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Homework Help: Charge and normal ordering in QFT

  1. Nov 30, 2017 #1
    1. The problem statement, all variables and given/known data
    Express the charge Q in terms of the creation and annihilation operators.

    2. Relevant equations
    $$\phi_{(x)}=\int \dfrac {d^3 p} {(2\pi)^3} \dfrac {1} {2 \omega_p} (a_p e^{i x \cdot p} + b^{\dagger}_{p} e^{-i x \cdot p})$$
    $$\pi_{(x)}=\dfrac {-i} {2}\int \dfrac {d^3 p} {(2\pi)^3} (b_p e^{i x \cdot p} - a^{\dagger}_{p} e^{-i x \cdot p})$$
    $$Q=-i\int d^3 x(\pi\phi - \phi^* \pi^*)$$

    3. The attempt at a solution

    Hey guys, little bit stuck with the normal ordering procedure. So i've basically plugged in my expressions for pi and phi into the expression for Q and arrived at the following:
    $$Q=-i \int d^3x \int \dfrac {d^3 p} {(2\pi)^3} \dfrac {-i} {2\omega_p} (b_p b^{\dagger}_p - a^{\dagger}_p a_p)$$
    So I know that I shouldn't have the spatial integral, but i'm not sure how to get rid of it and I know I need to normal order the operators but I'm stuck there to :/ any guidance would be massively appreciated :)
     
  2. jcsd
  3. Nov 30, 2017 #2

    TSny

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    The ##p## in the integral for ##\pi## should be distinguished from the ##p## in the integral for ##\phi##. Use different symbols for these ##p##'s. Thus, the product ##\pi \phi## will be a double integral over the two different ##p##'s. The integration over ##x## is used to eliminate the exponentials and to introduce delta functions involving the two different ##p##'s. The delta functions allow you to integrate over one of the ##p##'s so that you end up with a single integral over the remaining ##p##.
     
  4. Nov 30, 2017 #3
    Oh ok, thanks very much :) Looks like I have it now.
     
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