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Klein Gordon Field

  1. Apr 30, 2014 #1
    Hi everyone! Im' a new member and I'm studying Quantum Field Theory.

    I read this:

    "The interpretation of the real scalar field is that it creates a particle (boson) with momentum p at the point x."

    and :

    [itex]\phi[/itex][itex]\left(x\right)[/itex] [itex]\left|0\right\rangle[/itex] = [itex]\int \frac{d^3p}{(2\pi)^3(2\varpi_p)}[/itex] [itex]e^{-ipx} |p\rangle[/itex] (1)

    but I didn't understand this equality... i know that:

    [itex]\phi (x) = \int \frac{d^3p}{(2\pi)^3(2\varpi_p)} (a_p e^{ipx} + a^+_p e^{-ipx})[/itex] (2)

    So... where it goes the term [itex]a_p e^{ipx}[/itex] in the expression (1) ???

    Can someone kindly show me all the steps?
    I know it's a stupid question, but I can not understand.

    thank you very much!!!!!
     
  2. jcsd
  3. Apr 30, 2014 #2

    WannabeNewton

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    The vacuum is annihilated by ##a_p## by definition.
     
  4. Apr 30, 2014 #3
    Thank you for your reply.
    But in theory... [itex]a_p[/itex] it should not destroy the particle created by [itex]a^+_p[/itex]??

    What am I doing wrong? :confused:

    Thank you.
     
  5. Apr 30, 2014 #4

    WannabeNewton

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    We don't have ##a_p a^{\dagger}_p## in the free KG field. We have ##a_p## attached to the negative frequency modes and ##a^{\dagger}_p## attached to the positive frequency modes so they act independently of one another.

    As such ##\phi(x)|0\rangle## simply creates a particle at ##x##.
     
  6. Apr 30, 2014 #5
    Thanks!! :)
     
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