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Klein Gordon eqn, decoupling degrees of freedom

  1. Oct 24, 2011 #1
    Having some trouble following my notes in QFT. Any help greatly appreciated.

    We have the Klein Gordon equation for a real scalar field [itex]\phi\left(\overline{x},t\right)[/itex]; [itex]\partial_{\mu}\partial^{\mu}\phi + m^{2}\phi = 0[/itex].

    To exhibit the coordinates in which the degrees of freedom decouple from each other, we take the Fourier transform, [itex]\phi\left(\overline{x},t\right)= \int\frac{d^{3}p}{\left(2\pi\right)^{3}}e^{i \overline{p} .\overline{x}}\phi\left(\overline{p},t\right)[/itex].

    Then [itex]\phi\left(\overline{p},t\right)[/itex] satisfies [itex]\left(\frac{\partial^{2}}{\partial t^{2}}+\left(\overline{p}^{2} + m^{2}\right)\right)\phi\left(\overline{p},t\right) = 0[/itex].

    If you do it by brute force you get [itex]\int\frac{d^{3}p}{\left(2\pi\right)^{3}}e^{ i \overline{p} .\overline{x}}\frac{\partial^{2}}{\partial t^{2}}\phi\left(\overline{p},t\right) - \int\frac{d^{3}p}{\left(2\pi\right)^{3}}\partial^{2}_{i}e^{ i \overline{p} . \overline{x}}\phi\left(\overline{p},t\right) + m^{2}\int\frac{d^{3}p}{\left(2\pi\right)^{3}}e^{ i \overline{p} . \overline{x}}\phi\left(\overline{p},t\right) = 0[/itex]

    then [itex]\int\frac{d^{3}p}{\left(2\pi\right)^{3}}e^{ i \overline{p} .\overline{x}}\frac{\partial^{2}}{\partial t^{2}}\phi\left(\overline{p},t\right) + \int\frac{d^{3}p}{\left(2\pi\right)^{3}} \overline{p}^{2} e^{ i \overline{p} . \overline{x}}\phi\left(\overline{p},t\right) + m^{2}\int\frac{d^{3}p}{\left(2\pi\right)^{3}}e^{ i \overline{p} . \overline{x}}\phi\left(\overline{p},t\right) = 0[/itex]

    Now I dont see how to get rid of the intergrals. I can see its similar to a delta function, but you cant just take the [itex]\phi\left(\overline{p},t\right)[/itex] out of the integrals because the measure is [itex]p[/itex].

    Thanks for helping me with this, please let me know if I haven't been clear.
    Last edited: Oct 24, 2011
  2. jcsd
  3. Oct 25, 2011 #2
    Your last line can be rewritten as
    [itex]\int\frac{d^{3}p}{\left(2\pi\right)^{3}}[\frac{\partial^{2}}{\partial t^{2}}\phi\left(\overline{p},t\right) + \overline{p}^{2} \phi\left(\overline{p},t\right) + m^{2}\phi\left(\overline{p},t\right)]e^{ i \overline{p} .\overline{x}} = 0[/itex]
    Each term of e^{ip.x} is linearly independent (as they form a basis) and thus each term must be identically 0, giving the desired relation. Identically you can say the fourier transform of [itex]\frac{\partial^{2}}{\partial t^{2}}\phi\left(\overline{p},t\right) + \overline{p}^{2} \phi\left(\overline{p},t\right) + m^{2}\phi\left(\overline{p},t\right)[/itex] is 0. So it must be 0 as well.
  4. Oct 25, 2011 #3
    Ah, thats why. Thanks very much, much appreciated.
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