equation (IV.5) of gr-qc/0207106 you are looking for
    a way to express the notion of 'coherent state' in the LQG-like language,
    where you characterize a coherent state by its property of being an
    eigenstate of the annihilation operator

    a ~ x + ip

    (which is one of several possibilities of characterizing coherent states).

    This is an object in the algebra generated by p and q. It happens to have no
    operator ordering problems so the usual quantization of this guy is obvious.

    But now in your framework p is not representable, so something else has to
    be done. Some object in the algebra of operators that you do have must be
    identified as being the 'quantization' of the annihilation operator.

    Due to the Weyl-algebra context, what you really want is the exponentiation
    of the adjoint of the annihilation operator, i.e. an analog of

    exp(a^dag) .

    This has to be expressed somehow using the operators x and V that do exist
    on your Hilbert space. You do this by looking at the usual quantization of
    this object and read off the expression above equation (IV.5) from this,
    e.g.

    exp(a^dag) -> exp(x)V(-1/2)e^(-1/4) .

    But why don't you use for instance

    exp(a^dag) -> exp(x)V(-1/2)

    or

    exp(a^dag) -> exp(x)V(-1/2)e^(42) ?

    This is the factor which I am talking about and which is put in by hand. It
    is _not_ put in by hand in the _usual_ quantization, because there the
    e^(-1/4) comes from the BCH theorem. But in your rep there is no p and q
    which could enter the exponent in the BCH theorem. You are just copying this
    plausible looking result to your representation.

    And from reading the paper I got the impression that you would agree that
    here you are fixing an ambiguity by hand. Because on p.14 it says:

    "Collecting these ideas motivated by results in the Schroedinger
    representation, we are now led to seek the analog...".

    I do agree that this is indeed the most natural choice for this simple
    system. My problem is that in more difficult cases, where there is mayber no
    working Schroedinger representation from which ideas can be motivated, how
    do you fix such ambiguities then?

    I noted that in the 'LQG-string' as well as for the diffeo-constraints of
    3+1d LQG gravity the analog of this ambiguity is fixed by using ideas
    motivated by results not of the Schroedinger rep but of the classical
    Poisson algabra.

    Here is why:

    When quantizing the Nambu-Goto action by LQG-like methods we are faced with
    a problem very similar to the representation problem of the annihilation
    operator "a" in the above example. Namely, there we have the classical
    Virasoro constraints L_m, which are objects bilinear in the canonical p and
    q.

    In the standard quantization these p and q are promoted to operators and no
    matter how you deal with the operator ordering ambiguity the resulting
    quantum versions of the L_m feature the anomaly, which is a commutator
    effect very much like the factor discussed above.

    But now in Thomas Thiemann's 'LQG-string' quantization, as in your paper
    above, the p and q are not represented as operators. Hence he has to look
    for an _analog_ of the usual quantization, much like you construct an analog
    of exp(a^dag), as discussed above.

    So Thomas Thiemann considers the classically exponentiated Virasoro
    generators and models his quantum operators U(phi) on these. He _could_ have
    proceeded along the lines of your paper and "collected ideas motivated by
    the Schroedinger representation to seek an analog of" the exponentiated
    Virasoro generators in the usual quantization instead. This way he would
    have had the anomaly (in terms of the group instead of the algebra, but
    still).

    But he chooses not to. He instead collects ideas motivated by the classical
    action of the Virasoro generators. And that's how the anomaly disappears.

    I'd very much enjoy hearing your opinion on these assessments. If I am
    wrong, I'd very much want to be corrected. Thanks.
     
  8. Mar 14, 2004 #7

    Urs

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    I very much appreciate it that Josh Willis has posted a reply. Many thanks to him!

    I haven't got an answer from A. Ashtekar so far, which I had asked by private email about his opinion on the LQG string. But he will give a talk here at the DPG Symposium tomorrow here in Ulm. With a little luck I find a chance to speak to him during this weak. Let's see.
     
  9. Mar 14, 2004 #8

    marcus

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    I am happily looking forward to our hearing more from Urs about the
    Spring Conference of the German Physical Society in Ulm----where he must be at this moment. Here is an exerpt from Urs post on CT:

    ---quote---
    I am on my way to the spring conference of the German Physical Society...

    Frühjahrstagung der Deutschen Physikalischen Gesellschaft

    ...
    ...

    Ok, who else will bet there? There are many LQG people. A. Ashtekar will give a general talk on LQG for non-specialist. Bojowald of course will talk about what is called ‘Loop Quantum Cosmology’. With a little luck I find an LQGist willing to discuss the ‘LQG-string’ with me.

    I would also like to talk to K.-H. Reheren, who has announced a talk on algebraic boundary CFT, about Pohlmeyer invariants, but I am not sure if he considers it worthwhile talking to me… :-/
    ----end quote---
     
  10. Mar 15, 2004 #9

    Urs

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    The latest news from my quest to understand LQG can be found here.
     
  11. Mar 15, 2004 #10

    marcus

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    Urs report, part 1

    Urs report from the DPG Spring Conference is exciting and I copy here his reports from Sunday and Monday sessions:
    --------quote from Urs CT post---------
    Posted by: Urs Schreiber on March 14, 2004
    Re: Sunday

    Heard a very interesting and even moving, while unostentatious, talk by C. Yang about Albert Einstein, his ideas and his life.
    Encountering Yang reminded me of the joke

    Q: ‘Who invented the sledge hammer?’

    A: ‘Mr. Sledge.’

    It is almost like meeting somebody called ‘car’ or ‘lightbulb’, if you know what I mean. It’s kind of amazing.

    Before the talk I met Rüdiger Vaas, who is science journalist for the german popular science journal Bild der Wissenschaft and is working a lot on reporting about research in quantum gravity. I had first met him at the Strings meet Loops symposium last year. His report on ‘Strings meet loops’ will appear in the next issue of BdW.

    He tells me that more than half of the 12 pages long article will be concerned with LQG and in particular with M. Bojowald and his ‘Loop Quantum Cosmology’. Considering that also the recent issue of Scientific American had an article by Lee Smolin on LQG, which of course can be found translated in ‘Spektrum der Wissenschaft’, and considering that Spektrum and BdW are the two leading german journals for popular science, this gives an impressive amount of public attention for LQG here in Germany. Maybe there is a general tendency. The DPG Symposium here in Ulm is clearly dominated by LQG contributions. Kind of amazing when one is involved in the current discussion about the conceptual viability of LQG.

    I mean, ok, it is not established that string theory will survive experimental tests and everybody is free to believe that it will not. But at least it is clearly about theoretical physics. All kinds of concepts in string theory will definitely survive in and enrichen theoretical physics even if it might turn out that gravitons are not excitations of some string.

    But currently I am not so sure that LQG is even theoretically about physics.

    But of course Bojowald’s claim to be able to connect quantum gravity with experimental MBR data is enchanting.

    It would be great if for instance string cosmology could come up with a similarly nice cosmological model which removes and clarifies the initial singularity. Most of what I have heard so far about string cosmology was pretty disappointing. Veneziano’s pre-big bang model and similar scenarios always reminded me of the ‘then a miracle occurs’ mechanism. The interesting transition of the two classical branches remains a mystery.

    Maybe Jacques Distler can increase my faith in string cosmology by reporting interesting results from the conference Cosmology and Strings that he mentions in his latest musings.
    ---to be continued---
     
    Last edited: Mar 15, 2004
  12. Mar 15, 2004 #11

    marcus

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    Urs report part 2

    Urs report is spritely and entertaining. One sees what the purpose of these conferences is (for the grad student----perhaps the most essential perspective)
    I quote in full:

    -----Urs report for Monday----
    Monday
    This has been a very intensive day.

    When I arrived (late) in A. Ahstekar’s talk this morning I had only five hours of sleep behind me, but I made it to the first coffee break without major casualties and resupplied myself with caffeine.

    Ashtekar gave a general introductory lecture on LQG. Afterwards Peres talked about ‘Quantum Information and Relativity Theory’, being concerned with problems such as a quantum measurement in one frame may come before the measured event in another frame.

    After the coffee break Clifford Will gave an extremely enjoyable talk on experimental tests of gravity. As I have said here, I learned that LISA will not be able to see primordial gravitational wave backgrounds due to the noise made by binary star systems in our galaxy.

    After lunch the parallel sessions started. I had a problem, because I wanted to listen to the NCG stuff which was parallel to a session in which Folkert Müller-Hoissen gave a talk. Now, Eric and I have spent a lot of time with extending and generalizing the work by Dimakis and Müller-Hoissen and I had never met these authors before, so I decided to ditch the NCG session and sit in on ‘Symmetries, Integrability and Quantization’. It was sort of interesting, though I later learned that this way I missed a talk about NCG on Lorentzian spacetimes, which I really regret to have missed. I’ll need to talk to those people in private tomorrow.

    Anyway, my hope was to get hold of Müller-Hoissen after the talks and get on his nerves by talking about discrete differential geometry. But unfortunately he was inolved in a discussion with somebody else about something else, which went on and on and on…

    Finally I decided not to wait any longer and ran to the lecture auditorium to catch at least the second half of C. Fleischhack’s talk on ‘Progress and Pitfallls of LQG’. This was a very technical talk with lots of formulas with a huge amount of indices and symbol decorations. I have made a photograph of the point where he puts on the transparancy which says that now the spatial diffeomorphism constraints are solved. I believe that the field of LQG would maybe profit from deemphasizing technical details at this point and instead emphasizing the big crucial point: The diffeomorphism constraints are ‘solved’ without imposing Dirac constraint quantization.

    I saw that Thomas Thiemann was in the audience, A. Ashtekar was, Hermann Nicolai was, and decided that it would be nice to continue our Coffee Table discussion about this point, which some people feel is a little problematic.

    So as the talk was over I asked the lecturer about this point. He answered that this method is simpler than Dirac quantization and also has the advantge that also ‘large’ diffeomorphisms can be included, i.e. those that cannot continuously be connected to the identity (such as coordinate reflections).

    When I began to argue that the method may be simpler but is not what Dirac tells us to do (for good reasons, like path integral and BRST formalism), A. Ashtekar approached me. He was very nice and helpful, as usual, and we went outside to further discuss things.

    He checked if I am the one who had pointed him to the Coffee Table discussion and told me that he didn’t answer my email partly because he found some statements of the Coffee Table discussion overly offending. I think he is right about that and highly appreciate that he still very patiently talked and listened to me. Many thanks to Abhay Ashtekar, indeed.

    First he said that the way LQG deals with the gauge constraints is not different from what one does in gauge theory. I replied that that has to do with the fact that the anomalies of the standard model happen to cancel, while it is not clear that those of canonical gravity would (without the ghost sector). I think he agreed.

    I suggested that maybe LQG should then perhaps try to handle the ghost-extended Einstein-Hilbert action instead of the pure EH action. At least for 1+1 dimensional gravity this does indeed remove the anomaly, as is well known. I got the impression that A. Ashtekar found this idea is maybe worth considering (but I am not sure).

    He told me that what he found problematic with the ‘LQG-string’ is that the method does not in any way seem to involve the fields on the background spacetime. I found this remark interesting, because it resonates with my own feelings concerning this point, which I have expressed here. A. Ashtekar hinted at some alternative approaches by himself and somebody else which are apparently under investigation, but I feel that I should not report on that here in public.

    After this very illuminating discussion I was asked by somebody if I am working on LQG, because he had seen me pipe up in Fleischhack’s talk. After I had answered that to the negative we exchagned personal and scientific identities, and I was delighted to meet in Thorsten Prüstel somebody working on - guess what - discrete field theory on Lorentzian graphs.

    We immediately had lots of things to talk about. I showed him Eric Forgy’s and mine pre-pre-print and he was very interested and invited me to give a talk about that at University of Hamburg. He himself is working on an interesting approach to get gravity from the nonunitary part of an extended gauge group on a graph field theory, roughly. I hope that when I am in Hamburg I will get the chance to learn more about that.

    We already had to hurry to get to the ‘Welcome Party’. There we happened to sit next to Prof. Kastrup from Aachen. Kastrup was the thesis’ advisor of two of the leading figures in current LQG, namely Thomas Thiemann and Martin Bojowald. We learned how Thomas Thiemann as a student originally wanted to work on string theory and was later convinced to look into LQG.

    I found it very interesting that Kastrup agreed with my assessment that LQG is not ‘canonical’ in the usual sense, because it does not represent the classical canonical coordinates and momenta as operators on a Hilbert space.

    Indeed, in the afternoon I had heard the very interesting talk by Kastrup about quantization of integrable systems in angle/action variables. This is an interesting and subtle issue of canonical quantization, which can already be studied for the ordinary harmonic oscialltor.

    The point is that by the naive correspondence rule one would think that, since the angle &omega; and the action S are a pair of canonically conjugate coordinates and momenta (like the ordinary q and p are, too) there should be self-adjoint operators &omega;^; and S^; which satisfy [&omega;^,S^]=i .

    But it is easy to convince oneself that this cannot work, which has to do with the fact that the angle is defined only modulo 2&pi; , or, equivalently, that &omega; and S do not provide global coordinates on phase space, because the origin has the usual coordinate singularity of polar coordinates in the plane.

    Kastrup showed how with taking much care one can instead construct two other operators K + and K - such that together with K 0 = S^ they give the Lie algebra of SO (1,2) and that from these the standard q^ and p^ can be reobtained. (This can be understood heuristically by thinking of the 2d phase space of the oscillator with the origin removed as a (‘light’-)cone.)

    He concluded by saying that, while being equivalent to the quantization with q and p , this could have experimental consequences in quantum optics. (I asked him about how this can be true, but unfortunately failed to understand his answer.)

    Anyway, this is a nice example for how subtle ordinary canonical quantization itself can already be.
    ------end quote--------
     
  13. Mar 15, 2004 #12

    selfAdjoint

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    Wow! I almost felt I was there. Thanks to you Marcus for posting it, and thanks to Urs for writing it!
     
  14. Mar 16, 2004 #13

    jeff

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    Did you guys know that audio and (in some cases) video of lectures can be downloaded at the ITP? For example:

    rovelli

    ashtekar

    baez
     
  15. Mar 16, 2004 #14

    Urs

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    Following the report on A. Ashtekar comments on the 'LQG-string', here is now the summary of an interview with H. Nicolai on LQG, anomalies and all that.
     
  16. Mar 16, 2004 #15

    marcus

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    This is turning out to be a very interesting conference
    (the German Physical Society Spring 2004) and includes even
    the world premier performance of an opera about Albert Einstein!
    I shall copy Urs CT post so that we have a complete account.

    One may surmise that the role of Einstein is sung by the tenor and that the bass-baritone sings the part of Michele Besso, his friend for over 50 years. They may thus conveniently sing duets.



    -----quote from Urs---
    Tuesday
    The talks today didn't interest me much (things like "Einstein and art") and I spent the time reading and answering my mail as well as preparing my own talk.

    But I did have lots of very valuable conversations.

    At lunch I met Hermann Nicolai and we had a long discussion about Pohlmeyer invariants, DDF invariants, LQG, the "LQG-string", diffeomorphism anomalies in string theory, string field theory, Matrix Models and prejudices in quantum gravity.

    To begin with, I was kind of surprised to learn that H. Nicolai, together with K. Peeters, is currently thinking about Pohlmeyer invariants himself. We discussed the known results regarding their quantization and I mentioned that I think that there is a solution to the apparent problems. H. Nicolai was interested and invited me to visit the AEI in April to talk about these ideas.

    Then of course we discussed the "LQG-string" and what can be learned from it about LQG itself. H. Nicolai said that he had hoped that the LQG people would find the anomaly for the string, which, as he said, he would have considered a breakthrough for the whole LQG field. But what has now actually been done, he said, reminded him more of certain artificial constructions in axiomatic field theory, which are also mathematically well defined but physically empty.

    I asked him about what this now means for full LQG and he said that, similarly, he would expect that there should be an anomaly and that he finds the constructions done in LQG problematic.

    I tried to understand how the same issue could be understood from within string theory, and he basically said that one would have to understand closed string field theory in order to tackle this question. But currently a satisfactory definition of closed string field theory is of course not known.

    I said that I am wondering if we cannot learn anything in this regard from BFSS/IKKT Matrix Models. The authors of the IKKT/IIB model at least claimed that the permutation subgroup of the full U (N-> oo) gauge group of the model becomes the diffeomorphism group in the limit. Heuristically, we can think of the matrices as describing discrete spacetime points and the permutation subgroup permutes these points, so is related to diffeos in some sense.

    Hermann Nicolai didn't know the details of this claim (me neither :-) and remarked that most permutations would give rather pathologic diffeos in the continuum limit. In any case, my uneducated guess is that the issue of diffeo anomalies and the like is hidden in the N -> oo limit of the matrix models, which is not well understood at all, as far as I know.

    Since the time for the afternoon talk sessions was drawing near we stopped at this point. But as I was about to leave the cafeteria K.-H. Rehren approached me. I very much appreciated this, because from our previous online conversation I had gotten the, apparently wrong, impression that he wasn't interested in my comments on Pohlmeyer invariants.

    I was delighted that we immediately sat down, pulled out pens and paper and began doing algebraic calculations. I think that in a couple of minutes we could clarify issues that would have taken weeks by email, at the previous speed.

    But then we really had to hurry, because I was the one supposed to give the next talk!

    I am not sure if it is a good or a bad sign, but at this DPG spring conference of the faculties "Gravitation and Theory of Relativity" and "Theoretical and Mathematical Foundations of Physics" my talk is apparently the only one directly concerned with strings! But I couldn't complain. With H. Nicolai, K.-H. Rehren and F. Mueller-Hoissen in the audience I knew I was talking to people who I would have liked to ask about their opinion on my stuff at any rate.

    Unfortunately, there wasn't much time for questions and feedback. Let's see what tomorrow brings! If nothing else, my talk will probably enter the annals of the DPG as the only one based on chalk and blackboard in the age of PowerPoint. ;-)

    After the afternoon sessions I took care to catch one of the speakers on NCG whose talks I had missed the day before. I was lucky to get hold of a mathematical physicist of the name Paschke, who had given a talk on NCG on pseudo-Riemmanina manifolds. He had presented a technique where you slice a globally hyperbolic manifold in compact spatial leaves, perform ordinary NCG a la Connes on these spatial slices and then figure out how to glue the resulting spectral triples together to get a spectral quadruple. At first it might seem that this way time is commutative while only space is noncommutative, but the crux is apparently that it turns out that this is not the case and that in some sense also time becomes noncommutative. But I didn't see this in any detail.

    After having understood how Paschke is proposing to deal with NCG on pseudo-Riemannian spaces I tried to make him tell me what he thinks of Eric's and mine approach which is supposed to deal, among other things, with pseudo-Riemannian discrete spaces.

    Paschke emphasized that he finds it very problematic to break the compactness assumption of Connes' approach, which technically means that one is (compact) or is not (non-compact) dealing with a C * algebra which has a unit, because then many of Connes' theorems won't hold if there is no unital C * algebra. That may be true, but I am under the strong impression that one can do interesting physics on non-compact noncommutative algebras nevertheless. But this will be a crucial point to be worked out if I want to communicate with the Connes school of NCG.

    I have to run now to get to the debut performance (really, the official dress rehearsal) of Dirk D'ase's Einstein opera.
    ----end quote----
     
    Last edited: Mar 16, 2004
  17. Mar 20, 2004 #16

    marcus

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    the Josh/Urs conversation continues

    today on SPR Josh replied to Urs:
    ----------quote--------

    J:But it's also not necessary, because a self-adjoint representation of a Lie algebra exponentiates to a unitary representation of the Lie group. So what one should check instead is that one has an honest group representation, that is, that:

    U(g_1)U(g_2) = U(g_1 * g_2)

    And this *does* hold for the diffeo group in LQG.

    U:Yes, as it does for Diff(S^1) x Diff(S^1) in Thomas Thiemann's 'LQG-sting'. By construction. I.e. operators U can be found which do satisfy this relation. I do understand and agree that these operators can be constructed.
    The problem is, that these operators don't capture the usual quantum
    effects.

    J:I completely disagree. The point is that one could do *Schroedinger* quantum mechanics this way, if one wanted to; i.e., for the kinds of things we consider in the paper, one need never consider the self-adjoint generators. And you would get exactly the same physical effects as if you did consider them, because you're looking at exactly the same representation of exactly the same algebra. We are therefore just taking the unitary group rep, rather than the self-adjoint algebra rep, as the "jumping off" point.

    U:They do not come from what ordinarily is called 'canonical quantization'. That's because the ordinary prescription of 'canonical quantization' at some point says that we are to promote the classical canonical coordinates and momenta to self-adjoint operators on some Hilbert space. This step is explicitly violated in LQG, since half of these objects are not represented as operators at all.
    So with this step violated, something else has to be found to replace it. Among other things, I claim that this replacement introduces a large ambiguity.

    J:Well, if you don't like ambiguity, you're going to find quantization
    in general very frustrating :)

    U:Namely whenever there is now an object in the classical theory in the enveloping algebra of p and q, you don't know precisely how to quantize it (even modulo operator ordering issues) since one of p and q is not represented on the Hilbert space.

    J:Here is a good example of what I just said about ambiguity. You're
    used to constructing the space of observables from the enveloping
    algebra of the quantization of p and q. But of course this is hardly
    ideal, from the point of view of "quantization," since the enveloping
    algebra will not agree with the Poisson algebra of polynomials in p
    and q, in general. So there is already ambiguity, in choosing what
    subalgebra of the Poisson algebra one represents. People who study
    the general mathematical problem of quantization worry a lot about
    this sort of thing.

    U:I think that this is a problem both in gr-qc/0207106 as well as in the 'LQG-string'. Here is why:
    In the text leading to equation (IV.5) of gr-qc/0207106 you are looking for a way to express the notion of 'coherent state' in the LQG-like language, where you characterize a coherent state by its property of being an eigenstate of the annihilation operator
    a ~ x + ip
    (which is one of several possibilities of characterizing coherent states). This is an object in the algebra generated by p and q. It happens to have no operator ordering problems so the usual quantization of this guy is obvious.
    But now in your framework p is not representable, so something else has to be done. Some object in the algebra of operators that you do have must be identified as being the 'quantization' of the annihilation operator. Due to the Weyl-algebra context, what you really want is the exponentiation
    of the adjoint of the annihilation operator, i.e. an analog of

    exp(a^dag) .

    This has to be expressed somehow using the operators x and V that do exist on your Hilbert space. You do this by looking at the usual quantization of this object and read off the expression above equation (IV.5) from this, e.g.

    exp(a^dag) -> exp(x)V(-1/2)e^(-1/4) .

    But why don't you use for instance
    exp(a^dag) -> exp(x)V(-1/2)
    or
    exp(a^dag) -> exp(x)V(-1/2)e^(42) ?

    This is the factor which I am talking about and which is put in by hand. It is _not_ put in by hand in the _usual_ quantization, because there the e^(-1/4) comes from the BCH theorem. But in your rep there is no p and q which could enter the exponent in the BCH theorem. You are just copying this plausible looking result to your representation.

    J:You are misunderstanding several things here. First, as I indicated in my original post, this is not the point in the paper where we define the physics. That has already been done, in choosing the algebra and representation; the rest of the paper (more or less) is about finding out what the physics of this rep is---not specifiying it.
    Even though there is no p operator (there is in fact a q operator) we
    still know what the commutation relations should be, and therefore how
    to apply the CBH theorem, even if we had never heard of the
    Schroedinger rep. That's because we know the what the commutation
    relations should be from the *classical* theory, the Poisson algebra.
    This alone tells us that

    exp(a^dag) -> exp(x)V(-1/2)e^(42)

    is wrong, provided we know that a^dag = q +ip classically. But how do
    we know that? In other words, what does a state being an eigenstate
    of this particular operator have to do with that same state being a
    semiclassical state?

    But we could just as well ask the same question in Schroedinger
    quantum mechanics! The point is that as far as semiclassicality is
    concerned (because of course as you know coherent states have many
    other interesting and important properties), it's just a fact that the
    states which satisfy this eigenvalue equation also have minimal
    uncertainty and saturate the Heisenberg bound. But it is the latter
    that makes them semiclassical, not the fact that they are eigenstates
    of the annihilation operator. Likewise, in our paper the states are
    semiclassical because they meet the criteria on pp. 20-21. It just
    turns out that states which meet this criteria also satisfy the
    eigenvalue equation (IV.5).


    U:And from reading the paper I got the impression that you would agree that here you are fixing an ambiguity by hand. Because on p.14 it says:
    "Collecting these ideas motivated by results in the Schroedinger
    representation, we are now led to seek the analog...".

    J:You're misunderstanding our point, I think.
    We've introduced in this paper a non-standard rep of the
    Weyl-Heisenberg algebra, which we claim should be thought of as a
    viable (at least for a non-relativistic theory) quantization of the
    particle on the line. If we wanted to do the analagous thing to
    quantum gravity, all we would be looking for is the semiclassical
    limit: that is, to show that Newton's laws are recovered in an
    appropriate sector of the theory. Of course, for quantum mechanics
    (as opposed to quantum gravity, at present) this is not nearly enough;
    we must also show that the quantum effects that are confirmed by
    experiment are also reproduced.

    But in fact we aim to do slightly more even than that. We are trying
    to show that a great deal of the structure of the Schroedinger
    representation can be carried over to this polymer particle
    representation. A priori that might seem impossible, since the two
    Hilbert spaces aren't even unitarily equivalent, much less the
    representations. The "conventional wisdom" would therefore dictate
    that these must be physically different theories. And indeed they
    are: there is some regime in which their physical predictions will
    disagree, as there must be. But not only is there a large regime in
    which their predictions are physically indistiguishable, but there is
    much more similarity in structure than that superficial observation of
    unitary inequivalence might have suggested. And in order to compare
    the structure of the polymer particle rep to the Schrodinger rep, we
    obviously have to say what the latter is. But if we didn't know or
    care about the Schrodinger rep, there would be no need to rely on it.

    But we do, and that is why the emphasis on constructing an analogue of
    the coherent state eigenvalue equation; if all we cared about was
    semiclassicality we could have gone directly to positing the state and
    verifying its properties. This might have been inelegant, but there's
    nothing per se that requires an eigenvalue equation to select
    semiclassical states. And indeed most semiclassical states will not
    satisfy any particular such equation, because the class of
    semiclassical states is much larger than the class of coherent states.
    But in fact these states, which are exactly the states one might have
    "guessed," also satisfy the same criterion as coherent states in the
    Schrodinger representation.

    -----end quote----
     
  18. Mar 20, 2004 #17

    marcus

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    a few more bits of dialog

    -----quote from today's SPR post----
    U:I do agree that this is indeed the most natural choice for this simple system. My problem is that in more difficult cases, where there is mayber no working Schroedinger representation from which ideas can be motivated, how do you fix such ambiguities then?

    J:Two things: again, I don't really think there's an "ambiguity," as I explained above. Second and more importantly, there are guidelines to what one should be looking for in other quantum theories, including gravity. Specifically, one would be looking for the "best approximation" to Fock space, among other things. If you're really interested in this then you should read the papers that led up to the one we wrote, namely:

    gr-qc/0001050, gr-qc/0104051, and gr-qc/0107043

    Some other stuff has been put on the web since then as well.

    U:I noted that in the 'LQG-string' as well as for the diffeo-constraints of 3+1d LQG gravity the analog of this ambiguity is fixed by using ideas motivated by results not of the Schroedinger rep but of the classical Poisson algabra.

    J:As I said ealier, you're comparing apples to oranges if you compare the selection of coherent states in the polymer particle paper to the representation of the diff-algebra in LQG. The former isn't representing any algebra at all; the choice of algebra has already been made, and it was made based on the classical Poisson algebra. In this respect the polymer particle is not a complete model because we didn't look at constraints, but if we had they too would be dictated by the Poisson algebra; you would fix that before you ever looked for semiclassical states, coherent or otherwise.

    --Josh
    --------------end quote---------
     
    Last edited: Mar 20, 2004
  19. Mar 20, 2004 #18

    Haelfix

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    I am now, thoroughly confused again. Proffessors I've talked too are as well it seems (eg the answer depends on who you ask).

    One thing is for sure, the LQG approach to this particular problem is subtle in many ways. In many ways its conjecturing correspondances between classical regimes and quantum ones, and inserting them by hand.

    I don't buy really buy it, but again the mathematical formalism deep at the heart of this is abstracted away from the language they use.

    Back to the blackboard!
     
  20. Mar 20, 2004 #19

    jeff

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    Hardly anyone does.
     
  21. Mar 23, 2004 #20

    marcus

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    the Josh/Urs dialog continued on SPR yesterday

    ----from yesterday's post on spr---
    JOSH: In the case of the diffeo group in LQG, one can explicitly check that the Hilbert space used, which carries an appropriated rep of the Weyl-Heisenberg algebra and other observables, like the area operator, does in fact have a unitary rep of the diffeo group as well.


    URS: This is apparenly a crucial point for the communication between LQG people and others. In LQG there is a lot of emphasis on finding such unitary reps. For instance Thomas Thiemann shows how such a rep exists for the string. But I believe that it is important to emphasize that the mere existence of such reps is not the issue. They do exist, and it is not hard to see why and how.


    JOSH: I don't think this is nearly as easy as you claim. The key point is finding all of the things I mention in the quoted para. in one rep.

    To back up a little, it's best to outline some of the criteria normally needed for a "quantization" of a given classical system. Normally, for a prequantization one needs:

    (1) Poisson brackets go over to commutators,
    {f,g} --> (i/\hbar)[\hat{f},\hat{g}]
    (2) The constant function on phase space is mapped to the identity operator.
    (3) If the Hamiltonian vector field of f on phase space is complete, then \hat{f} is essentially self-adjoint.

    If we have only these three postulates, then there are "quantizations" such that the entire Poisson algebra can be promoted to an operator representation preserving the classical Poisson brackets.

    However, even for finite dimensional systems this is not enough, and so for a prequantization to be a true quantization, one demands another criteria, usually either that the representation of the "basic observables" (i.e., the p's and q's) be represented irreducibly, or that for instance a certain class of observables (functions only of q, for instance) be multiplicatively represented. This severely constrains allowed quantizations, and in general one can then only implement (1) for a certain subalgebra of the classical symplectic algebra.

    Now consider the case where one is quantizing a constrained system by the Dirac quantization "program." Then it is essential that the constraints be among the classical observables that can be promoted to operators respecting the Poisson algebra, and in general this is nontrivial.

    So far, everything I have written is for what I think you would call "canonical quantization." What we are doing is now demanding instead of (1) and (3) that a subgroup of the symplectic group on the manifold---corresponding to the subalgebra that is represented by e.s.a. operators---be unitarily represented on the Hilbert space. Appropriate irreducibility criteria must still hold. When the unitary representation is continuous, one can obtain a representation of the self-adjoint operators from the given unitary representation; when it is not, no such rep exists, and we are now allowing for this possiblity

    Now, it seems that you think that this relaxation makes it easy to satisfy the criteria I list above. Why do you think this? Can you give a general construction? I'll tell you why I don't think it's so easy, again using LQG as an example.

    The way that these criteria are met in LQG is as follows. One takes as the basic algebra of observables the "flux-holonomy algebra"; this is supposed to be analagous to the Heisenberg algebra in ordinary QM. One then demands that the holonomy observables be implemented multiplicatively. This is done by constructing a certain space Abar of "generalized connections" on which the holonomy algebra is the algebra of all bounded, continuous functions. Then the Hilbert space you have is an L_2 space on Abar with respect to some measure, and the choice of representation is dictated by the choice of this measure. *Any* regular, borel measure by construction will represent the holonomy algebra faithfully. But that's not enough. We must represent the entire flux-holonmy algebra, and to do Dirac quantization we must also represent the diffeomorphism group unitarily.


    Take the latter restriction first. This means we can no longer consider any measure on Abar, but just diffeomorphism invariant measures. That already throws out a lot of possibilities.


    However, there are still loads of diff-invariant measures on Abar. But there is only *one* for which it is true that one gets a representation of the full flux-holonomy algebra. This is the measure usually referred to as the Ashtekar-Isham-Lewandowski measure (or sometimes just Ashtekar-Lewandowski). That this is the unique measure Abar allowing one to represent both the flux holonomy algebra and the diffeomorphism algebra is shown in recent work of Sahlmann, Thiemann, Lewandowski, and Okolow.


    So, given this I am highly dubious that one can make a general construction for any subalgebra of the Poisson algebra; it is not, for instance, even clear that one can enlarge the flux-holonomy + diffeomorphism algebra in LQG without breaking the construction.


    URS: The crucial problem is rather that many people don't want to call the procedure of finding such a unitary rep a 'quantization' of the system at hand. Rather, they would want to see a canonical quantization in the ordinary strict sense where the symmetry generators are represented on some Hilbert space.

    JOSH: I suppose some people may think that, but it's worth pointing out that people other than the LQG community (for instance many working in various rigorous approaches to QFT) often focus on unitary reps, for several reasons.

    (1) Unitary operators, being bounded, are much nicer to deal with than unbounded self-adjoint operators.

    (2) The unitary criteria is easier to justify physically, since it is what corresponds to preservation of expectation values by the action of a symmetry. When the rep is continuous and the phase space finite-dimensional, the Stone-von Neumann theorem guarantees that the only continuous unitary rep is unitarily equivalent to the Schroedinger rep. But in infinte dimensions the Stone-von Neumann theorem fails, and it is for this reason that in QFT one often wants to look at the unitary algebra. Haag mentions this early on in _Local_Quantum_Physics_, IIRC.

    (3) When the symmetry group is infinite dimensional, as for instance Dif(S^1), any neighborhood of the identity contains group elements that cannot be obtained by exponentiating the Lie algebra. Thus, the algebra doesn't carry the full information about the group.


    JOSH: But there's no claim that for any classical system
    with a Lie algebra of symmetries, there must exist some quantum rep in
    which the corresponding group action is unitarily implemented.


    URS: On the other hand, it seems that when you allow non-seperable Hilbert spaces then constructions analogous to those used in the 'LQG-string' do give you unitary reps for virtually everything. Can you give an example of a symmetry group which, by the methods used in LQG and in the 'LQG-string' could _not_ be represented unitarily?


    JOSH: Again, I would instead challenge you to give examples of how you can do this for "virtually everything."


    JOSH: Even if there is such a rep, there can of course be other pathologies of the quantization.


    URS: Could you please explain what kind of pathologies you have in mind here?


    JOSH: Failing one or more of the criteria I mentioned above, for starters.


    URS: For instance, do you think that the method of 'shadow states' could be applied to Thomas Thiemann's 'LQG-string', thus showing that his approach does reproduce the usual quantization in some limit?


    JOSH: I don't know---we haven't even tackled full GR yet in this framework. And I haven't looked at Thomas's paper.


    URS: I believe that this won't be possible, because all the information about the usual quantum effects have been eliminated in the 'LQG-string' and they won't reappear in any limit.


    JOSH: I don't understand this statement at all. See my other response (that I hope to write :) ) to your other reply to my post. But at various points you seem to me to have implied that the absence of anomalies means that one won't get the "usual quantum effects." But there are lots of quantum effects besides the existence of anomalies! After all, plenty of systems don't show anomalies in standard quantum prescriptions. Surely you wouldn't say that these show no quantum effects, that the classical and quantum systems are physically indistinguishable?


    For instance, one of the first quantum effects one learns about is the inability to measure momentum and position simultaneously, because they are not commuting observables. This in turn arises because one is quantizing using the Poisson brackets, rather than the usual commutative algebra structure of real-valued functions on phase space. And this effect is certainly preserved in these "unitary group" quantizations, precisely because (for a particle on the line, for instance) one chooses to represent the Heisenberg group, rather than the commutative group R^2.


    URS: Is this the kind of pathology that you are referring to above?


    JOSH: No.

    ----endquote---
     
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