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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 $\phi(t,\vec{x})$:

$$\phi(t,\vec{x}) = e^{iH\tau}\phi(t_0,\vec{x})e^{-iH\tau}\tag{5.49}$$

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")

$$i\partial_t S(t,t_0) = H(t_0)S(t,t_0).\tag{7.28}$$

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 $t$.

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Maggiore is assuming there that the field interacts with itself or with other fields in a stationary way. In other words, there is no direct appearance of time in the total Hamiltonian.

This is a quite fair hypothesis (homogeneity of time) when describing evolution of closed systems in an inertial reference frame.

In this case the total Hamiltonian operator $$H = \int_\Sigma :\hat{T}_{00}(x): d^3x$$ is a constant of motion, that is $H$ is time-indipendent, if $T_{\mu\nu}$ includes the contributions of all parts of the system, assuming that the total system is isolated.

Schwartz instead considers a case where there is also an external system $S'$ whose evolution is given and that interacts with the studied system $S$.

Here time in the Hamiltonian of $S$ shows up due to the external system.

This situation is typical in the case of a quantum field or a quantum system in external background. The price to pay is that, looking at $S$ only, Poincaré covariance is broken, whereas Maggiore deals with a case where there is a full representation of the Poincaré symmetry.

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  • $\begingroup$ About homogeneity of time, does that mean $H$ indirectly depends on the 4-vector $x$ and then on t, through the field $\phi(x)$, but in the end it's a constant of motion? In other words, does in this case the Hamiltonian can still depend indirectly on time? $\endgroup$
    – Andrea
    Commented Jun 24 at 13:04
  • $\begingroup$ Asking cause a common example to get Feynman rule in the books is to use the interaction $\phi^4$ so in this case it depends on $\phi(x)$ and then on $t$ indirectly through $x$ $\endgroup$
    – Andrea
    Commented Jun 24 at 13:10
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    $\begingroup$ YES: When you integrate $T_{00}$ over the rest space $\Sigma$ in $d^3x$, the $\vec{x}$ dependence disappears and the $t$ dependence disappears as well because the integral is a constant of motion. That is true also for the $\phi^4(x)$ model, of course. $\endgroup$
    – Valter Moretti
    Commented Jun 24 at 15:05

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