1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Homework Helper
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Charge and normal ordering in QFT
  1. Normal order of A + B (Replies: 1)

Loading...