Deriving the Hamiltonian of the Klein-Gordon field in terms of ladder operators (<PERSON> and <PERSON> 2.31)
In <PERSON> and <PERSON>'s QFT book they give \begin{align} H &= \int d^3x\int \frac{d^3p d^3 p'}{(2\pi)^6}e^{i(\mathbf{p+p'})\cdot \bf x}\left{-\frac{\sqrt{\omega_{\bf p}\omega_{\bf p'}}}{4} (a_{\bf p}-a_{\bf -p}^\dagger )(a_{\bf p'}-a_{-\bf p'}^\dagger) +\frac{-\bf p \cdot \bf p' + m^2}{4 \sqrt{\omega_{\bf p} \omega_{\bf p'}}}(a_{\bf p}+a_{-\bf p}^\dagger)(a_{\bf p'}+a_{-\bf p'}^\dagger)\right} \ &=\int \frac{d^3 p}{(2\pi)^3}\omega_{\bf p}\left(a_{\bf p}^\dagger a_{\bf p}+\frac 1 2 [a_{\bf p},a_{\bf p}^\dagger]\right) \end{align} Where $$\omega_{\bf p} = \sqrt{|{\bf p}|^2+m^2}$$ I was trying to get from the first line to the second line but I'm getting stuck and was wondering if someone could nudge me in the right direction. 281