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Inner product on boson one-particle space

  1. Feb 9, 2009 #1
    Hi,

    I'm interested in constructive QFT and I'd like to pose a question about construction of the one-particle Hilbert space of states for bosons.

    For fermions satisfying the Dirac equation, the inner product is
    [tex]
    \langle \psi, \phi \rangle = \int d^3 x \; \psi^\dagger(x) \phi(x).
    [/tex]
    The inner product is a constant of motion (and Lorentz invariant) because the integrand is the 0th component of a conserved current. The inner product is positive definite, and defines the Hilbert space L^2.

    Is there a similar inner product for bosons satisfying the Klein-Gordan equation?

    I previously thought that boson state spaces required an indefinite inner product (i.e., existence of negative-norm states), which is a problem because the triangle inequality is broken. This basically ruins any chance of applying Hilbert space theory.

    Recently I found a paper [1] which refers to an answer, but doesn't give it explicitly. To quote from the introduction:
    Can somebody help me out here?

    Thanks very much,

    Dave

    [1] Shale, Trans. Am. Math. Soc. 103 (1962) 149, Linear Symmetries of Free Boson Fields.
     
  2. jcsd
  3. Feb 9, 2009 #2
    I say put all derivative operatorings with gauge covariant derivative operatorings, then get answer.
     
  4. Feb 9, 2009 #3
    Could you be a little more direct in your answer? Suppose H is the space of solutions to the K-G equation, and [itex]\psi, \phi \in H [/itex]. What is the inner product [itex] \langle \psi, \phi \rangle [/itex]? Is it a constant of motion and Lorentz invariant? Is it positive definite?
     
  5. Feb 9, 2009 #4
    Positive definite ONLY if positive frequency solutions of the K-G. (time look like vector field).
     
    Last edited: Feb 9, 2009
  6. Feb 9, 2009 #5

    Fredrik

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    This one might help: http://home.uchicago.edu/~seifert/geroch.notes/ [Broken]
     
    Last edited by a moderator: May 4, 2017
  7. Feb 9, 2009 #6

    Fredrik

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    I might as well ask it here instead of starting a new thread: Is there a book that explains these things? The sort of "things" I'm talking about is e.g. how to define the Hilbert space of one-particle states from the solutions of the field equation, and how to define operators on that space. Geroch does it pretty well, but there are things in his notes that I just don't get, so it might help to read something written by someone else.
     
  8. Feb 10, 2009 #7
    Cheers for the help, in fact, I had read parts of Geroch's notes previously and forgot his take on the problem. I'm a little dissatisfied: restriction to the positive-frequency solutions is equivalent to specifying the equation of motion
    [tex]
    i\frac{\partial}{\partial t} \psi = \sqrt{k^2 + m^2} \; \psi
    [/tex]
    in Fourier space, and this isn't equivalent to a differential equation on spacetime. I'd prefer having a differential equation as conceptually more appealing, and it might suggest some way of making it generally covariant. The latter would be necessary if we consider quantizing the scalar field on a curved spacetime.

    Or to phrase the same objection in a different way, does the solution space of a generally covariant K-G equation (replace partial derivatives with covariant derivatives) also admit the same separation into positive and negative-frequency solutions?

    Any comments?
     
  9. Feb 10, 2009 #8

    Fredrik

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    The covariant derivative and the partial derivative operator have exactly the same effect on a scalar.

    [tex]\nabla_{\partial_\mu}\phi=\partial_\mu\phi[/tex]

    So there's no difference.
     
  10. Feb 11, 2009 #9
    Yes, for the first covariant derivative, but no, not for the second derivative. The laplacian of a scalar on a Riemannian manifold is

    [tex]
    \square \psi = \psi_{;i}{}^i =
    \frac{1}{\sqrt{\det g}}
    \left( g^{ij} \psi_{,i} \sqrt{\det g} \right)_{,j}
    [/tex]
     
  11. Feb 11, 2009 #10

    George Jones

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    If spacetime is stationary, this separation can be made; see Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics by Wald.
     
  12. Feb 11, 2009 #11
    Stationary spacetime is a heavy restriction! But I'll check it out. I can't get the book you mention but I found some papers by the same author with similar titles on SPIRES and the arxiv. Thanks.
     
  13. Feb 11, 2009 #12

    Fredrik

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    D'oh...the posts I write immediately before I go to bed seem to be much dumber than then ones I write at other times of the day.
     
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