Hello! I am reading Peskin's book on QFT and I reached a part (in chapter 4) where he is analyzing the two-point correlation function for ##\phi^4## theory. At a point he wants to find the evolution in time of ##\phi##, under this Hamiltonian (which is basically the Klein-Gordon - ##H_0## - one plus the interaction one). Anyway, when he begins his derivation he says that for a fixed time ##t_0## we can still expand ##\phi## in terms of ladder operators in the same way as we did in the free (non-interaction) case (this is on page 83), i.e. ##\phi(t_0,\mathbf{x})=\int{\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_\mathbf{p}}}(a_\mathbf{p}e^{i\mathbf{p}\mathbf{x}}+a_\mathbf{p}^\dagger e^{-i\mathbf{p}\mathbf{x}})}##. I am not sure I understand why can we do this, for a fixed time. When we wrote this in term of ladder operators for the free case, we used the KG equation in the free case, which resembled to a harmonic oscillator in the momentum space, and hence we got the ladder operators. But now, the equation of motion is different (it has ##\phi^3## term, instead of 0, as before), so can someone explain to me why we can still use the same formula as before, even if the equation of motion is different? Thank you!(adsbygoogle = window.adsbygoogle || []).push({});

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# I Two point correlation function in Peskin book

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