Maxwell field commutation relations(adsbygoogle = window.adsbygoogle || []).push({});

I'm looking at Aitchison and Hey's QFT book. I see in Chapter 7, (pp. 191-192), they write down the canonical momentum for the Maxwell field [itex]A^\mu(x)[/itex]:

[tex]

\pi^0=\partial_\mu A^\mu \\

\pi^i=-\dot{A}^i+\partial^i A^0

[/tex]

and then write down the commutation relations

[tex]

[\hat{A}_\mu(\boldsymbol{x},t),\hat{\pi}_\nu(\boldsymbol{y},t)]=ig_{\mu\nu}\delta^3(\boldsymbol{x}-\boldsymbol{y})

[/tex]

and state that if you assume the commutation relations

[tex]

[\hat{A}_\mu(\boldsymbol{x},t),\hat{A}_\nu(\boldsymbol{y},t)]=[\hat{\pi}_\mu(\boldsymbol{x},t),\hat{\pi}_\nu(\boldsymbol{y},t)]=0

[/tex]

we see that the spatial derivatives of the [itex]\hat{A}[/itex]'s commute with the [itex]\hat{A}[/itex]'s, and with each other, at equal times.

They state it as if it's obvious, so maybe I'm missing something, but I don't see why the spatial derivatives of the [itex]\hat{A}[/itex]'s commute with the [itex]\hat{A}[/itex]'s, and with each other.

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# A Maxwell field commutation relations

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