<PERSON> formula vs <PERSON> formula for 1pt functions
I have a basic confusion about the relation between the in-in and the in-out 1pt functions, in perturbation theory. Or in other terms: I am not able to resolve a conflict between the <PERSON> formula and the <PERSON> expansion. I would like to understand how to make compatible these two well known perturbative expansions when applied to the 1pt functions.
Let's recall the <PERSON> perturbative expansion for the split of the hamiltonian $H=H_0+H_{int}$ (where in general $H_{int}$ hence $H$ can depend on time via external currents, but I will assume $H_0$ does not), specialized to 1pt functions $$ \langle \Omega|\phi(x)|\Omega\rangle=\frac{\langle0|T\left(\phi(x)S\right)|0\rangle}{\langle0|S|0\rangle}\,,\qquad S\equiv T\mathrm{exp}-i\int_{-\infty}^{+\infty}H_I(\tau)d\tau $$ where ${}{I}$ stands for the interaction picture relative to $H_0$, that is $H_0|0\rangle=0$, $H_I=H{int}(\phi_I)$ and $\phi_I(t,\vec{x})=e^{i (t-t_0) H_0 }\phi(t_0,\vec{x})e^{-i (t-t_0) H_0}$. 281
By $|\Omega\rangle$ I mean the true vacuum of the full Hamiltonian $H$.
Since the Heisenberg field evolves as $\phi(t,x)=V(t,t_0)^{-1}\phi(t_0,\vec{x})V(t,t_0)$ where $V(t,t^\prime)$ is the exact time-evolution operator, $i\partial_t V(t,t^\prime)=H(t) V(t,t^\prime)$, the relation to $\phi_I(t,\vec{x})$ is $$ \phi=U(t,t_0)^{-1}\phi_I(x)U(t,t_0)\qquad U=e^{i(t-t_0)H_0}V(t,t_0)%\,,\qquad V(t,t_0)^{-1}e^{-i(t-t_0)H_0}\phi_I(x)e^{i(t-t_0)H_0}V(t,t_0)\,. $$ The $U$ solves the evolution equation $i\partial_t U(t,t_0)=H_I(t)U(t,t_0)$ with initial condition $U(t_0,t_0)=1$ so the solution being $$ U(t,t_0)=T\mathrm{exp}-i\int_{t_0}^t d\tau H_I(\tau)\,. $$ Therefore, the 1pt function can as well be written as as in-in correlator $$ \langle\Omega|\phi(t,\vec{x})|\Omega\rangle=\langle\Omega |U(t,t_0)^{-1}\phi_I(t,\vec{x})U(t,t_0)|\Omega\rangle=\langle\Omega|\left(\bar{T}\mathrm{exp}i\int_{t_0}^t d\tau H_I(\tau)\right) \, \phi_I(t,\vec{x}) \, \left( T\mathrm{exp}-i\int_{t_0}^t d\tau H_I(\tau)\right)|\Omega\rangle $$ where $\bar{T}$ is the anti-time ordering. 281