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The connection in Loop Quantum Gravity

  1. Jun 3, 2003 #1

    marcus

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    In the Physics forum, jby started a LQG thread which at one point included some discussion of Connections. It might be helpful to have a thread focused on the connection which is taking over the central role from the metric for some purposes.

    GR used to be the study of the metric on a smooth manifold. A metric would give rise to an idea of curvature and an idea of parallel transport of vectors along a path from one point to another. Imagine a sphere and a tangent vector at the equator pointing in some direction and imagine scooting that vector up to the north pole. What direction does it point then?

    The simplest way to imagine a connection, I believe, is as a machine to parallel translate vectors along paths from one point to another. If you have a metric and derive a connection from it and then your disk crashes and you have forgotten the metric, then you can RECOVER the metric from the connection (except for scale). So the connection-----the definite proceedure for parallel transport----has the same information in it, including an idea of curvature.

    Rovelli begins his brief history of LQG with 1986 when it occured to someone to quantize the connection instead of the metric

    http://www.livingreviews.org/Articles/Volume1/1998-1rovelli/index.html

    in the index of Rovelli's review of LQG you see section 3: History.
    Just jump to "3 History of Loop Quantum Gravity" and you find the first few sentences are about this 1986 move to connections.

    [edit: the 1986 paper was Ashtekar's seminal reformulation
    of GR in terms of connections, also called the "new variables", if interested, look at the brief history in section 3 for yourself]

    Anyone is welcome to take over the job of telling the story. I am just summarizing from Rovelli's LivingReviews article at this point
     
    Last edited: Jun 3, 2003
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  3. Jun 3, 2003 #2

    chroot

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    marcus,

    Can you recommend any introductions to LQG? I am decently capable in GR, but cannot seem to bridge the gap to LQG.

    - Warren
     
  4. Jun 3, 2003 #3

    marcus

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    the essential steps in building a quantum theory are
    you have a configuration space, say the real line showing, for example, positions of some particle. And you make the space of
    (lets say square integrable) functions on R.

    That is called L2(R)-----square integrable functions on R.

    The functions ψ in L2(R) are called "states" and they form a vectorspace (infinite dimensional but nice) called a Hilbert space and they represent a blend of knowledge and uncertainty about the particle's position.

    One can modify L2(R) in various ways to get related Hilbert spaces and construct operators (observables) on whatever Hilbert space you end up using.

    the main thing is that when you realize that Nature wont let you pin her down to a particular point in R, then you build a space of functions on R and work with those. And physical observations correspond to operators on that space.

    Now in studying gravity we have a manifold and a collection A of all the possible connections A that can be on the manifold.

    And the main breakthru will be, how do we construct a Hilbert space out of L2(A)? How do we give that meaning?

    All the possible connections is taking the place of the real line R or of any other space of positions or configurations or what have you. All the possible connections includes the idea of all the possible metrics and curvatures and geometries there can be on this underlying manifold (which so far has no particular character except a smooth topology but which you are free to stretch and bend everywhichway.)

    A Hilbert space requires an inner product. How can we set up functions f( A ) and h( A ) and arrange to integrate their product to get the usual sort of Hilbertspace inner product?

    After that, how do we set up operators on the hilbertspace?

    It was so easy in 1920 for Schroedinger and those guys to set up
    of L2(R) the hilbertspace of functions defined on the real line and suchlike simple stuff----and so hard to set up L2(A) the hilbertspace of functions defined on connections.

    In the real line case the inner product was just ∫ f*(x)h(x) dx, taking the complex conj of one of the two functions if you were using complex numbers but basically just integrate the product of the two functions!

    How to do this with f( A ) and h( A ) two functions defined on the space of possible connections?
     
    Last edited: Jun 3, 2003
  5. Jun 3, 2003 #4

    marcus

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    I am going thru Rovelli's easy beginning parts in this thread.
    Have you looked at Rovelli?

    We could write an introduction to LQG right here at PF, if a good one does not exist.

    Yell and scream for Instanton to return to PF. He was here a few days ago and recommended Thiemann's review of LQG.
    arXiv:gr-qc/0110034

    Also Baez has one arXiv gr-qc/9504036

    I guess "gr-qc" means general relativity and quantum cosmology.

    But the short answer chroot is sadly NO I do not know of any
    entrylevel intro to LQG.

    How familiar is the idea of a hilbert space to you? If you focus on one thing at a time, the problem now is to build a hilbert space---how to define the inner product. I would be happy just to get that down solid, for the moment.
     
  6. Jun 3, 2003 #5

    chroot

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    Hilbert spaces are fine. I know QM, jeez. :P

    - Warren
     
  7. Jun 3, 2003 #6

    Tom Mattson

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    Try this:

    Introduction to Modern Quantum General Relativity

    I want to go through it, if I can ever get done with that @$#%*! book on Fields by Siegel.
     
  8. Jun 3, 2003 #7

    marcus

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    BTW chroot,
    you know some GR so you know this

    that if you are on a spherical map of the earth, a globe, and you parallel translate a tangent vector from a point on the equator up to the north pole and then down another longitude line to the equator and then back along the equator you end up with a different vector!

    If you know this you are way ahead because most people have not realized it.

    And so doing that loop of parallel transport generates a shuffling around of the tangent space at the point you come back to.

    It is a symmetry on it.

    So every loop induces a transformation of the tangent space.

    And the curvature enclosed by that loop is reflected in the rotation of tangent vectors that going around the loop causes.

    Intuitively this is why loops are so valuable for extracting information about geometry from connections (the machines of parallel transport). You run the machine on a loop and it tells you part of what is happening.

    A loop is almost an "observable", an operator on connections and also a physical observation defined on the space of connections.

    Is any of this intuitive for you or is it a disaster that I am talking like this?
     
  9. Jun 3, 2003 #8

    marcus

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    Yeah I thought so. So we will have done something if we just see how to define the inner product on A the space of connections on the manifold. Rovelli explains how on one page so lets go ahead and do that.
     
  10. Jun 3, 2003 #9

    chroot

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    Well, duh, marcus... the failure of parallel transport to return a vector to itself is the very definition of curvature. The Riemann metric is just an expression of what happens to vectors when parallel-transported around infinitesimal loops.

    - Warren
     
  11. Jun 3, 2003 #10

    marcus

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    Rovelli defines the Hilbert space concisely, he says:

    We can start ``à la Schrödinger'' by expressing quantum states by means of the amplitude of the connection, namely by means of functionals Ψ(A) of the (smooth) connection. These functionals form a linear space, which we promote to a Hilbert space by defining a inner product. To define the inner product, we choose a particular set of states, which we denote ``cylindrical states'' and begin by defining the scalar product between these.

    Pick a graph Γ , say with n links, denoted γi, immersed in the manifold M. For technical reasons, we require the links to be analytic. Let Ui((A) be the parallel transport operator of the connection A along γi. Ui((A) is an element of SU(2). Pick a function f(...gi...) on [SU(2)]n . The graph Γ and the function f determine a functional of the connection as follows

    ψΓ, f(A) = f(... Ui((A)...)


    That is enough for one chunk of Rovelli. So concise. He is going to define the inner product on these functionals. We need to pause for intuition to catch up. What is one of these functionals?

    The graph is just a collection of n paths (n "links") from one place to another. And parallel transporting along each one gives you a matrix-group element g. And doing all n paths gives you n separate group elements g1, g2, .....gn. And f just happens to be a numerical valued function defined on n-tuples of group elements like that!

    So you give me a connection A and I give you back a number.

    Which I get from running the graph Γ and using the function f on the n-tuple of things that result.

    ********
    He is going to tell us how to do an inner product of two of these
    ψ things.


    This is a page labeled 6.2 in Rovelli's review. I may not have said it so clearly but it does not seem fundamentally so hard. Or?
     
  12. Jun 3, 2003 #11

    chroot

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    Which is the definition of a functional on the connections: a map from the connections to the reals.

    - Warren
     
  13. Jun 3, 2003 #12

    marcus

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    Indeed chroot! It seems to be very elementary. You are the perfect person to discuss this with and it makes me hope that we can bring LQG on board here in a fairly concrete way.

    As you know with inner products the job is two fold. You have to define the inner product of any two of a spanning set of functions and you have to verify it is bilinear and satisfies some sensible conditions like an inequality or two. Lets omit that stuff!
    Lets just see how he defines it.

    I have to go look how to write the integral sign in PF. be back shortly.

    The essence is that with any two graphs Γ and Γ' you can merge them into a union where both
    f and f' are defined on their respective subgraphs as before and as zero elsewhere. So then you just integrate f* multiplied by f' on the [SU(2)]n. They do it every time. It is always how the innerproduct on these spaces of functions is defined!
     
  14. Jun 3, 2003 #13

    marcus

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    Hello again, my computer seems to have slowed down
    considerably. By this time you may be happily reading
    Rovelli directly. It is a bother copying Rovelli equations into PF
    because I have to redo all the subscripts and superscripts.

    However selfsufficient you chroot may be we still must think of other readers! So I will write out the integral that defines the innerproduct.

    That will create the initial hilbert space and in some sense LQG takes off from there with a process of
    refining the hilbertspace and choosing a BASIS for it in a clever way (this brings in "spin networks" as a better basis than just any old loops) and defining operators like the area operator andvolume operator.

    It is amusing that in the quantum theory of a particle what one wants immediately is position and momentum operators.
    But in the case of geometry what one wants are area and volume operators. they turn out to have discrete eigenvalues---that is, area and volume are quantized in steps of planck area and planck volume. wow.

    Well now I know how to write the integral so here is the inner product between

    the psi of Γ f and the psi of Γ', f'.

    The recipe is you merge the graphs into their union, which involves a new larger n. And you extend the definition of f and f' so they are just defined as zero where they werent defined before. So now you have f and f' defined on n-tuples of group elements.

    And any compact group has a natural left and right invariant Haar measure---basically the uniform measure one expects. like on the unit circle for rotations. it is even a probability measure, can take it to sum to one.

    So there is a natural uniform measure on the n-tuples
    [S(2)]n and

    you just integrate the mothers!

    ∫ f*(.....gi......)f'(......gi......) dg1......dgn


    Integrate with respect to invariant Haar measure on the group!


    That gives a number for the two psi's. These psi's are functionals on the connections. The inner product so defined is bilinear etc etc.

    Then we take limits of linear combinations of these "cylindrical" psi's upon which the inner product has been defined. But that is a standard proceedure in defining hilbert spaces.






     
    Last edited: Jun 3, 2003
  15. Jun 3, 2003 #14

    jeff

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    Marcus,

    Since you started this thread to teach LQG, it seems odd that given that the point of departure of LQG is the ashtekar reformulation of GR, you haven't said much about it beyond some vague or general reference to metric and connection. For example, in words, describe the correspondence between the degrees of freedom as they appear in the ashtekar versus conventional formulations and in particular explain the physical significance of su(2) in all of this. Again, since you didn't start this thread without pretension, I don't think my request is unfair.
     
    Last edited: Jun 4, 2003
  16. Jun 3, 2003 #15

    jeff

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    What does "decently capable in GR" mean?
     
  17. Jun 4, 2003 #16

    marcus

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  18. Jun 4, 2003 #17

    marcus

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  19. Jun 4, 2003 #18

    jeff

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    It means that in this thread I want to tailor my explanations to fit the poster's level of understanding. Just because you've recognized one or two obvious analogies between LQG and what little you know about quantum theory doesn't mean you really understand LQG, after all, the paper you're looking at is at a somewhat technical level. As usual, I answer your questions but you don't answer mine. So what about su(2)?
     
    Last edited: Jun 4, 2003
  20. Jun 4, 2003 #19

    marcus

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    If anybody is displeased that we are going over parts of Rovelli's review of LQG and wants us to stop, just say so----I and chroot (if he is still around) will be happy to stop.

    If anybody wants us to interrupt going thru Rovelli in order to
    discuss personalities or rank or STATUS or qualifications or manners or whatever-----instead of just Rovelli and LQG, which is sort of the topic of the thread----then, gosh, I dont know what to do. Just leave the thread? Create a new thread for deciding who the authority is and who should test other people's knowledge?

    Or if anybody wants to argue about the ORDER in which we pick up bits of subject matter----whether to discuss this before that---I really have no answer except that you can always make your own thread to have a workshop or studyhall on whatever you like and do in whatever order you believe best.

    This is said without rancour to anybody reading the thread.
    I dont know now whether or not to proceed because I dont want to have to argue or listen to arguments about side issues.

    The person who originally asked for help understanding essentials of LQG is jby. He posted a question about it on Physics. I'm actually trying to tune this to what I think
    is his receiver-band, whether he is listening or not. If he is around
    he can say if he feels like our going ahead on this track or quitting---that might have some bearing.
     
  21. Jun 4, 2003 #20

    marcus

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