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LOST theorem found (important result for LQG)

  1. May 1, 2005 #1

    marcus

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    http://arxiv.org/abs/gr-qc/0504147

    this has been promised for a couple of years now.
    here at PF we studied the papers leading up to it

    there was one paper we discussed here by Lewandowski and Okolow (LO)
    and several by Sahlmann and Thiemann (ST) or by Hanno Sahlmann solo.
    The four of them have gotten together to prove the most general form of the LOST uniqueness theorem.

    Uniqueness of diffeomorphism invariant states on holonomy-flux algebras

    Jerzy Lewandowski, Andrzej Okolow, Hanno Sahlmann, Thomas Thiemann
    38 pages, one figure
    AEI-2005-093, CGPG-04/5-3

    "Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms.
    While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory."
     
    Last edited: May 1, 2005
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  3. May 1, 2005 #2

    marcus

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    HAPPY MAY DAY TO ALL!

    In other news today, Sheldon Glashow posted a paper proposing the existence of a whole other class of fermions which he calls
    "terafermions"

    http://arxiv.org/abs/hep-ph/0504287
    A Sinister Extension of the Standard Model to SU(3)XSU(2)XSU(2)XU(1)
    Sheldon L. Glashow
    9 pages, adapted from talk at XI Workshop on Neutrino Telescopes, Venice

    "This paper describes work done in collaboration with Andy Cohen. In our model, ordinary fermions are accompanied by an equal number `terafermions.' These particles are linked to ordinary quarks and leptons by an unconventional CP' operation, whose soft breaking in the Higgs mass sector results in their acquiring large masses. The model leads to no detectable strong CP violating effects, produces small Dirac masses for neutrinos, and offers a novel alternative for dark matter as electromagnetically bound systems made of terafermions."
     
  4. May 1, 2005 #3

    marcus

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    one thing of interest here is they introduce a new category of manifolds.

    the semianalytic manifolds.

    It will probably get studied in Differential Geometry.

    semianalytic essentially means PIECEWISE analytic

    it is a bit like what Rovelli and Fairbairn were doing but they were just saying smooth rather than analytic

    it is an interesting extension of the category of analytic manifolds

    For a mapping Rm -> Rm to be semianalytic the exceptional set where it is not analytic has to be defined as the zero set of another analytic function h. Or more generally the exceptional set has to be of the form {h = 0} or {h < 0} or {h > 0}.

    Rovelli and Fairbairn made the exceptional set be just a finite set of points. Their paper was already pretty interesting and we discussed here at PF in a long thread. But this extends that in a certain sense because given any finite set of points it seems clear you can define an analytic function h which is zero exactly on that finite set. So the Rovelli Fairbairn finite exceptional set is also an exceptional set of the semianalytic theory.
     
    Last edited: May 2, 2005
  5. May 1, 2005 #4

    marcus

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    I heard a crash in the back room.

    it is a good paper
     
  6. May 1, 2005 #5

    marcus

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    the main theorem is called theorem 4.2 and it is on page 19
    it says this:

    There exists exactly one invariant state on the quantum holonomy flux star-algebra A


    that is the thing to remember and then you go back and pick up background details like there is this semianalytic manifold [itex]\Sigma[/itex]
    and a compact group G
    and a principal bundle P on [itex]\Sigma[/itex] which bundle is also semianalytic
    and there are connections defined on [itex]\Sigma[/itex]
    (representing the possible geometries that we we can have quantum uncertainty about which geometry it is)
    and one feels these connections with one's eyes shut by doing HOLONOMIES which just means to run around loops and networks and stuff feeling one's gyroscope writhing as one goes around the loop or along pathways in the network

    oops I have to go out for a moment, back soon

    but that only sounds complicated, morally it has a simple enough meaning. you need some paraphernalia to catalog all the possible geometric configurations of space so that you can be uncertain about what shape it is---and so then you can embody your uncertainty, your incomplete knowledge, in a hilbert space, for such is the custom of men and nations.

    it was partly selfAdjoint's intuition that we should study the papers of LO and ST carefully just about 2 years ago.
    it turns out that was a good idea. the notation is essentially unchanged, the concepts have been streamlined, the theorem has gelled and looks like it will be a central one in LQG
     
    Last edited: May 1, 2005
  7. May 2, 2005 #6

    selfAdjoint

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    Wonderful news Marcus! I like the generality. Any theory built on a compact group over a manifold with a connection defining the field strength along with its dual flux, will obey this theorem. Mmm, What does that say about Thiemann's quantisation of the closed string?
     
  8. May 2, 2005 #7

    marcus

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    yes, they make the generality explicit in the abstract but they do not mention an important detail there----the manifold is not just m-times-differentiable (Cm) it is semianalytic.

    I believe this is why the LOST paper was a year and a half delayed, so that people began joking that it was really "lost". And it is why they thank Christian Fleischhack twice (in the acknowledgments and in the appendix) for personal communication "drawing our attention to the theory of semianalytic sets". And, I suspect, why the second and third references, right after [1] Ashtekar "Lectures", are

    [2]
     
  9. May 2, 2005 #8

    marcus

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    yes, they make the generality explicit in the abstract but they do not mention an important detail there----the manifold is not just m-times-differentiable (Cm) it is semianalytic.

    I believe this is why the LOST paper was a year and a half delayed, so that people began joking that it was really "lost". And it is why they thank Christian Fleischhack twice (in the acknowledgments and in the appendix) for personal communication "drawing our attention to the theory of semianalytic sets". And, I suspect, why the second and third references, right after [1] Ashtekar "Lectures", are

    [2]Lojasiewicz, S. (1964): Triangulation of semianalytic sets. Ann. Scuola. Norm. Sup. Pisa 18, 449-474
    [3]Bierstone, E. and Milman, P. D. (1988): Semianalytic and Subanalytic sets. Publ. Maths. IHES 67, 5-42

    Everything is pointing to semianalytic as being very important, so the first thing I am trying to do with this paper is understand why.
     
  10. May 2, 2005 #9

    marcus

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    you remember Fairbairn Rovelli of just a year ago
    http://arxiv.org/gr-qc/0403047
    "Separable Hilbert Space in LQG"
    there it made an enormous difference what diffeomorphisms one actually used

    now with this paper, when they say manifold they mean semianalytic manifold
    when they say bundle they mean semianalytic bundle
    when they say diffeomorphism invariant they mean semianalytic invariant.

    it is bound to make a considerable difference (I mean it already has---they say the main point of their paper is this difference) so right now i am trying gradually to understand what the differences could be.
     
  11. May 2, 2005 #10

    marcus

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    selfAdjoint, it looks to me like the spin networks are different.

    in the old LQG one had a [itex]C^m[/itex] or smooth manifold [itex]\Sigma[/itex]

    and one imbedded spin networks in there to feel the connection-geometry and those imbedded spinnetworks were quantum states of geometry and formed a basis for the kinematic Hilbertspace.

    and then one identified spinnetworks that were equivalent by a smooth or almost smooth mapping.

    but now [itex]\Sigma[/itex] is supposed to be a semianalytic manifold.

    the imbeddings of the spinnetworks, I presume, are to be semianalytic.
    this means intuitively that the Hilbert space will be SMALLER to begin with.

    maybe that is a good thing. but will it be separable?

    I mean after imbedded spinnetworks are identified by semianalytic maps and formed into equivalence classes (which in Rovelli's case were knots) is there a similar reduction to a separable Hilbert space?

    I am looking for reassurance that the semianalytic diffeomorphisms are a good class to be using.

    so far I find little research done on semianalytic sets or functions

    indications are the best reference is
    E. Bierstone and P.D. Milman, Semianalytic and subanalytic sets, Publ. Math. I.H.E.S. 67 (1988), 5–42.

    but this is not online and I have not checked it. If I restrict to what is available online then I find very little:

    wolfram mathworld has a short entry, only one reference, to a 1997 paper
    http://mathworld.wolfram.com/Semianalytic.html

    an arxiv search using keyword "semianalytic" came up with only two papers, one differential geometry paper by a man at oxford
    http://arxiv.org/abs/math.DG/9706227

    one algebraic geometry paper which was only remotely connected with the topic
    http://arxiv.org/abs/math.AG/9910064
     
    Last edited: May 2, 2005
  12. May 2, 2005 #11

    selfAdjoint

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    Semianalytic sets

    Marcus, it is not the manifolds which are required to be semianalytic, but the diffeomorphisms. And they are indeed still diffeomorphisms, that is [tex]C^{\infty}[/tex] functions. But instead of being analytic (convergent Tayor series) everywhere, they are so on a hierarchy of subsets of successively lower dimensions. Analytic on U except on n-1 dimensional U1, and restricted to U1 analytic except on n-2 dimensional U2, etc. These are apparently the semianalytic sets they base their theory on.
    Or that's how I read the paper. Notice that the class of semianalytic functions is bigger than that of analytic ones.
    But the hypotheses of the theorem are still general: principal bundle over manifold, compact group, curvature of connection forming field strength as one-form. Dual flux convertable to something that can be integrated over dual one forms, i.e.faces.
     
    Last edited: May 2, 2005
  13. May 2, 2005 #12

    marcus

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    I think you have it about right. However see the definition of a semianalytic manifold on page 34.

    "...A semianalytic structure on [itex]\Sigma[/itex] is a maximal semianalytic atlas. A semianalytic manifold is a differential manifold endowed with a semianalytic structure."
     
    Last edited: May 2, 2005
  14. May 2, 2005 #13

    marcus

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    you know how everybody these days finds it convenient to work with categories-----a lot of times when doing setup and definitions it really does save time and makes things clearer----well I see them doing that with the semianalyitic category.

    on page 34 section A.2 they are giving basic definitions they need for their main theorem. the section is called "Seminanalytic manifolds and submanifolds"

    first they define what a s.a. manifold is (that is def.A10)

    then given two s.a. manifolds, they define what is a s.a.map between them is (definition A11)

    then they define what is a s.a. submanif. of a s.a. manifold. (def.A12)

    then they define a semianalytic manifold with boundary

    then they prove a property of intersections of s.a. submanifolds (proposition A14) , which is crucial for their main theorem.
    what they say is special about semianalytic submanifolds is that when two of them intersect the intersection is locally a finite disjoint union of connected s.a. submanifolds.

    intersections between C-infinity submanifolds can be more complicated, may not be able to write as a finite union of connected pieces, they may wiggle too much so they may intersect in pathological sets. (they hint at one example on page 35)
     
  15. May 2, 2005 #14

    marcus

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    from equations A15 and A16 on page 35, I gather that what they mean by "edge" is semianalytic edge-----this would make a difference in what a spin network is.

    and a "face" they clearly say to be a semianalytic submanifold (with other properties like codimension 1, oriented)---so it would be a s.a. face

    they seem to want to work in a whole S.A. CATEGORY where everything is s.a.

    and then immediately after that they give the "partition of unity" thing, at the top of page 36. they use the partition of unity (equation 121) in the proof of their main theorem (theorem 4.2 on page 19)
     
  16. May 4, 2005 #15

    marcus

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    I am going to potter around with their notation some to see how to write it in LaTex
    [tex]\mathfrak{A} \text{ is the *-algebra}[/tex]

    [tex]\omega \text{ is a positive normalized linear functional on }\mathfrak{A}[/tex]

    [itex]\mathfrak{J}[/itex] is the ideal consisting of all the elements of omega-norm zero, that is all a for which

    [tex]\omega (\text{a*a}) = 0[/tex]


    then we get a Hilbert space by completing

    [tex]\mathfrak{A}/\mathfrak{J}[/tex]
     
    Last edited: May 5, 2005
  17. May 4, 2005 #16

    selfAdjoint

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    Why the scare quotes, Marcus? Do you have an issue with partitions of unity? They are quite a common tool in topology, especially in regards to getting refinements of coverings.
     
    Last edited: May 4, 2005
  18. May 4, 2005 #17

    marcus

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    :smile: no problem
    seeing the words recalls happy times
    "partition of unity" may be new to one or two other PFrs, though, if anyone is reading this besides you and me
     
  19. May 4, 2005 #18

    marcus

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    we have an explanation problem here. why is quantum mechanics so often done with a Hilbertspace? What is this *-algebra ("holonomies and fluxes")?

    Whenever you have a *-algebra and you have a positive normalized linear functional (a *-morphism to the complex number plane with some simple properties) then that gives you a Hilbertspace. The functional provides the inner product.

    And it gives you a representation of the *-algebra on that Hilbert space. And if the funtional is unique then the hilbertspace and the rep are unique.

    How do we talk about this so it is intuitive?
     
  20. May 5, 2005 #19
    The *-algebra is actually too general to be of practical use in quantum mechanics. In the usual GNS construction one usually proceeds with a specific kind of *-algebra, a C*-algebra. The self-adjoint part of the C*-algebra is the algebra of observables, and is formally a JB algebra. There is a general theorem from the 70's by J.D. Wright establishing that every JB algebra is the self-adjoint part of a C*-algebra, and also that the self-adjoint part of every C*-algebra is a JB-algebra. Thus, whenever you have a C*-algebra, you automatically get a JB algebra of observables for free.

    Now it is possible to formulate quantum mechanics purely in terms of observables, meaning we base the GNS-construction on the JB self-adjoint part of a C*-algebra. In this construction, the Hilbert space is built from projection elements of the JB algebra under a trace norm. In the case of a finite dimensional JB algebra, this Hilbert space can geometrically take the form of a smooth manifold with isometries corresponding to automorphisms of the JB algebra. The isometries form a group, which has the holonomy group as a subgroup. As the JB algebra is also a *-algebra, it is possible view the holonomy group as a subgroup of the automorphisms of the JB *-algebra. This is likely how the name "holonomy flux *-algebra A" came about.

    Regards,

    Mike
     
    Last edited: May 5, 2005
  21. May 5, 2005 #20

    marcus

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    Mike, it would be great having your company while looking over this paper.

    Please say what you mean by "JB"

    IIRC a *-algebra is just an algebra with involution----a unary operation analogous to complex conjugation, or to taking the conjugate transpose when it's matrices.

    How about writing down some definitions? Share the labor of getting the LaTex to work?

    I will put the URL for the LOST theorem in my sig, if it will fit, to keep it handy.
     
    Last edited: May 5, 2005
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