- #1
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Homework Statement
I want to show that
[tex]\mathbf{P} = -\int d^{3}x}\pi(x)\nabla\phi(x) = \int{\frac{d^{3}p}{(2\pi)^3}\mathbf{p}a_{p}^{\dagger}a_p[/tex]
for the KG field.
Homework Equations
[tex]\phi(x) = \int{\frac{d^{3}p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_p + a_{-p}^{\dagger})e^{ipx}[/tex]
[tex]\pi(x) = -i\int{\frac{d^{3}p}{(2\pi)^3}\sqrt{\frac{\omega_k}{2}}(a_p - a_{-p}^{\dagger})e^{ipx}[/tex]
The Attempt at a Solution
I'm having trouble seeing why this is true. What happens to the [itex]a_pa_k[/tex] and [itex]a_p^{\dagger}a_k^{\dagger}[/itex]-like cross terms?