I am trying to work through Peskin and Schröder and am a little stuck in Chapter 4 [section 4.2 p. 83 below eq. (4.13)], when he first treats interacting fields. The subject is the quartic interaction in Klein-Gordon Theory. They claim that:
"At any fixed time, we can of course expand the field (in the interacting theory) as before (in the free theory) in terms of ladder operators."
I don't see why this should be possible in general. Their argument for the ladder operators and the expansion in plane waves in the case of the free theory was that from the Klein-Gordon equation we get Fourier modes independently satisfying harmonic oscillator equations. However as far as I can see, once I add an interacting term I don't get these equations anymore.