Relativistic particles and quantum fields and time like cone Considering that a "classical" relativistic particle remains inside the time-like cone, does it guarantee that a quantum field must also do that for each of its path history? or it has to be do it on average?
 A: Fields extend through all of space so one can easily get confused talking about their lightcones. But as far as I can tell you are thinking about quantum field theories of  two different types.

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*Those in which $\left < \psi_f \right | U(0, t) \left | \psi_i \right > = 0$ whenever $\left | \psi_i \right >$ is a wave packet of compact support (call it $S$) and $\left | \psi_f \right >$ is a wave packet supported on points that cannot be reached from $S$ by light rays in time $t$.

*Those in which the amplitudes above (obtained by summing over histories) are non-zero but suppressed compared to causal ones.

We are usually much happier dealing with the first type of QFT. In practice, what we usually look for in a theory is micro-causality, the property that field operators commute at space-like separated points. The micro-causality property is very similar to the statement above phrased in terms of wave packets but there are a lot of subtleties that make me unsure if they're rigorously equivalent. One of these is the Reeh-Schlieder theorem, discussed in Witten's notes, which states that the full Hilbert space can be constructed by acting on the vacuum with operators supported in an arbitrarily small open set. However, this does not contradict micro-causality because these operators which allow you to get everything are not necessarily unitary.
Anyway, the first part of your question is whether this follows from the classical limit. If the theory is not free, it probably doesn't. Ensuring that causality is still preserved by quantum corrections restricts the types of interactions you can add when building an effective field theory. But finding equivalent rephrasings of causality so that checking it eventually becomes less complicated is something that's currently being pursued.
