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LQG and Category Theory

  1. Feb 25, 2004 #1

    marcus

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    Alejandro recently conducted a poll on Category Theory
    and the response was remarkably positive
    out of 7 respondents to the question
    "Do you know what a Category is?"
    there were 6 who said they could quote the definition from memory.

    the poll was in the "Isham New Quantization" thread
    https://www.physicsforums.com/showthread.php?s=&threadid=14371
    ------------------------------

    A new paper by the Portuguese mathematician Jose Manuel Velhinho
    updates LQG by giving it a Category Theory formulation.

    I think it is kind of elegant and maybe more conceptual. It impresses me as more than putting old wine in new bottles (more than a simple act of translation into a new language).

    Also what he does is not obvious----the Category Theory is not
    mere glitter----or so it seemed to me. You may judge of this differently. I believe he put some thought into it, and it represents creative mathematics: more than just razzle-dazzle.

    Jose Manuel Velhinho
    "On the structure of the space of generalized connections"
    http://arxiv.org/math-ph/0402060

    It is a 30 page paper.
    The first 8 pages are a quick summary of LQG
    the next 12 are putting LQG into Category Theory terms
    the last 10 are "Representations of the holonomy algebra"
    which treats a modern development of LQG due in part to
    work by Sahlmann and by Lewandowski and Sokolow that we
    read here at PF a few months back.
     
  2. jcsd
  3. Feb 25, 2004 #2

    marcus

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    ---------exerpt from page 4-----
    The natural sup norm C*-completion of this algebra, introduced by Ashtekar and Isham [14], is called the holonomy algebra Cyl(A). One then requires that the quantization of kinematical variables produces a representation of the unital commutative C*-algebra Cyl(A).

    By the Gelfand–Naimark fundamental characterization of commutative
    C*-algebras, we know that every commutative unital C*-algebra is (isomorphic to) the algebra C(X) of continuous complex functions on a unique (up to homeomorphism) compact Hausdorff space X. In the case of the holonomy algebra Cyl(A), the corresponding compact space is naturally realized as the space of all morphisms from a certain groupoid of paths (equivalence classes of curves) to the gauge group. This is the space of generalized connections A-bar.

    --------end quote----

    some of us may have acquired the habit of laughing at those who use words like "groupoid", here's another exerpt

    ---------quote---------
    The study of the structure of A-bar...and of measures thereon was
    done in the first half of the nineties [15, 22, 23, 24, 25, 16]. After the seminal work of Ashtekar and Isham [14], Ashtekar and Lewandowski introduced the analytic framework and succeeded in the construction of a very natural measure, distinguished by its simplicity and invariance properties [15]. This is the so-called Ashtekar–Lewandowski, or uniform, measure µ0....

    ... the uniform measure still stands as the only known measure that gives rise to a representation of the full kinematical algebra of holonomies and flux variables. In other words, the corresponding kinematical Hilbert space H0 := L2(A-bar, µ0) supports the only known representation of the kinematical algebra of loop quantum gravity [17].

    This H0 representation is cyclic with respect to C(A-bar), of course, and is irreducible, as it should, under the action of the kinematical algebra [26].

    Moreover, the measure µ0 is gauge invariant and invariant under the action of analytic diffeomorphisms. This immediately leads to the required unitary representations of both groups. Thus, as far as the Gauss and diffeomorphism constraints are concerned, the H0 representation seems to qualify as an intermediate, or auxiliary, representation of kinematical variables and constraints, as required in the Dirac method.

    ---------end quote--------
     
    Last edited: Feb 29, 2004
  4. Feb 26, 2004 #3

    selfAdjoint

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    John Baez has some GREAT stuff about categories and quantizations in his latest TWF (#202). And Marcus, I am sorry now I didn't go into the category section of that paper. I'll make it up today. This is the way to go!
     
  5. Feb 27, 2004 #4

    arivero

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    A groupoid (er, I wrote "grupoid" years ago in my first article) is a fairly more standard object than a category. It is a category with inverses, ie for each arrow it can be given an inverse one such than both compositions AB and BA give the respective units (A unit, remember, is an arrow with equal source and range, and such that composition with a unit perfoms as expected: A.1= 1' A =A. Note 1 and 1' are different arrows).

    If over a set of n points you build a groupoid with arrows all the possible ordered pairs of points, then the canonical construction of an algebra over this groupoid is the one of n xn matrices. If you use the tangent groupoid to a manifold, the procedure of building an algebra is very similar to Weyl quantization.

    On other hand, any foliation of a manifold can be given a groupoid structure, very much as a equivalence class relationship. Then foliations can be studied categorically, and we go to Moerdijk and sheaf theory (and topoi) or algebraically and then we go to Connes and the classification of factors and all that.

    ---------------

    Thus, really the news on Isham approach were that he was boldly taking a generic category, not a groupoid.
     
    Last edited: Feb 27, 2004
  6. Feb 29, 2004 #5

    marcus

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    I need to review categories

    a groupoid is a kind of category
    and would it not be a natural question to ask
    "why are the arrows of a groupoid not forming a group?"
    and the answer to that might be contained in Alejandro's post
    namely
    "note 1 and 1' are different arrows"

    So I see that this post is dehydrated
    like one of those
    things that looks like an aspirin tablet which one drops
    into a glass of water and it then makes
    a small multicolored flower garden


    Anyway, I need to teach myself Introductory Categories
    (slow remedial version) and go down into low-low gear

    I am not completely sure I understand why the arrows of
    a groupoid do not form a group unless it could be that
    there are a lot of identities (since 1 and 1' are different).

    And what, pray tell, is the canonical construction of an algebra
    over a groupoid?

    well you dont really have to say. I will find the answer without assistance one way or another. But I just thought I would let you know what the level of part of your audience is.
    It is intriguing to be told that the canonical construction of an algebra over that (rather primitive-seeming) groupoid you mentioned
    is a matrix algebra
    but where does the field come from? doesnt there has to be a field to pick the matrix-elements from?

    are we talking about two "overs"?
    canonical construction of an algebra over(1) the groupoid over(2) some field?
     
  7. Feb 29, 2004 #6

    selfAdjoint

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    The arrows of a category do not usually form a group because the arrows (morphisms) do not generally have inverses that are morphisms. Think of the category where the objects are topological spaces and the morphisms are continuous functions. A continuous function whose inverse is also a continuous function is a very special thing, a homeomorphism.
     
  8. Feb 29, 2004 #7

    marcus

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    I agree. And my understanding was that a groupoid is a category where the morphisms do have inverses that are morphisms. if that is the definition (as I believe it is) then the morphisms form something very much like a group because the associative rule holds

    but there is no unique identity

    maybe you can confirm?
     
  9. Feb 29, 2004 #8
    the identity morphism on any object is always unique.

    what makes a groupoid different from a group is that a group is a category with only one object, so all morphisms can be composed with all other morphisms.

    in a groupoid, there maybe many objects, and then you can only compose morphisms with the appropriate domains and codomains.

    the associative rule holds in a groupoid whereever it is defined, but, for example, if f is a morphism from A to B, and g is a morphism from C to D, and h is a morphism from E to F, then the associativity law doesnt even make sense, unless we happen to have B=C and D=E. only then does the equation h(gf)=(hg)f make any sense.
     
  10. Feb 29, 2004 #9
    for any groupoid, if you restrict yourself to only those morphisms that go from A to A (for any object A in the groupoid), then you will have a group. thus there may be many groups inside a groupoid
     
  11. Feb 29, 2004 #10

    marcus

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    splendid, now the groupoid business is getting clearer, thankyou Lethe
     
  12. Feb 29, 2004 #11

    marcus

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    BTW selfAdjoint mentioned a paper called
    "Variations on a theme of Gelfand and Naimark"
    http://arxiv.org/math.FA/0402150
    by a student of John Baez called
    Miguel Alvarez
    and it combines abstract nonsense with
    functional analysis in an intriguing way

    I keep involuntarily going back to it and looking
    at it and being intrigued for some reason

    it comes out of Baez Quantum Gravity seminar I suppose
    and it connects C*algebras with Category Theory

    Has anyone looked at this paper? Any thoughts?

    A friend of mine got a bunch of Bitter Seville oranges
    so I'm spending the afternoon making marmelade.
     
    Last edited: Feb 29, 2004
  13. Mar 1, 2004 #12

    arivero

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    Ok, but if the thread becomes too philosophical perhaps it should be moved to the Phyl. section.

    To speak about the world, one must be aware there are two restrictions: the language and the experience. Similarly, to describe Nature we are limited by Mathematics and Experiments.

    The abstruse mathematics has two values: helping us to organize our experiments, and helping to determine the reach of our theory.

    When taking a mathematical object, one considers (or one should) if it helps in the above sense. It was clear time ago about [pseudo]riemannian geometry, and it is becoming to be a possibility for groupoids; and perhaps for whole cathegory theory.
     
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