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The rotations, SO(3), in Loop Quantum Gravity

  1. Jun 10, 2003 #1

    marcus

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    this is a collective effort to hit the easy parts of LQG
    for anyone who has shown an interest in this approach to
    quantum gravity to add constructive input

    Lubos Motl has been a critic of LQG and has just published
    a clear presentation of it, plus made an important contribution to and he recommends a certain 7 page paper by Rovelli and Upadhya as "an efficient review of Loop Quantum Gravity".

    Only 7 pages plus a couple of appendices and it introduces the subject! they call it a LQG "primer".

    he also recommends Thiemann LivingReviews for "a more extensive" presentation. I looked at the Rovelli-Upadhya primer and was very impressed by the clarity and conciseness.
    I think it is really the best paper to begin with of all I have seen
    (except it is 1998 so it uses SU(2) instead of SO(3))

    Rovelli-Upadhya:
    arXiv:gr-qc/9806079

    And Motl himself is exceptionally clear and concise in style, so
    his recommending Rovelli-Upadhya as a way into the subject carries a lot of weight with me.

    Motl:
    arXiv:gr-qc/0212096

    I dont take Motl's bad-mouthing LQG seriously because he is bright and obviously very interested all of a sudden in LQG
    and he knows why and he contributes a solid mathematical result.
    His actions speak louder than his heckling.

    Motl's paper is also quite short, only 26 pages in all, and if you read it you see immediately that we need to review some things about SO(3).

    So I am hoping this thread can do some of that, and also
    recap some of the stuff treated in the previous two LQG threads.
     
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  3. Jun 10, 2003 #2

    marcus

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    Here is how Rovelli-Upadhya 1998 LQG primer begins its development. Except that I substitute SO(3) in for SU(2):

    [[Let M be a fixed three-dimensional compact smooth manifold. ...
    Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3).

    We denote by A the space of smooth so(3) valued 1-forms A on M.

    The space A equipped with the supremum norm is a topological space. We denote by L the space of
    continuous functions on A. Equipped with the pointwise topology, L is a topological vector space...]]

    Rovelli is a great guy but he never wrote this IMHO. It must have been Upadhya.

    So then Upadhya, if it is he, introduces the cylindrical functions---a special subset of L

    And before you know it he has an inner product defined on the cylindrical functions. And presto there is a Hilbert space.
    It is all very brief and easy and "comme il faut". No confusion, waste effort or false moves.

    Then, in short order, he gets an orthonormal basis for the hilbert space and (within 7 pages) defines the AREA operator on the Hilbert space. Where was this paper when I was trying to read Thiemann!!! OK Thiemann is very good and careful and thorough but this is good too in a different way.

    Now the riddle in this approach up until this year has always been the Immirzi number. That number not being pinned down has been considered a crisis in LQG. Suddenly this year Olaf Dreyer appears to have found out that it is 1/8.088.
    Motl says this is the size of the "bare" G as compared with the
    macroscopic Newtonian G we are familiar with, and Motl is very interested in this (which is a favorable omen) and he points out as Dreyer also did that it means you have to look at SO(3)

    Now as good luck would have it this is the worlds simplest least complex most intuitive Lie group and its Lie algebra is also as simple as can be. So this is actually entry-level!!! Someone who is curious to learn about classical Lie groups can actually step in here and make it their first adventure along those lines. This is extremely lucky explanation-wise. So I will try to pull together some basic facts.
     
  4. Jun 10, 2003 #3

    marcus

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    SO(3) is a compact Lie group of dimension 3.

    Its Lie algebra so(3) is the space of real skew-symmetric 3x3 matrices
    with bracket [A,B] = AB - BA.

    The Lie algebra so(3) can be identified with R3
    the 3-tuples of real numbers by a vectorspace isomorphism
    called the"hat map"

    v = (v1,v2,v3) goes to v-hat, which is a skew-symmetric matrix
    meaning its transpose its its NEGATIVE, and you just stash the three numbers into such a matrix like:

    +0 -v3 +v2
    +v3 +0 -v1
    -v2 +v1 +0

    v-hat is a matrix and apply it to any vector w and
    you get vxw.

    Everybody in freshman year got to play with v x w
    the cross product of real 3D vectors (in May I remember someone wrote to PF about freshman physics and rotations and v x w)
    and R3 with ordinary vector addition and cross product v x w is kind of the ancestral Lie algebra from whence all the others came.

    And the hat-map is a Lie algebra isomorphism:smile:

    EULER'S THEOREM (just following CalTech's Marsden which is the vanilla ice cream of Lie groups)

    Every element A in SO(3) not equal to the identity is a rotation
    thru an angle θ about an axis w.

    SO SO(3) IS JUST THE WAYS YOU CAN TURN A BALL---it is the group of rotations (this will drive chroot nuts since he is used to much harder stuff, its really basic)

    THE EIGENVALUE LEMMA is that if A is in SO(3) one of its
    eigenvalues has to be equal to 1.
    The proof is just to look at the characteristic polynomial which is of degree three and consider cases.

    Proof of Euler is just to look at the eigenvector with eigenvalue one----pssst! it is the axis of the rotation. It takes three sentences to prove.

    A CANONICAL MATRIX FORM to write elements of SO(3) in
    is

    +1 +000 +000
    +0 +cosθ -sinθ
    +0 +sinθ cosθ

    For typography I have to write 0 as +000
    to leave space for the cosine and sine under it
    maybe someone knows how to write handsomer matrices?

    EXPONENTIAL MAP
    Let t be a number and w be a vector in R3
    Let |w| be the norm of w (sqrt sum of squares)
    Let w^ be w-hat, the hat-map image of w in so(3), the Lie algebra. Then:

    exp(tw^) is a rotation about axis w by angle t|w|


    It is just a recipe to cook up a matrix giving any amount of rotation around any axis you want.

    Hope no one thinks it is bad taste to go over such elementary stuff. I think it is nice and I believe that historically it is the
    paradigm of Lie groups and Lie algebras---a kind of ancestor of the huge proliferation of groups in physics we have today.

    Anyway, Dreyer found that this ancestor belongs in LQG
    ---instead of the more usual SU(2). So now I or somebody has to tell that story and say what
    SO(3) does in Loop Quantum Gravity

    hint: connections....and how tangent vectors get rotated by parallel translation around loops
     
    Last edited: Jun 10, 2003
  5. Jun 10, 2003 #4
    Quite interesting overview. Although the facts about so(3) and su(2) are quite standard, I don't know whether other members will find it as evident as it is (basing on past experience). We will see if anyone gives some answer.
     
  6. Jun 10, 2003 #5

    marcus

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    Yes rutwig, we will see!

    and I think you might approve of my adding here that
    SO(3) is diffeomorphic to the real projective sphere RP3

    proof: about any axis there are always two senses in which
    one can turn the ball to get the same effect.

    wait, one should be more deliberate----they are both diffeomorphic in fairly obvious ways to the solid ball in R3 with antipodal points on the boundary identified. QED:smile:
     
  7. Jun 10, 2003 #6

    marcus

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    it now seems quite possible that the LQG people have discovered the quantum of area and the size of the bare gravitational constant G---many people I guess, including
    an outspoken critic Lubos Motl, now see this.

    What I am describing is something else: namely that I am just beginning to see how "fast" the theory is. One can go thru it like a hot knife thru butter and in a few pages one can get to the premier results---the area operator and the Immirzi number 1/8.088.
    It is a "streamlined" mathematical theory and this (according to an age-old prejudice going back perhaps to Pythagoras :wink: )
    could be a good omen.
    It seems that when the theory was first being developed it was not so efficient.

    Here is a partial recap of the initial segment, following Rovelli-Upadhya's primer. I refer to them collectively as Upadhya because I like the name.




    Let M be a fixed three-dimensional compact smooth manifold. ...
    Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3).

    We denote by A the space of smooth so(3) valued 1-forms A on M.

    The space A equipped with the supremum norm is a topological space. We denote by L the space of
    continuous functions on A. Equipped with the pointwise topology, L is a topological vector space.

    CYLINDRICAL FUNCTIONS

    Let f be a function defined on SO(3)n----the cartesian product of many copies of the group of rotations. Confidentially it is going to play the role of "trace"---a numerical function defined on matrices by summing the diagonal. But we want it to be defined more generally----on an n-tuple of group elements.

    Let Γ be a network of n piecewise analytic curves γ meeting at nodes denoted "p" if we need to mention them. Γ is simply a graph embedded in the 3D manifold M.



    Now the cylindrical function ΨΓ,f

    is the following beautiful and sexy object. It is defined for every connection A in A.

    So you choose a connection A for it to work on and what do you do? You run parallel transport on each leg of the graph.
    That gives you n ways to roll the ball!

    There are n legs "γ" so by running "holonomy", which is a faintly pretentious word for parallel transport, you get n elements of SO(3). So.....you just apply f(.......) to them and presto you have a plain old number.

    this is really boiling it down fast. You start with a delicate complicated thing A, a connection expressing the curvature of the manifold and useful for transporting tangents from one point to another.
    And you hit this connection A with the cylindrical function
    ΨΓ,f and bang you have a plain old number, written ΨΓ,f(A).

    It is suggestive that these network-based cylindrical functions span the function space, L, described earlier ----to me privately it confirms my respect for Roger Penrose who had a hunch that networks were the essence of space---space is all the possible networks. He had that hunch a long time ago. Now I am seeing that some network-based Ψ functions, which will turn out to be quantum states, span a certain interesting linear space. Excuse me if I sound excited---I am, just now.
     
    Last edited: Jun 10, 2003
  8. Jun 10, 2003 #7

    marcus

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    Re: Re: the rotations, SO(3), in Loop Quantum Gravity

    From arXiv-----

    arXiv:gr-qc/0212096

    Jeff you have undoubtably already seen his website---he is a cool guy with a sense of humor---anybody else who hasnt yet, just google.

    Dont be put off by the title of his paper!!!

    "An analytical computation of asymptotic Schwarzschild quasinormal frequencies."

    It is really a clearsighted assessment of LQG from an outsider and sometimes critic.

    maybe I will get links later to the arXiv paper and his site, but I think you (Jeff) most likely can go from here

    I got them and edit them in here:

    Link for Lubos Motl
    http://www.arxiv.org/abs/gr-qc/0212096

    Link for Rovelli-Upadhya nice concise primer
    http://www.arxiv.org/abs/gr-qc/9806079

    personally I like olives, or the branches anyway
    this might be a favorable combination of people
    jeff, sauron, instanton, rutwig, chroot---- if they stay around
    or at least persist in dropping in
     
    Last edited: Jun 10, 2003
  9. Jun 10, 2003 #8

    marcus

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    if possible ask rutwig about this

    to the extent that it is possible to ask help from rutwig this might be a point where we could use advice

    In the Upadhya primer it is 1998 and they use SU(2) and
    the function space L
    is complex valued functions on the connections
    and each cylindrical function Ψ is defined
    using some function f taking values in the complex numbers.

    The inner product on cyl. functions is easy to define
    using the haar measure of the group,
    and one gets a complex hilbert space.

    But here we dont have to use complex valued functions if
    we dont want to (but very likely for future convenience we
    will want to!). Still, for the initial construction this could be a real Hilbert space.

    What is the feeling about this? If I dont hear anything I will just make it complex instead of real.
     
    Last edited: Jun 10, 2003
  10. Jun 10, 2003 #9

    marcus

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    Extending the initial segment to inner products:

    Here is a partial recap of the initial segment, following Rovelli-Upadhya's primer. I refer to them collectively as Upadhya because I like the name.

    Let M be a fixed three-dimensional compact smooth manifold. ...
    Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3).

    We denote by A the space of smooth so(3) valued 1-forms A on M.

    The space A equipped with the supremum norm is a topological space. We denote by L the space of
    continuous complex valued functions on A. Equipped with the pointwise topology, L is a topological vector space.

    CYLINDRICAL FUNCTIONS

    Let f be a function defined on SO(3)n----the cartesian product of many copies of the group of rotations. Confidentially it is going to play the role of "trace"---a numerical function defined on matrices by summing the diagonal. But we want it to be defined more generally----on an n-tuple of group elements.

    Let Γ be a network of n piecewise analytic curves γ meeting at nodes denoted "p" if we need to mention them. Γ is simply a graph embedded in the 3D manifold M.



    Now the cylindrical function ΨΓ,f
    is the following beautiful and sexy object. It is defined for every connection A in A.

    ΨΓ,f(A) = f(... U(γi, (A)...)

    You choose a connection A for it to work on and what do you do? You run parallel transport on each leg of the graph.
    That gives you n group elements
    Ui = U(γi, A)
    Then you apply the function f to those n group elements
    To make it a little easier on the eyes,
    ΨΓ,f(A) = f(...Ui...)

    So given any network with n legs, you run parallel transport on each leg and get n elements of SO(3) and apply f(.......) to them and voila a complex number.

    Networks really can feel any connection A out and tell everything about it. Because doing parallel transport around loops shows what any connection is made of. So these network-based Ψ functions are a fully adequate to generate the function space. Actually the only drawback is that they are more than you need. They span but arent linearly independent. So a later refinement "spin networks" is a way of selecting out a linearly independent basis for the space.


    DEFINING THE INNER PRODUCT OF TWO CYLINDRICAL FUNCTIONS

    Now suppose we have two of these psi functions

    ΨΓ1,f(A) = f(...Ui...)

    ΨΓ2,g(A) = g(...Ui...)

    We merge the two graphs into a combined piecewise analytic graph Γ with some larger number of legs, say n.
    We define a new f and g on SO(3)n to just be equal to the old f and g where they were defined and otherwise zero. It is a trivial construction just to get an integral to be defined. So then the inner product is this:

    (ΨΓ1,f, ΨΓ2,g) =

    ∫ f*(.....Ui......)g(......Ui......) dU1......dUn

    There is a uniform measure on SO(3) which is just what you think it is, uniformly spread out, that is called "Haar" measure and the integral is a straightforward multiple integral with that uniform measure on the group.

    **********

    Now we have the inner product and the Hilbert space of quantum states, except for some technical proceedures to get a nicer basis.
    So we should pause and think about where we are going, which is the area operator.

    The area and volume operators have discrete spectrum which is to say that there are steps of area, for example, and that it is quantized. And Olaf Dreyer found out that the size of the step is 4 ln(3) times the conventional planck unit area. Have to keep this target in sight.

    I'll try posting this to see how it looks
     
    Last edited: Jun 10, 2003
  11. Jun 10, 2003 #10

    marcus

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    Jeff,

    any keyword would help, any author's name or word in the title
    you think might help find it? hope it turns up.
    was it discussed by anyone on Usenet? might find it thru
    a usenet thread. thank you for mentioning it. dont think
    I will go hunting for it myself but wait to see if it turns up here at PF---either that or some keyword clues
     
  12. Jun 10, 2003 #11

    marcus

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    We should get the goal in clearer focus
    Here is a short quote from near the front of the thread
    this post is going to spell stuff out very methodically and
    deliberately----many if not all readers will have noticed
    these things already----but I will still shift down into low gear
    to include as many potential readers as I can.

    The two numbers that LQG has unexpectedly come up with are this 8.088 and the area quantum 4 ln3

    the numbers are exact predictions---I just happen to write the 8.088 as if it were approximate. The physical meaning of the 4 ln 3 number is that area is quantized in steps which are that number times the conventional planck area.

    If you multiply 4 ln 3 by 8.088 and divide by sqrt 2, you get 25.13. This 25.13 is the 8 pi that appears in the Einstein equation of GR. So these two numbers 8.088 and 4 ln 3 are related by the theory. Originally these numbers seem to have arisen semiclassically from Bekenstein/Hawking black hole entropy and Hod's black hole vibration frequencies, later refined by Motl. Black holes have always motivated quantum gravity development ever since Bekenstein 1974. To sum up:

    1/8.088 Gnewton is the bare gravity constant Gbare

    4 ln 3 plancklength2 is the smallest step of area

    The LQG area formula (in form given by Motl) is as follows. There is an orthogonal basis for the Hilbert space which consists of a countable set of spin network states.
    Given any 2D surface S in the 3D manifold M we can see which of the spin networks intersect the surface and sum over those that intersect with it! Typically an intersection contributes sqrt 2 to the summation Σ. What people call the LQG area formula is essentially this, with a symbol for the Immirzi parameter in place of the dorky 1/8.088 which it is my irritating custom to write.

    AreaS = 8pi (1/8.088) Gnewton Σ sqrt 2

    I guess you could also write this as

    AreaS = 8pi Gbare Σ sqrt 2

    And also 8pi (1/8.088) sqrt 2 = 4 ln 3.

    I should say that although I am reporting what seems to me innovative work by people I respect--a good many things here
    baffle me and the idea of quantizing area runs contrary to my intuition and prejudices. Good luck to any who can understand these matters
     
    Last edited: Jun 10, 2003
  13. Jun 11, 2003 #12
    Re: if possible ask rutwig about this

    When I have a close look to the reference given I will probably give a more concise answer, but the main reason to work usually on the complex is because of the representation theory of su(2). Since it is the compact form of usual sl(2,C), all complex representations are recovered from it. Now these representations of su(2)=so(3) (isomrophisma at Lie algebra level) are also complex, and this is the classical source for the failure of the exponential mapping. If the representation is integer-spin, then we can find a transformation such that the induced representation is also real, but for half-spin-integer representations this is no longer true, and the corresponding real representation is of double size. So for example the spin j=1/2 representation of su(2) is complex, and to obtain the corresponding real one we have to consider a four dimensional space. However, thanks to some classical results, if we work on C we can later adapt all to the real case, without being forced to make the previous distinction. This is why usually the complex is more comfortable.

    P.S: If there is some question about convergence, the thing is more serious, because conclusions on R can easily become false.
     
  14. Jun 11, 2003 #13

    marcus

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    a little eureka about labeling the graphs

    rutwig thanks, your occasional comment is more helpful than you may realize.

    I had a small eureka this morning about how to do "spin networks" (essentially embedded graphs labeled by representations of the group) in the SO(3) case.

    One can simply use the double cover and consider the SO(3) representations used to label the edges as "pulled back" to be SU(2) representations.

    So the labels are merely drawn from a particular restricted class of labels. And then one can proceed as before to construct the quantum state corresponding to that particular network.

    I have nothing more of any concern to say at the moment and my wife is insisting that I go get gas for the car----that is, she is reminding me that it is necessary that this be done.

    But I will try to get back soon and carry the exposition a little further.

    BTW what is the basic reason to move to "labeled network" states, when we already have a nice collection of "cylindrical functions" defined earlier, which span the hilbert space. The answer is this, I think:

    Network labeling is a clever way of using the graph Γ to restrict the choices of the function f. In the previous construction
    of cylindrical functions, f was allowed to be essentially anything.
    That introduces a lot of redundancy.

    Now we are going to fix on a graph Γ and set up some rules for labeling it and consider all the different ways we can label it (conforming with those rules, which restrict the possibilities) and then derive a numerical valued function from each legal labeling. This way we get an efficient selection of functions and avoid the redundancy.

    So the "labeled network" (aka "spin network") states will
    turn out not only to span the Hilbert space but to provide an orthogonal basis. I will have to edit this later.

    ***************************

    Extending the initial segment to inner products and then to "spin" or labeled networks:

    Here is a partial recap of the initial segment, following Rovelli-Upadhya's primer.

    Let M be a fixed three-dimensional compact smooth manifold. ...
    These things always have an analytic structure available if one needs it, so we might sometimes use piecewise analytic embeddings. But the basic idea is just a compact smooth manifold. Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3).

    We denote by A the space of smooth so(3) valued 1-forms A on M.

    The space A equipped with the supremum norm is a topological space. We denote by L the space of continuous complex valued functions on A. Equipped with the pointwise topology, L is a topological vector space.

    CYLINDRICAL FUNCTIONS ("cylindrical" is just a time-honored customary terminology, they dont look at all cylindrical to me but I have to conform to other people's usages)

    Let Γ be a network of n piecewise analytic curves γ meeting at nodes denoted "p" if we need to mention nodes. Γ is simply a graph embedded in the 3D manifold M.

    Let f be a function defined on SO(3)n, the cartesian product of many copies of the group of rotations. It is incredibly general---pretty much any nicely behaved function defined on n-tuples of group elements.

    Let's define a quantum state of gravity (!)
    The cylindrical function ΨΓ,f
    is the following beautiful and sexy object, defined for every connection A in A.

    ΨΓ,f(A) = f(... U(γi, A)...)

    You choose a connection A for it to work on and what do you do? You run parallel transport on each leg of the graph.
    That gives you n group elements
    Ui = U(γi, A)
    Then you apply the function f to those n group elements
    To make it a little easier on the eyes,

    ΨΓ,f(A) = f(...Ui...)

    So given any network with n legs, you run parallel transport on each leg and get n elements of SO(3) and apply f(.......) to them and voila a complex number.

    Networks can feel any connection A out and tell everything about it, because doing parallel transport around loops tests out the connection and detects curvature. These network-based Ψ functions represent enough information to generate the function space. Actually the only drawback is that they are more than you need. They span but arent linearly independent. So a later refinement, "spin" or labeled networks, is a way of selecting out a linearly independent basis for the space.

    DEFINING THE INNER PRODUCT OF TWO CYLINDRICAL FUNCTIONS

    Now suppose we have two of these psi functions

    ΨΓ1,f(A) = f(...Ui...)

    ΨΓ2,g(A) = g(...Ui...)

    We merge the two graphs into a combined piecewise analytic graph Γ with some larger number of legs, say n.
    We define a new f and g on SO(3)n to just be equal to the old f and g where they were defined and otherwise zero. It is a trivial construction just to get an integral to be defined. So then the inner product is this:

    (ΨΓ1,f, ΨΓ2,g) =

    ∫ f*(.....Ui......)g(......Ui......) dU1......dUn

    There is a uniform measure on SO(3) which is just what you think it is, uniformly spread out, that is called "Haar" measure and the integral is a straightforward multiple integral with that uniform measure on the group.

    **********
    "SPIN" OR LABELED NETWORKS

    Hurkyl helped me get to my present stage of understanding of the kind of labeled network Γ, j, v which is described here. (H. not responsible, however, for errors.)


    We now have the inner product and the Hilbert space of quantum states, but we need to go through a technical proceedure to get a nicer basis.

    A "spin" or labeled network is a graph which is designed and equiped to self-destruct, when you give it a connection, and yield a number. It collapses by a great crashing tensor contraction.
    It has to be set up right to do this.

    We fix on a graph Γ and consider all the possible ways we can "color" the edges and nodes. The edges will be labeled (or "colored" as they sometimes say) with representations of the group and the nodes will be labeled with multilinear forms on the representation spaces.

    There is going to be a new psi function defined on the configuration space which is A the space of connections.

    ΨΓ, j,v

    Here the ji label the edges----i = 1,...,n---with reps.
    And the vr label the nodes----r = 1,...,m---with multilinear forms.


    This is all just a plot to obtain a number! We are going to grab a connection A out of the configuration space A and evaluate Ψ on it. The connection A will give us a group element by running parallel transport along any edge. Then the rep will interpret that group element as a linear operator on a vector space (the space of the representation).

    Each node with valence k will give us a k-linear form. And the whole works consisting of the operators and multilinear form will collapse down to a number. So there will be a way to evaluate Ψ

    HURKYLS TRIANGLE EXAMPLE

    Consider a network which is simply a triangle with three nodes, each of valence 2, and three edges. Suppose the nodes are labeled with 2-linear forms (3 x 3 matrices) L,M,N and that the representations, applied to the group elements resulting from parallel transport by the connection, give linear operators X,Y,Z.
    Then the tensor contraction of the whole shebang is

    trace(LXMYNZ)

    Maybe we should consider the labeling of the graph in a bit more detail, but this is the rough idea anyway.
     
    Last edited: Jun 11, 2003
  15. Jun 11, 2003 #14

    marcus

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    Re: a little eureka about labeling the graphs

    Originally posted by marcus

    This short summary of LQG follows Rovelli-Upadhya's primer.
    However because of Dreyer's result SO(3) is preferred to SU(2)
    in some places.

    This installment recaps the preceding and tries to go a bit farther. I have edited it to incorporate some valuable clarification by Hurkyl.


    Let M be a fixed three-dimensional compact smooth manifold.
    Such things have an analytic structure available if one needs it, so we might sometimes use piecewise analytic embeddings. But the basic idea is just a compact smooth manifold. Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3). We denote by A the space of smooth so(3) valued 1-forms A on M.

    The space A equipped with the supremum norm is a topological space. We denote by L the space of continuous complex valued functions on A. Equipped with the pointwise topology, L is a topological vector space. Using "cylindrical functions" an inner product has been defined, making a Hilbert space of quantum states of gravity.

    LABELED NETWORK STATES

    A labeled network, in this context, is a graph which is equiped to self-destruct, when you give it a connection, and yield a number. It collapses by a great crashing tensor contraction.
    It has to be set up right to do this.

    We are going to fix on a graph Γ and set up some rules for labeling it. We consider all the different ways it can be labeled conforming with those rules, which restrict the possibilities and allow us to derive a numerical valued function. This way we get an efficient selection of quantum states of gravity and avoid the redundancy.

    The labeled network states will turn out not only to span the Hilbert space but to provide an orthogonal basis.



    The edges of the graph Γ will be labeled (or "colored" as they sometimes say) with representations of the group and the nodes will be labeled with multilinear forms on the representation spaces.

    There is going to be a new psi function defined on the configuration space which is A the space of connections.

    ΨΓ, j, v

    Here the ji label the edges----i = 1,...,n---with reps.
    And the vr label the nodes----r = 1,...,m---with multilinear forms.
    Here Γ has m nodes and n edges.


    This is all just a plot to obtain a number! We are going to grab a connection A out of the configuration space A and evaluate Ψ on it. The connection A will give us a group element by running parallel transport along any edge. Then the rep will interpret that group element as a linear operator on a vector space (the space of the representation).

    Each node with valence k will give us a k-linear form. And the whole works consisting of the operators and multilinear form will collapse down to a number. So there will be a way to evaluate Ψ

    HURKYLS TRIANGLE EXAMPLE

    Consider a network which is simply a triangle with three nodes, each of valence 2, and three edges. Suppose the nodes are labeled with 2-linear forms (3 x 3 matrices) L,M,N and that the representations, applied to the group elements resulting from parallel transport by the connection, give linear operators X,Y,Z.
    Then the tensor contraction of the whole shebang is

    trace(LXMYNZ)

    A FIXED CHOICE OF REPRESENTATION MACHINERY

    A choice of irreducible representations of the group is made once and for all at the outset---finite dimensional vectorspaces with inner product, with linear operators (matrices) to represent elements of the group.

    These can be unrelated to the Hilbert space of quantum states. But the finite dimensional vectorspaces on which the reps act are themselves Hilbert spaces, with inner product. So we have one big Hilbert space of quantum states defined on the connections---and also a whole bunch of little finite dimensional Hilbert spaces defined on the side, which are just machinery to crank out numbers with.

    The whole reason for this is that going around loops with parallel transport ROTATES tangent vectors (that is what curvature is about) and we need ways to boil rotations down to plain old numbers so we can define numerical valued functions on our space of connections. At least for the moment, that is why this extra "irreducible representations" machinery is sitting around.

    Any representation of SO(3) can be considered a representation of SU(2) by the covering map.

    A little notation:
    Hi is the finite dimensional Hilbert space which the irreducible representation ji acts on.
    All that "irreducible" means is that Hi is not any bigger than it has to be----it doesnt have a nontrivial subspace left invariant---everything in it moves under the ji action, except the zero vector.

    Now we assume that all the edges of a graph Γ,embedded in the manifold M, have been labeled with irred. reps ji and we proceed to the nodes. We look at some k-valent node in the graph, call it p, and the k edges that meet a p. There will be a subset of indices Ip that tells which edges γi meet there.
    And the set of reps will be {ji for i ε Ip}

    The crafty Upadhya, with Rovelli looking over his shoulder, tells us to take the tensor product of all the finite dimensional hilbert spaces {Hi for i ε Ip} and to define Hp to be the subspace invariant under the combined action {ji for i ε Ip}.


    Upadhya discusses how to ensure that this subspace Hp is non-trivial and he assumes that an orthonormal basis has been chosen for it ahead of time once and for all. That probably should have been mentioned at the beginning.

    Now we can write a labeled graph Γ ji, vp
    or more simply Γ, j, v
    where ji is an irreducible rep labeling each edge γi, and
    and vp, is a chosen for each node p from the basis of Hp.

    Now at last we can write the new quantum gravity state
    based on the labeled graph Γ, j, v

    ΨΓ, j, v (A)

    this is a numerical valued function of connections A where
    you get the number by a gigantic orgasmic tensor contraction.

    The recipe for this well-nigh catastrophic tensor contraction is to first run A on each edge to get a group element. And then apply the label (a group rep) to get a linear operator. So now each leg of the graph has an operator sitting on it.

    And each node, you suddenly notice, has a toad sitting on it (no I mean a k-linear form:wink:). You snap your fingers and everything begins eating everything else. The edges disappear as their operators apply themselves to the multilinear forms---producing new multilinear forms---and the nodes disappear as those are eaten in turn by other operators. The network "contracts" or consumes itself until finally the only thing left is a number. This number is the value of the function

    ΨΓ, j, v

    on the connection A.

    Care must be taken to set up the graph properly so there are no loose ends that might cause uneaten scraps to be left over! In fact some of the review articles tell you to manipulate the graph first so all the nodes are "tri-valent" -----have 3 edges meeting at them. According to Upadhya one may eliminate all univalent and bivalent nodes without loss of generality. And the labeling around trivalent nodes should satisfy "Clebsh-Gordan conditions" which ensure there is one and only one possible choice of vector at such a node. But rather than get into such fine detail, I want to stop here with the unrigorous and figurative image of the labeled network, once it has been provided with a connection to use in parallel transport along its links, consuming itself and producing a number.
     
    Last edited: Jun 11, 2003
  16. Jun 11, 2003 #15

    Hurkyl

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    I wouldn't be surprised if I misunderstood the paper, but didn't it say that the labels for each node were multilinear forms? The labels on the nodes would be vectors only if the node had only one edge incident with it... in general a node with order n would be labelled with an n-linear form.


    For example, suppose the graph is simply a cycle with 3 nodes. Each node has 2 edges incident upon it, so it would be labelled with a bilinear form. If L, M, and N are matrices representing the bilinear forms at each node, and X, Y, and Z are matrices representing parallel transport in the connection A along the edge joining the nodes labelled L & M, M & N, and N & L respectively, then:

    Ψ(A) = trace(LXMYNZ)

    It is just a scheme for yielding a number, but it's not as simple as you seem to make it.


    edit: I wrote this before I saw your most recent post
     
    Last edited: Jun 11, 2003
  17. Jun 11, 2003 #16

    marcus

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    You are right that the nodes are labeled by multilinear forms!
    Yet for some reason in several places Upadhya refers to the
    labels on the nodes as "vectors". This must mean that he
    thinks of the multilinear forms as a vector space and is able
    to call a large tensor product a "vector" without his hair falling out. Your post helps much to clarify. Glad to know at least
    someone read my latest attempt to cover their LQG primer.

     
  18. Jun 11, 2003 #17

    marcus

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    Hurkyl, this usage of "vector" when one ought
    to say multilinear form is bad and I should edit it out
    and give you credit.

    Before I get around to it, you may have other suggestions!

    Let me know of any paragraphs that you think should be
    improved or scrapped.
     
  19. Jun 11, 2003 #18

    Hurkyl

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    Yah, the set of n-linear forms with the usual addition and scalar multiplication forms a vector space, so it is accurate to call them vectors, if confusing!! It's probably better to call them multilinear forms to avoid confusion until we're more comfortable with the idea. :smile:


    I'm gonna do some paraphrasing of the first initial ideas, both because I think I can put it into simpler concepts, and to make sure I really understand what I think I do. :smile:


    For the sake of simplicity, let's presume that the graph in question, Γ, lies entirely inside a single coordinate chart. If j is a link in our graph, then the idea of parallel transport along j simply becomes the rotation a traveller experiences (with respect to the coordinate chart) as he traverses the link.

    Now, if f is a function of n rotations and our graph has n links, then we can form the pair:

    ΨΓ,f(A)

    which simply applies the function f to the rotations corresponding to the individual links. Of course, f has to be suitably defined so switching coordinate charts doesn't change the resulting value.


    Hypothesis: Can we completely ditch the manifold at this point and abstract the idea of a connection merely to something that acts on edges to give rotations?


    I'm not up on representation theory, so I don't have too much to say about the spin networks... but the idea of "consuming" the graph to compute the big tensor product was quite informative. I was having a hard time swallowing just what it could mean, but iteratively performing pieces of the product as you shrink the graph made more sense (and I think I can go from there to explicitly writing the effect that the graph manipulations have on the actual product).
     
  20. Jun 11, 2003 #19

    marcus

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    I believe one just says [ sub ] and writes whatever one wants and then [ / sub ]

    So that one can put whatever one wants, upper or lowercase, in for the subscript.

    But I think that there is only one level of subscripts. And likewise for super. Let me know if you discover more, please.

    I am writing spaces in [ sup ] to fool it but you eliminate the spaces to get it to work.
     
  21. Jun 12, 2003 #20

    marcus

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    thanks Jeff
    BTW do you happen to know a code for tensor product
    that would work here at PF?
     
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