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Amplitudes, metrics, and measures

  1. Nov 12, 2007 #1
    This is a question about how the amplitudes of QM might be generalized. I'm not sure this post belongs here or perhaps some other forum. But I believe I am talking about extending QM beyond the standard.

    Correct me if I'm wrong, but what we have in flat space is states, [tex]\[
    |\psi> \][/tex], projected onto eigenvectors of the position operator [tex]\[|x > \][/tex] to give us a wavefunction, [tex]\[ < x|\psi> = \psi(x)\]
    [/tex], whose modulus squared is a probability density, [tex]\[
    < \psi|x > < x|\psi> = |\psi(x)|^2 = p(x)\][/tex].

    These position eigenvectors form an orthonormal basis, [tex]\[ < x_i |x_j > = \delta (x_i - x_j )\][/tex], so that the inner product is preserved, [tex]\[ < x_i |x_j > = < x_k |x_l > = 1\][/tex]. This give a unitary transformation from one place to another and allows a probability to be calculated to go from some initial state to a final state.

    But it occurs to me that a probability is just one type of measure, and the orthonormal inner product, [tex]\[ < x_i |x_j > = \delta (x_i - x_j )\][/tex], is just one type of metric. So quantum mechanics establishes a relationship between the metric and the measure.

    My question is can this relationship between the metric and the measure be generalized to curved spacetime in such a way that as the metric becomes flat, the measure becomes a probability?

    I suppose that if the metric were to change like this, then the evolution of states would cease to be unitary, it would be more difficult to describe states in a basis of position eigenstates, and the measure would cease to be the probabilities we are familiar with. New interpretations would have to be employed to describe such a situation, no doubt.

    I believe that the study of QFT in curved spacetime might have something to say about this. But I am hoping to find something more fundamental. Is there anything in the study of metric spaces, measure theory, of functional analysis that would justify this kind of general relationship between the metric and the measure? Thanks.
  2. jcsd
  3. Nov 14, 2007 #2
    Has anyone thought of getting at quantum gravity by more accurately defining the measure on the space of paths in the Feynmann path integral? It would seem that this measure involves the metric. And a better understanding of how the metric is involved might go a long way to understanding how quantum gravity works.
  4. Nov 15, 2007 #3
    I've tried to Google this issue of undefined measure of paths in Feynmann path integrals, and I'm not finding much. I did find the following.


    Am I correct that the first paragraph after the abstract indicates that quantum physics of curved spacetime is determined by defining the measure in the path integral?

    Where can I find more info on this measure of the path integral?

  5. Nov 15, 2007 #4
    I don't know if I agree with that last idea "quantum mechanics establishes a relationship between the metric and the measure" because the position eigenvectors you are using are defined on a Hilbert space...not "real" spacetime.

    On the Hilbert space, the eigenvectors for position are orthonormal with respect to the inner product.

    In "real" spacetime, that may not be so. So the metric you would be taking about would be a metric of the Hilbert space (or, I think more precisely, the position subspace of the Hilbert space describing the system).

    Interesting approach, but I think it's off a wee bit. Look up Dirac's Lectures on Quantum Mechanics, he has an approach to quantum theory on curved spacetime that is moderately accessible.
  6. Nov 16, 2007 #5
    Yes, I'm sure it needs some work. But isn't [tex]\[ < x_i |x_j > = \delta (x_i - x_j )\][/tex] a metric. And isn't the probablitiy a measure? I have seen papers that use the delta function of the inner product to derive the probabilities of normal QM. I think that means there is a relationship.

    I have to wonder if there is not some more intuitive reasons to think that quantum gravity involves a better understanding between this metric and the measure. I mean in normal QM or QFT there is no attempt to "measure" the metric. In quantum gravity we are trying to measure the metric. So it seems a better understanding of the measure is needed. Or am I using the word "measure" in an equivocal fashion. Does "measuring" QM observables have nothing to do with a measure defined in measure theory? I thought I remember seeing something about operators in Hilbert space defined in terms of a measure or a metric,... something like that.
    Last edited: Nov 16, 2007
  7. Nov 17, 2007 #6
    Perhaps I should try to explain the background of QM more fully to justify my explanation.

    You have a "phase space" in classical mechanics where the state of a classical body is described by a 2*D dimensional vector for D spatial dimensions and D "conjugate" momentum dimensions. So that's in (e.g.) cartesian coordinates x,y,z coordinates in space, and the [tex]p_{x}[/tex], [tex]p_y[/tex], [tex]p_z[/tex] dimensions (the x, y, z components of the momentum vector).

    In quantum mechanics, we replace this phase space with a special type of space called a Hilbert space. The orthogonality of the position eigenvectors is a property of the Hilbert space, not the "actual" spatial dimensions.

    So the metric you are looking at is the metric of the Hilbert space, not of spacetime itself.

    I'm not sure how much sense that makes, I haven't slept too much after finding out about this "Saturn's Revenge" drink at my local cafe (4 shots of espresso in a mocha :S).
  8. Dec 8, 2007 #7
    In my opinion........

    The Hilbert space metric is:

    [tex]< x|x' > = \delta (x - x') = \frac{1}{{2\pi \hbar }}\int_{ - \infty }^{ + \infty } {e^{ip(x - x')/\hbar } dp} = \frac{1}{{2\pi \hbar }}\int_{ - \infty }^{ + \infty } {e^{ip\dot x\Delta t/\hbar } dp}[/tex]

    The [tex]{e^{ip\dot x\Delta t/\hbar } }[/tex] is a plane wave. The [tex]p
    [/tex] is the momentum in the cotangent bundle of configuration space, or the phase space, and the [tex]\[{\dot x}\][/tex] is a velocity in the tangent bundle of the configuration space. This would make [tex]\[p\dot x
    \][/tex] the metric that converts vectors to covectors. It is called the "mass metric" in Geometry of Physics by Frankel, page 55. This shows how the metric in configuration/phase space is connected to the metric in Hilbert space.

    In Special Relativity the inner product in Minkowski space is between the four-velocity and the four-momentum. And this is an invariant in SR, namely -E. So the Hilbert space metric remains a delta function.

    My question is: how does the inner product change when we go over to General Relativity? Would the Hilbert space metric remain a delta function in GR? Or in other words, how is the configuration space metric constrained if the Hilbert space metric must remain a delta funciton? Any help is appreciated. Thanks.
    Last edited: Dec 9, 2007
  9. Dec 13, 2007 #8
    By the way,

    I found where [tex]\[\delta (x - x')\][/tex] is called a "Dirac measure". Search for "dirac measure" at wikipedia.org.

    Question: Is this dirac measure a well defined measure in every sense used for other measures? Thanks.
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