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Recent developments in LQG

  1. Jul 31, 2003 #1

    marcus

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    I was just reading a paper co-authored by Jerzy Lewandowski
    which is dated June 24 2003----but also dated February 2003, I dont know which applies.

    http://arxiv.org/gr-qc/0302059

    It begins "the quantum holonomy operators and the quantum flux operators are the basic elements of quantum geometry..."

    "...Recently, Sahlmann [1] proposed a new, more algebraic point of view. It opens the door to a representation theory of quantum geometry. The main idea is to spell out a definition of a *-algebra constructed from the holonomies and fluxes, that underlies all the loop quantum gravity framework, and to study its representations..."

    Lewandowski calls a certain kind of holonomy-flux *-algebra a "Sahlmann algebra". In his papers Hanno Sahlmann is more modest and does not call the algebra that. (naming things after oneself is not customary in mathematics). Hanno is pretty young to have something named after him. I am trying to understand this a little better.

    A good many of the key papers (1994--present) in loop quantum gravity (quantum-general-relativity in effect) have been by Ashtekar and Lewandowski---if there are central figures in LQG then those two would be my pick of the lot. Hanno Sahlmann just appeared---his thesis date is, I think, 2002.

    In November 2002 Ashtekar, Lewandowski, and Sahlmann posted
    http://arxiv.org/gr-qc/0211012
    "Polymer and Fock representations for a Scalar field"
    here is the abstract:
    "In loop quantum gravity, matter fields can have support only on the polymer-like excitations of quantum geometry, and their algebras of observables and Hilbert spaces of states can not refer to a classical, background geometry. Therefore, to adequately handle the matter sector, one has to address two issues already at the kinematic level. First, one has to construct the appropriate background independent operator algebras and Hilberts spaces. Second, to make contact with low energy physics, one has to relate this polymer description of matter fields to the standard Fock description in Minkowski space. While this task has been completed for gauge fields, imporatant gaps remained in the treatment of scalar fields. The purpose of this letter is to fill these gaps."

    edit: forgot earlier to include link to the first paper mentioned
     
    Last edited: Jul 31, 2003
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  3. Jul 31, 2003 #2

    selfAdjoint

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    Is the first paper you mentioned, the one coauthored by Lewandowski, online? If so, could we have a link to it? If this is the Ashtekar school's approach to putting quantized matter into their quantized spacetime, it is very important indeed.
     
  4. Jul 31, 2003 #3

    marcus

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    selfAdjoint thanks for flagging my omission. I intended to include a link to that paper but forgot as I was typing. It is:
    http://arxiv.org/gr-qc/0302059

    At almost the same time (Feb 2003) that paper by Okolow and Lewandowski appeared, there also came out one by Sahlmann and Thiemann which is doing essentially the same thing:
    http://arxiv.org/gr-qc/0302090
    "On the Superselection Theory of the Weyl Algebra for Diffeomorphism Invariant Quantum Gauge Theories"

    Here is a quote from the beginning to give an idea:

    <<Abstract

    Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Lewandowski representation, has been constructed. This representation is singled out by its mathematical elegance, and up to now, no other diffeomorphism invariant representation has been constructed. This raises the question whether it is unique in a precise sense.

    In the present article we take steps towards answering this question. Our main result is that upon imposing relatively mild additional assumptions, the AL-representation is indeed unique. As an important tool which is also interesting in its own right, we introduce a C*-algebra which is very similar to the Weyl
    algebra used in the canonical quantization of free quantum field theories.


    1. Introduction

    Canonical, background independent quantum field theories of connections [1] play a fundamental role in the program of canonical quantization of general relativity (including all types of matter), sometimes called loop quantum gravity or quantum general relativity. For a review geared to mathematical physicists see [2], for a general overview [3]).

    The classical canonical theory can be formulated in terms of smooth connections A on principal G-bundles over a D-dimensional spatial manifold &Sigma; for a compact gauge group G and smooth sections of an associated (under the adjoint representation) vector bundle of Lie(G)-valued vector densities E of weight one. The pair (A, E) coordinatizes an infinite dimensional symplectic manifold (M , s) whose (strong) symplectic structure s is such that A and E are canonically conjugate.

    In order to quantize (M , s), it is necessary to smear the fields A, E. This has to be done in such a way, that the smearing interacts well with two fundamental automorphisms of the principal G-bundle, namely the vertical automorphisms formed by G-gauge transformations and the horizontal automorphisms formed
    by Diff(&Sigma;) diffeomorphisms. These requirements naturally lead to holonomies and electric fluxes, ...>>
     
    Last edited: Jul 31, 2003
  5. Jul 31, 2003 #4

    selfAdjoint

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    Funny thing is, I had found the Sahlmann-Thiemann paper just browsing the archive back when it came out. All that G-bundle theory you quoted brought it back to me. How are you on that stuff? I can do part of it but the vertical homomorphisms are out of my orbit. I have another paper where a very similar construction is used to define the BRST transformation in terms of topology.
     
  6. Jul 31, 2003 #5

    marcus

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    I'm willing to try to connect with Sahlmann's work, by going thru the easiest most pedagogical of his recent papers. I've looked over everything from early 2002 onwards (his PhD thesis is 2002 and called "Coupling Matter to Loop Quantum Gravity") and the most accessible, I think, is

    http://arxiv.org/gr-qc/0207112
    "When Do Measures on the Space of Connections Support the Triad Operators of Loop Quantum Gravity"

    Have a look at it, selfAdjoint, and see if you'd be interested in reading some of it with me
     
  7. Aug 1, 2003 #6

    selfAdjoint

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    Yes, I printed off and I would very much like to go through it with you and work on questions either of us might have along they way. I am currently ready to start on section 2, Measures on thespace of generalized Connections.

    Do you want to do this here on the boards or by email?
     
  8. Aug 1, 2003 #7

    marcus

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    I have it printed out too, and section 2 seems like a good place to start (right after the introduction). I see no harm in proceeding with it in this thread (maybe Hurkyl, who has a longterm interest in LQG, or Lethe will join in) and we can always retire into a corner
    (either with PF's personal messaging, or email) if there are too many interruptions.

    The "projective limit" mechanism excites me. It is a beautiful device for constructing (not just measures but other) utilities on infinite dimensional spaces. The projective limit is how A-L originally defined the measure &mu;AL on the space of
    connections in the landmark 1994 paper "Projective techniques and functional integration for gauge theories"
    http://arxiv.org/gr-qc/9411046

    What we must understand is two things----invariant Haar measure, which is just the analog of the uniform measure on the circle, and which every compact group has-----and how A-L promoted Haar measure (by projective limit) up to a measure on the space of connections.

    It is a neat business. to take the projective limit of a family of measures, they need a DIRECTED SET (basically a partial ordering with something inclusive of each pair) and the GRAPHS in the manifold are a directed set! Given any two you can always merge them to make a larger one containing both.

    And associated with each graph there is a bunch of numerical valued "cylindrical functions" c[A] of the connection. To specify a cyl function you give a graph----a set of N edges embedded in the manifold---and you give a function c(h1, h2, ...., hN) defined on GN the cartesian product of N copies of the group. (The group is probably just SU(2) and has an invariant Haar measure.)
    What could be simpler than one of these cylindrical functions?!
    To evaluate it on a connection A you just run holonomy on the edges of the graph and get an N-tuple of group elements, to which you apply the function c(h1, h2, ...., hN) and get a number c[A]. (By abuse of notation, i'm using the letter c in two related senses.)

    To define the measure on the whole of A we just need to know how to integrate cyl functions. A measure can be identified with a linear functional on a function space, corresponding to integrating those functions with that measure. So get ready to do the projective limit----for each graph we need a measure or linear functional defined on the cyl functions with that graph----but for that the old N-fold Haar measure on GN will work!

    In their section 2, Sahl/Thiemann, snuck a density function &fnof; into the picture for a bit more generality but then in the last paragraph they remark that the really classical application of it is with the density identically equal to 1, in which case you get the A-L measure. But you get John Baez measures and other variants by letting &fnof; vary.
     
    Last edited: Aug 1, 2003
  9. Aug 1, 2003 #8

    selfAdjoint

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    I am right with you on most of your exposition, and section 2, but I can't quite get my mind around the statement on page 5 where they have taken their projective limit of spaces of connections (bar_A) and say that the closure of the cylindrical funtions on bar_A under the sup norm is a C* algebra. OK so far, then they say the spectrum of the algebra can be identified with bar_A thus endowing it with a Hausdorf topology.

    I have been vaguely aware that this is a pretty standard move in function theory, to get the space back from some algebraic operation on the functions defined on it. But I don't have the foggiest how this is achieved. Could you give me a for dummies quicky? Also the statement of the cited Riesz-Markov theorem?

    As you say this graph projection approach is obviously powerful and very neat. Seems to be worth while to work a bit to get my head around it.

    I do thank you for pointing to the paper and your excellent presentation.
     
  10. Aug 1, 2003 #9

    marcus

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    Let's look at page 11 of the A-L paper they are drawing from:
    http://arxiv.org/gr-qc/9411046

    <<2.2 The Gel'fand spectrum of bar_Cyl...

    A basic result in the Gel'fand-Naimark representation theory assures us that every Abelian C*-algebra ...with identity is realized as the C*-algebra of continuous functions on a compact Hausdorff space, called the spectrum...Furthermore, the spectrum can be constructed purely algebraically...[as a space of maximal ideals]...>>

    We have to understand the space of maximal ideals---I, as you, have seen this over and over as a "pretty standard move" as you say. There must be something good about this move that we should understand. I will try to talk about it some---to get and share intuition.

    As far as that Riesz Markov theorem goes, I wont forget and will get around to it, but I want to focus on this maximal ideal space business for a while.

    As for thanks, mine to you likewise. This is topnotch math and its fun to have someone to talk to about it.
     
  11. Aug 1, 2003 #10

    selfAdjoint

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    Oh yeah, those maximal ideals. I know I have run into that somewhere recently - not as real exposition, but some kind of historical note in Mathematical Intelligencer. I'll try to find it.
     
  12. Aug 1, 2003 #11

    selfAdjoint

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  13. Aug 1, 2003 #12

    marcus

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    the maximal ideals of a ring

    this is something which Hurkyl and Lethe may have seen before and could join us on----it is a very cool trick (not being an expert I only halfway recall or understand it, I have to rediscover it, but I dimly know it to be cool)

    Some gypsies give you a ring with commutative multiplication and they conceal from you the fact that this ring is simply the continuous complex valued functions on the interval [0,1]. Those crafty bastards! They give it to you in abstract form as a bag of different colored M and M candy which you can add and multiply to get other M and Ms of different color. They dont tell you that the thing originally arose in all its algebraic splendor simply as a space of functions on [0,1]. The next morning the gypsies are nowhere in sight and the chickens are gone too.

    But each point x in the interval [0,1] corresponds to an IDEAL in the space of functions on [0,1] consisting of the set of functions which are zero at x

    Ix = {f such that f(x) = 0} is an ideal because it has the sticky quality that if you take any g at all and multiply g by some f in the ideal you wind up in the ideal.

    for any g in the ring, and any f already in the ideal, gf is in the ideal. Like a black hole or the tarbaby, you touch it, and you're in.

    And that Ix is maximal in that it is not contained in any larger ideal except the whole ring.

    So there is a one-one correspondence between points x in the interval [0,1] and the set of maximal ideals.

    so the gypsies gave you an abstract ring---a mere bag of M&Ms---but you suspect that it arose as the ring of continuous functions on some space X. And you are able to RECOVER the space X as the set of maximal ideals of the ring.

    so far its only heuristic. there is more: a way to put a topology on the space of maximal ideals that makes it compact. maybe I will post something, or someone else will, about that later.

    there's even a hint of a technique for "compactification" here. Start with a space that isnt compact, make a ring of functions that are well behaved on it in some fashion, take the maximal ideals, with some topology and it may turn out to be a compact space "including" the original. I seem to recall the "almost periodic" functions on the real line being handled this way, as functions on a compact ideal space. maybe someone has heard of this. Ashtekar used that method of compactification in a cosmology paper recently calling it "the Bohr compactification of the real line". I couldn't remember having heard of the Bohr compactification before but I harbor a deep suspicion that maximal ideals were being used there too.

    And an algebra? What that? just a ring with scalar multiplication. Lethe! Is that right? And a *-algebra is just an algebra with something analogous to complex conjugation.
    It if is the ring of continuous complex-valued functions on something, then it automatically has complex conjugate----f* is just what you think it should be. And if there is a norm too then it is a C*-algebra. so we are just talking rings----function rings, rings with norms, conjugation, addition & multiplication of functions. It does not get any more natural. You can define these things in your sleep without having ever seen the definition.

    OK and Hanno Sahlmann has taken the holonomy+flux operators of LQG and made a C*algebra of them (by taming the flux operators to make them bounded) and proved that there is essentially only one irreducible representation of that algebra as operators on a hilbertspace. And it does not even seem to be all that gnarly! Blows me away. Must get down and learn this stuff. "Get my mind around it" to use selfAdjoint's expression.
     
  14. Aug 1, 2003 #13

    marcus

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    Re: the maximal ideals of a ring

    selfAdjoint, I just saw your post with the link to
    discussion of GNS, for which thanks! I personally
    have not the skill to read PS, and read everything in PDF,
    but others very likely can profit from that file.
    I'm inclined to move ahead at catch up on GNS details
    later. but also if anyone is especially interested in
    this point or has questions about it we could dwell
    on it for a while.

    BTW they are going to take the projective limit of
    QUOTIENT spaces to get a measure on something
    they call A/G
    (notation gets awkward because of lack of symbols)
    which is the connections with the diffeomorphisms modded out

    I am referring a lot to the original A-L 1994 paper because
    it is less condensed than the one we are reading in its
    coverage of these things.
     
  15. Aug 2, 2003 #14

    selfAdjoint

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    Marcus, I have just gone through all the projective development in the 1994 paper, and surprize! I understood it. Ashtekar is a super expositer, and the only difficulty with jis relaxed approach in comparison to the lemma, lemma, theorem, corollary, corollary approach of my youth is that the onus of rigor falls on the reader. It's so easy to skim and say "Sure, sure, yes that's right". Well I worked through about half of their basic proofs and sorted of fuzzed the other half, but I feel that I am well grounded now and we can discuss the application of this projective technique to Sahlmann's theorem.

    BTW I finally twigged to the brand of Category thoeory that would apply here. Isham's toposes (or topoi if you want to be clever) include a partially ordered "index" component. They would be naturals for projective reasoning.
     
  16. Aug 2, 2003 #15

    marcus

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    You are ahead now and I need to catch up
    I agree with you about Ashtekar's writing style.
    In case anyone wants to join us the paper is

    http://arxiv.org/gr-qc/9411046
    "Projective techniques and functional integration for gauge theories" by Ashtekar and Lewandowski

    maybe this is a good clear place to start reading
    and this years papers by Sahlmann and others attach
    on to this as an extension

    Anyway I should make myself a strong cup of coffee and
    do my homework: give a more careful reading to the 1994
    paper you mention
     
  17. Aug 3, 2003 #16

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    I have now reviewed Sahlmann's paper down through the section of diffeomorphism-invariant measures. I need to re-review a lot of the stuff, starting in section 3 where he suddenly jumps from the principle bundle with group G to spacetime with a group SU(2). I think his graphs also suddenly morph into those simplex edges carrying spinor reps of SU(2), taken I suppose directly from his reference 13, the 1997 paper by A&L where they quantize the area operator. I don't really want to go back and review that paper too - it strarts to feel Like I'm Achilles and Sahlmann is the tortoise!

    What I need is some help unpacking his notation - what are the X's from and to, and in what sense are the f's "co-vectors"? Mappings from something to the complex numbers, but from what? G (aka SU(2)? Any help would be appreciated.
     
  18. Aug 3, 2003 #17

    marcus

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    this is great
    having some definite questions from you will help
    me get some traction
    so I wont keep slipping and sliding all over the place

    You are referring, I see, to

    http://arxiv.org/gr-qc/0207112
    "When do measures..."

    I have this in hand and will try to respond. I was proposing this
    paper earlier as a good place to start and it may turn out to
    be.

    WOAH! I forgot to respond earlier to your question about the X's.
    The clearest explanation is by Lewandowski on page 8 of
    http://arxiv.org/gr-qc/0302059
    This is the Okolow-Landowski paper which we talked a bit about earlier. It presents itself as a parallel presentation of Sahlmann's ideas with maybe a slight correction. Landowski is a senior person and it was this paper that put me onto Sahlmann's papers. Because it is more expository and takes more time with the definitions---slow and careful---it is easier in some ways to read than the original. On page 8 it says what those X's are.

    Tomorrow I will try to give an intuitive reading of page 8. This is the key step of exponentiating the flux operator thru a given surface S in order to "tame" it, and then including it in the holonomy-flux algebra. This may be the gnarliest place---hope
    once thru it we have smooth sailing.
     
    Last edited: Aug 3, 2003
  19. Aug 3, 2003 #18

    marcus

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    Im backing up and getting into low gear with this paper "When do measures..."

    To review what he is trying to do, he starts on page 2 with the standard Ashtekar or "new" variables of GR, namely the connection A and the triad field E and he writes two equations (1) and (2) which introduce holonomies and fluxes:

    he[A] is the holonomy along edge e using connection A

    ES,f[E] is the flux of E through surface S integrated with the help of a covector f that I think of as collapsing the triad so he can get a number (each choice of f gives a different value for the integral so he puts f into the subscript along with the surface S.

    "All this makes it worthwhile to study the representation theory of the observables (1) and (2) in somewhat general terms."

    He notes that representation theory of the algebra of HOLONOMIES is already well studied. In fact the cyclic representations are in 1 to 1 correspondence with measures
    on the space A-bar. What is missing is to also include
    the FLUXES in the algebra and then to characterize the representations of the larger algebra----if possible as before by
    finding they are in 1-1 correspondence with measures.

    The general theory of putting reps into correspondence with measures is an extremely efficient (powerful) math tool which I can describe. I should do this. It leads to uniqueness theorems
    and a good control of the reps. Stuff you can actually calculate with!

    But the section you asked about is where he begins to TINKER with the fluxes so that he can trick them into coming into the algebra and joining the holonomies in one big happy algebra.
    I just got a telephone call and must do something else for an hour or so but will be back to this soon
     
  20. Aug 3, 2003 #19

    marcus

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    BTW a cyclic representation is even more general than an irreducible one.

    Any irreducible representation is cyclic

    So this great GelfandNaimark theorem that makes the
    cyclic reps correspond to measures on a certain maximumidealspace applies very generally

    A cyclic rep of some algebraic "thing" (a normed ring, an algebra...) on a hilbertspace is one where there is at least one ONE point in the hilbertspace that goes everywhere under the action of the "thing"

    say it is a ring R and the rep is denoted &pi;:R --> GL(H) the operators on H
    Now &pi;(R) is a whole bunch of operators on H, the collective image of the whole ring
    and cyclic just means there is at least one x in H that gets moved around so much by the bunch of operators that it forms a dense set of points in the hilbertspace

    that is, the set of points &pi;(R)[x] is dense in H

    x is called a cyclic vector because it "cycles thru" the whole space as you apply successive ring elements to it

    I just pulled a book by Naimark himself off a bookshelf and it fell open at a page stating the theorem we need and it isnt even very hard to prove. just one of those good ideas that somebody has and then get used so much the become classics

    Naimark's book is called "Normed Rings", the first american edition 1964 with a brief introduction by Naimark himself written in Russian and on page 245 he states the theorem

    "Every cyclic representation of a complete completely regular commutative ring R with identity is equivalent to its representation, defined by means of the formula

    Ax &xi;(m) = x(m)&xi;(m)

    in some space L2(&fnof;) where &fnof; is an integral in the ring C(M) of all continuous functions on the space M of maximal ideals of the ring R."

    What he means by an integral defined on C(M) is what we call a MEASURE defined on M-----it is a means of integrating functions on M. So we normally call such a thing a measure &mu; and write the integral as &int; d&mu; But it is just a semantic difference.
    So his hilbertspace L2(&fnof;) we would write as
    L2(M, &mu;) the square-integrable functions defined on the maximalidealspace M using the measure &mu; on M.

    And the representation action is simple MULTIPLICATION.
    A point in the hilbertspace is just a function &xi;(m) defined on M
    and you want to know how a ring element x in R is going to act
    on &xi; to give another function on M, another point in the hilbertspace.

    And he refers to the main theorem on page 230 which says that
    a the ring R is isomorphic to the continuous functions on its maximal ideals. So there is a natural correspondence between x in the ring and x(m) a function on the maximal ideal space.

    So this at-first-mysterious equation
    Ax &xi;(m) = x(m)&xi;(m)
    finally becomes clear. A point x in the ring corresponds to
    a function x(m) defined on M
    and the generic representation of the ring goes like this:
    x acts on &xi; in the hilbertspace simply by multiplying
    the two functions x(m) and &xi;(m) together to get yet another
    function defined on M.

    It looks to me as if what Naimark was calling normed rings in 1963 are essentially what are now called C* algebras or
    *-algebras. And what he calls the maximalidealspace is also called the GelfandNaimark "spectrum" of the ring or algebra.
    Terrible how alternative terminology builds up like treebark
    wish we could slough it off once and for all and have the ideas
    in clean pristine beauty but it never happens.
     
  21. Aug 3, 2003 #20

    selfAdjoint

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    Thank you for both posts.

    I am still puzzled by the f's, they are co-vectors in the case of SU(2) but simple functions in the case of U(1). Is this because of the densities defined on the triads? That's the only place the group might come into the E's that I can see.

    Your second post had a lot of stuff I didn't know, and will have to ponder. So this is the "trading reps for measures" you wrote of in your first post?
     
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