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Why the term cylindrical functions for the states in LQG?

  1. Dec 5, 2009 #1
    Why the term "cylindrical functions" for the states in LQG?

    I just can't find a any reference that explain the why of the term "cylindrical" for the states in LQG. Thanks in advance for any comment.

    Z.
     
  2. jcsd
  3. Dec 5, 2009 #2

    marcus

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    Re: Why the term "cylindrical functions" for the states in LQG?

    The term means different things in different contexts. The term "cylinder function" comes up in constructing measures on infinite dimensional vector spaces.

    For example here is a reference to 1957 Proceedings of the National Academy of Sciences
    http://www.ncbi.nlm.nih.gov:80/pmc/articles/PMC528446/?page=1

    Kurt Friedrichs and Howard Shapiro
    Integration over Hilbert Space and Outward Extensions.

    The Hilbertspace H is infinite dimensional, and you want to be able to do integral calculus on it. You want to be able to integrate functions defined on the Hilbertspace.
    You start with a simple set of functions. A function f is a cylinder function based on E, if there is a finite dimensional subspace E, of H, such that if PE is the projection of H down onto E, then for all x in H we have f(x) = f(PE x).

    It is a beautifully simple idea, it means that the function f depends only on finite coordinates. You can project down which means throwing away all the other information and the function only depends on x thru its projection onto the subspace E.

    I have to go. But Weiner measure on random walks and all kinds of stochastic process measures on various function spaces can be constructed this way.

    It is a classic idea in topological vectorspaces.

    I think they just took it over. I will check in later and clarify as needed.
    ===EDIT==
    Checked in the next day.
    Well maybe the analogy with what happens with cylinder functions in Loop is obvious. I don't see more questions. So maybe this explanation is OK.
     
    Last edited: Dec 6, 2009
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