- #1

Milsomonk

- 96

- 17

## Homework Statement

Express the charge Q in terms of the creation and annihilation operators.

## Homework Equations

$$\phi_{(x)}=\int \dfrac {d^3 p} {(2\pi)^3} \dfrac {1} {2 \omega_p} (a_p e^{i x \cdot p} + b^{\dagger}_{p} e^{-i x \cdot p})$$

$$\pi_{(x)}=\dfrac {-i} {2}\int \dfrac {d^3 p} {(2\pi)^3} (b_p e^{i x \cdot p} - a^{\dagger}_{p} e^{-i x \cdot p})$$

$$Q=-i\int d^3 x(\pi\phi - \phi^* \pi^*)$$

## 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 :)