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Quantum Quandaries-Baez' new paper

  1. Apr 8, 2004 #1

    marcus

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    Quantum Quandaries---Baez' new paper

    quant-ph/0404040

    just out

    gets down on the key QG issues
    sample:

    "...propose another possibility, namely that quantum theory will make more sense when regarded as part of a theory of spacetime. Furthermore, I claim that we can only see this from a category-theoretic perspective — in particular, one that de-emphasizes the primary role of the category of sets and functions.

    Part of the difficulty of combining general relativity and quantum theory is that they use different..."
     
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  3. Apr 8, 2004 #2

    marcus

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    a moment of recognition
    it is not enough for humans to arrive at a
    quantum theory of gravity
    we have also to learn why it has
    been so hard to get there

    what can be learned from the 75 years
    of trying to merge quantum theory
    with general relativity and them acting
    like oil and water
     
    Last edited: Apr 9, 2004
  4. Apr 9, 2004 #3

    marcus

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    selfAdjoint had an earlier link to this paper
    I was just reading back in the "Rovelli's program" thread
    where John Baez had dropped in and sA mentioned he
    was reading this one.
    It could be important. The idea of a *-category
    Who invented the idea of a *-category?
    Baez does not attribute it or cite a reference
    It is possible that it is new (I mean in the mathematical sense
    of not having been already studied and all the easy theorems picked off).

    It is a good idea. one can see that it has possibilities.
    I wonder if Baez invented it himself.
    But perhaps not. It may be very common and wellknown to people
    who are more familiar than I am with category theory.

    It is interesting that both Hilbert spaces and cobordisms form
    categories which are *-categories. We should discuss this
     
  5. Apr 9, 2004 #4

    marcus

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    reading further, the definition of *-category is on page 10

    -----quote-----

    What is a *-category? It is a category C equipped with a map sending each morphism f:X → Y
    to a morphism f*: Y → X, satisfying
    1X* = 1X

    (fg)* = g*f*,
    and
    f** = f.

    To make Hilb into a *-category we define T* for any bounded linear operator T :H → H′ to be the adjoint operator T* :H′ → H, given by
    [tex]\langle T^* \psi,\phi \rangle = \langle \psi, T\phi \rangle [/tex]
    We see by this formula that the inner product on both H and H′ are required to define the adjoint of T.
    In fact, we can completely recover the inner product on every Hilbert space from the *-category structure of Hilb...
     
    Last edited: Apr 10, 2004
  6. Apr 10, 2004 #5

    marcus

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    It's nice how he recovers the inner product from the * relation

    its like wearing comfortable clothes instead of a suit
    if you define Hilb naively then since (,) is structure the maps have
    to preserve the innerproduct so only unitaries get to be morphisms
    what a drag, very restrictive

    but he says look physicists use selfadjoints and other kind of maps besides
    unitary, so lets include as morphisms all the bounded (complex-) linear
    operators
    so the category is basically just the complex vector spaces
    there is a low-level topological issue that they should have a topopology
    that is compatible with some innerproduct or other and the morphisms should
    be continuous in that topology (here already a few little theorems about
    hilbertizable topol. linear spaces, clarifying the meaning of hilbertizable, but still nothing hard or surprising)

    So we have this category of a kind of linear-space with cts linear mappings as morphisms----about as vanilla as one can get. And he says ahah I have a little trick to show you, notice that the complex numbers C are an object in this category!

    And for every space X and for every x in X
    there is a natural map Tx: C -> X
    injecting C into X, such that it takes the complex number one to the vector x.

    Now if this category is a *-category then each morphism
    Tx has a buddy-morphism Tx*
    satisfying the three axioms above

    an easy calculation shows
    that
    (x,y) = Tx* Ty
    where the rhs is just composition of mappings
    or to be precise the lhs number is what you get by applying
    the rhs map to the number one.

    that gives an inner product on the space X and in fact gives the
    original innerproduct that the * relation defined on the category came from.

    so if you have a *relation you can forget that it came from the objects
    being originally hilbertspaces, and then you can recover all the innerproducts
    of all the hilbertspaces from the *relation.

    the idea of *-category has a quality of gentle insistence, it may get in thru the door by sheer niceness

    and then we have to see that a category of Cobordisms which is basic to General Relativity is also a *-category even tho the morphisms are not
    even mappings (they are physicomental umbilicals like feynmann diagrams that connect up one situation to another by a kind of trunk or storyline, well you say it however you like)
     
    Last edited: Apr 10, 2004
  7. Apr 10, 2004 #6

    marcus

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    by sometime today (given the ways of the web)
    baez has probably put the term
    "star-functor"
    into mathematics-language

    a topological quantum field theory, TQFT,
    is among other things a *-functor
    from nCob to Hilb

    :cool:
     
  8. Apr 10, 2004 #7

    Janitor

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    "baez has probably put the term..."

    As well as being famous for using "tensor" as a transitive verb!
     
  9. Apr 10, 2004 #8

    marcus

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    :biggrin:

    seriously though, I couldnt find the idea of a star-category
    in any of the category-theory stuff I looked in
    had you heard or read of the term?

    he actually says that *-category talk makes ordinary category
    theorists nervous. I dont understand their alleged reluctance
    I'm curious about your take on this, please tell me straight (no irony)
    so I can be sure I understand
     
  10. Apr 10, 2004 #9

    matt grime

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    Using star category is just a nicely suggestive way of talking about the opposite category in this familiar setting (to physicists at least). Given any category C, there is the opposite category C^{op} whose objects are those of C and with Hom_{C^op}(X,Y) identified with Hom_{C}(Y,X) by reversing the arrows. The simplest case is Vect the category of vector spaces where op corresponds to taking transposes. It so happens that for Hilb we need not just transpose but conjugate transpose, often denoted by * or [tex]\dagger[/tex] (dagger in case the tex doesn't work). Star doesn't make us nervous, it's just that stars or daggers don't fit into the nice slogans about preserving structures that we often make.
     
  11. Apr 10, 2004 #10
    This is a neat trick and Urs used something like this to define a metric in our paper (something I had failed to do on my own for 6 years!). I think I'll point out a few things though. First, if you have another map [itex]T:\mathcal{H}\to\mathcal{H}[/itex], then

    [tex]
    T T_\phi = T_{T\phi}.
    [/tex]

    From this it follows trivially that

    [tex]
    \langle T \psi,\phi\rangle = (T_{T \psi})^* T_\phi = (T T_\psi)^* T_\phi = T^*_\psi T^* T_\phi = T^*_\psi T_{T^* \phi} = \langle\psi, T^* \phi\rangle
    [/tex]

    as you expect. Now, if you had another invertible map [itex]g:\mathcal{H}\to\mathcal{H} [/itex] that was self-adjoint with respect to this metric, we can define a g-modified metric

    [tex]
    \langle \psi,\phi\rangle_g = T^*_\psi g T_\phi
    [/tex]

    Then we've got

    [tex]
    \langle T\psi, \phi\rangle_g = (T_{T\psi})^* g T_\phi = T^*_\psi T^* g T_\phi = T^*_\psi g g^{-1} T^* g T_\phi = \langle \psi, g^{-1} T^* g \phi\rangle_g
    [/tex]

    so we could let

    [tex]
    T^{*_g} = g^{-1} T^* g
    [/tex]

    denote the adjoint with respect to the g-modifed metric in terms of the adjoint with respect to the original metric.

    This could be used, for example, to turn a Riemannian metric into a Lorenztian one. It could also be used to deform a flat metric into a curved one. Let your imagination wander :)

    Eric
     
  12. Apr 16, 2004 #11

    marcus

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    I found what I think may be the seed of the *-category idea
    on page 22 of this paper by Baez and Dolan:
    http://arxiv.org/q-alg/9503002
    "Higher-dimensional algebra and topological quantum field theory"
    This appeared in 1995 in Journal of Mathematical Physics.

    a *-relation on the morphisms of a category is described, but
    the term *-category isnt used, instead they use a
    circumlocution and dont really give that type of category a name.
    but they are clearly getting at the idea.
     
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