Related questions Expanding field operators at a fixed time $t_0$ (from Peskin/Schroeder) About an expression of Peskin and Schroeder
I saw this question has been asked a couple of times in different contexts. But I am still unsure about one aspect. Let me start from Expanding field operators at a fixed time $t_0$ (from Peskin/Schroeder) (a bit editting to be more similar to P&S page 83)
At any fixed time $t_0$, we can of course expand $\phi$ in terms of ladder operators: $$\phi(t_0, \textbf{x})=\displaystyle\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\mathbf{p}} }}(a_{\mathbf{p}} e^{i\textbf{p.x}}+a_{\mathbf{p}} ^\dagger e^{-i\textbf{p.x}}) (1) $$ Then to obtain $\phi(t, \textbf{x})$ for $t\neq t_0$ we just switch to the Heisenberg picture as usual: $$\phi(t,\textbf{x})=e^{iH(t-t_0)}\phi(t_0,\textbf{x})e^{-iH(t-t_0)} (2) $$
If I choose $t_1 \neq t_0$, then
$$\phi(t_1, \textbf{x})=\displaystyle\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\mathbf{p}} }}(a_{\mathbf{p}} e^{i\textbf{p.x}}+a_{\mathbf{p}} ^\dagger e^{-i\textbf{p.x}}) (3) $$
Eq. (3) will be the same as (1)? I may use Eq. (2) get $$\phi(t_1,\textbf{x})=e^{iH(t_1-t_0)}\phi(t_0,\textbf{x})e^{-iH(t_1-t_0)} (4) $$ ,which looks confusing if $ \phi(t_1,\textbf{x}) = \phi(t_0,\textbf{x}) $
Some comment from About an expression of Peskin and Schroeder suggest the mass in $E_{\mathbf{p}}$ may differ than the field theory, but this term may come from normalization (page 23 of P&S) Some comment from Expanding field operators at a fixed time $t_0$ (from Peskin/Schroeder) suggest the ladder operator will implicitly depends on time, but how?
I guess $1/\sqrt{2 E_{\mathbf{p}}}$ is kind of expansion coefficient of Fourier mode for the interacting $\phi$ which beyond normalizations, then it may differ than the free theory. And, later P&S choose $\lambda = 0$, do they assume any kind of adiabatic switching on interactions?
