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Klein Gordon Hamiltonian

  1. Jun 4, 2007 #1
    Suppose that we take the Klein-Gordon Hamiltonian to be of the form

    [tex]H = \int d^3x \, \mathcal{H}(x) = \frac{1}{2}\int d^3x\, (\pi^2(x) + (\nabla\phi(x))^2- m^2\phi^2(x))[/tex]

    If we want to compute, say, the evolution equation for [itex]\phi(x)[/itex] we use the Poisson bracket:

    [tex]\dot{\phi}(x) = \{\phi(x),H\} = \int d^3x'\,\{\phi(x),\mathcal{H}(x')\}[/tex]

    So, if we recall that the definition of the Poisson bracket for some functionals [itex]F[\phi,\pi;x)[/itex], [itex]G[\phi,\pi;x)[/itex] is

    [tex]\{F,G\} \equiv \int d^3y\,\left( \frac{\delta F}{\delta \phi(y)}\frac{\delta G}{\delta\pi(y)} - \frac{\delta F}{\delta\pi(y)}\frac{\delta G}{\delta\phi(y)}\right)[/tex]

    we then have

    [tex]\dot{\phi}(x) = \int d^3x'\, \int d^3y \frac{\delta\phi(x)}{\delta\phi(y)}\frac{\delta\mathcal{H}(x')}{\delta\pi(y)}
    = \int d^3x' \int d^3y\, \delta^{(3)}(x-y)\pi(x')\delta^{(3)}(x'-y)[/tex]
    [tex]=\int d^3x'\,\pi(x')\int d^3y\delta^{(3)}(x-y)\delta^{(3)}(x'-y)[/tex]

    The question I have is whether or not I can integrate the dirac distributions as follows:

    [tex]\int d^3x'\,\pi(x')\int d^3y\delta^{(3)}(x-y)\delta^{(3)}(x'-y)
    = \int d^3x'\,\pi(x')\delta^{(3)}(x-x')[/tex]

    so as to give me [itex]\dot{\phi}(x) = \pi(x)[/itex]? Have I got this right?
    Last edited: Jun 4, 2007
  2. jcsd
  3. Jun 4, 2007 #2

    George Jones

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    It might be a bit better to do the [itex]x'[/itex] integration first. Then, you don't have to use a delta function to evaluate a delta. In the standard theory of distributions, products of distributions aren't defined.
  4. Jun 4, 2007 #3
    There Are no problems with the two deltas product cause they are evaluated at different points. (x-y) and (x'-y). The theory of distributions see Schwarz (a fields medal) says that the problem is doing this inner product (d(x)*d(x),f(x)) where d stand for the delta compact support distributions. and these integral is obviously f(0)*infinite or some not well defined value.
    You got it right shoeron: the time derivative of field (not only scalars field) is its conjugate momentum. It is usefull to remebere in fields theory
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