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In [1] it is stated that the creation operators of eq. 2.78

\begin{equation}\label{eqn:scalarFieldCreationOpCommutator:20}

\alpha_k = \inv{2} \int \frac{d^3k}{(2\pi)^3} \lr{

\phi(x,0) + \frac{i}{\omega_k} \partial_0 \phi(x,0)

}

e^{-i \Bk \cdot \Bx }

\end{equation}

associated with field operator \( \phi \) commute. Let’s verify that.

\begin{equation}\label{eqn:scalarFieldCreationOpCommutator:40}

\begin{aligned}

\antisymmetric{\alpha_k}{\alpha_m}

&=

\inv{4}

\frac{1}{(2\pi)^6}

\int d^3 x d^3 y

e^{-i \Bk \cdot \Bx }

e^{-i \Bm \cdot \By }

\antisymmetric

{

\phi(x,0) + \frac{i}{\omega_k} \partial_0 \phi(x,0)

}

{

\phi(y,0) + \frac{i}{\omega_m} \partial_0 \phi(y,0)

} \\

&=

\frac{i}{4}

\frac{1}{(2\pi)^6}

\int d^3 x d^3 y

e^{-i \Bk \cdot \Bx }

e^{-i \Bm \cdot \By }

\lr{

\antisymmetric{\phi(x,0)}{\inv{\omega_m} \partial_0 \phi(y,0)}

+

\antisymmetric{\inv{\omega_k} \partial_0 \phi(x,0)}{\phi(y,0)}

} \\

&=

\frac{i}{4}

\frac{1}{(2\pi)^6}

\int d^3 x d^3 y

e^{-i \Bk \cdot \Bx }

e^{-i \Bm \cdot \By }

\lr{

\frac{i}{\omega_m} \delta^3(\Bx – \By)

–

\frac{i}{\omega_k} \delta^3(\Bx – \By)

} \\

&=

-\frac{1}{4}

\frac{1}{(2\pi)^6}

\int d^3 x

e^{ -i (\Bk + \Bm) \cdot \Bx }

\lr{

\frac{1}{\omega_m}

–

\frac{1}{\omega_k}

} \\

&=

-\frac{1}{4}

\frac{1}{(2\pi)^3}

\lr{

\frac{1}{\omega_m}

–

\frac{1}{\omega_k}

}

\delta^3(\Bk + \Bm) \\

&=

-\frac{1}{4}

\frac{1}{(2\pi)^3}

\lr{

\frac{1}{\omega_{\Abs{-\Bk}}}

–

\frac{1}{\omega_{\Abs{\Bk}}}

}

\delta^3(\Bk + \Bm) \\

&=

0.

\end{aligned}

\end{equation}

# References

[1] Michael Luke. *PHY2403F Lecture Notes: Quantum Field Theory*, 2015. URL https://s3.amazonaws.com/piazza-resources/i87nj8g7yie7nh/ihdwuk7wva13qq/lecturenotes.pdf?AWSAccessKeyId=AKIAIEDNRLJ4AZKBW6HA&Expires=1451803428&Signature=IF6qOjlKqOYL01FwqT%2FGV6BSDb8%3D. [Online; accessed 02-Jan-2016].