I am self studying QFT on the book "A modern introduction to quantum field theory" by Maggiore and I am reading the chapter about the Dyson series (chapter 5.3).
It states the following equation for the time-evolution of the field $\phi(t,\vec{x})$:
$$\phi(t,\vec{x}) = e^{iH\tau}\phi(t_0,\vec{x})e^{-iH\tau}$$$$\phi(t,\vec{x}) = e^{iH\tau}\phi(t_0,\vec{x})e^{-iH\tau}\tag{5.49}$$
where $\tau=(t-t_0)$, and $H = H_0 + H_{int}$.
Then I guess it assumes that $e^{-iH(t-t_0)}$ is the time-evolution operator.
However, as far as I know from QM this is valid only if $H$ doesn't depend on $t$, but this doesn't seem to be true, since $H$ depends directly or indirectly on the 4-vector $x$.
So I looked to the same subject in the book "Quantum field theory and the standard model" by Schwartz and he states that the time-evolution operator satisfy the following equation (page 84 "Hamiltonian derivation")
$$i\partial_t S(t,t_0) = H(t_0)S(t,t_0)$$$$i\partial_t S(t,t_0) = H(t_0)S(t,t_0).\tag{7.28}$$
Note that it's $H(t_0)$, not $H(t)$, and then the equation has the solution:
$$S(t,t_0) = e^{-iH(t_0)(t-t_0)}$$
which is different from the previous one, $e^{-iH(t-t_0)}$, from Maggiore book.
So I am really confused about what is in QFT the time-evolution operator, and if in that formula $H$ depends on $t$.
