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I haven't been able to find an answer online but this seems like a pretty basic question to me. What are the commutator relations between the position/momentum operators and the field operator?

I'm not even certain what the commutation relations between X/P and a single ladder operator are in the QFT context. With the harmonic oscillator this is trivial, but it QFT we don't really define the ladder operators in terms of P/X, but rather as Fourier components of a classical field (and then proceed with canonical quantization (at least that's how I learned it)).

A simple motivation for this can be to show that a 2nd quantization Hamiltonian commutes with P for example (thus deducing that momentum is conserved in the system), however this would require knowledge of the commutator relations with the field operator.

Thanks for the help!

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# QFT - Commutator relations between P,X and the Field operator

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