For the classical Klein-Gordon field, the motion of a wavepacket is constrained slower than the speed of light for $m^2 >0$ and constrained to exactly the speed of light for $m = 0$ (the equation of motion is the wave equation in that latter case). However, in quantum mechanics, in determining amplitudes of propagation between different field configurations, the path integral integrates over all paths, not just the classical path.
I expect the fields $\phi(x)$ and $\phi(y)$ at equal times commute, and so we can find simultaneous eigenstates of all the field operators $\phi$ at equal times. I will label these eigenstates by a function $f(\cdot)$: $\phi(0, \vec{x})|f(\cdot)\rangle = f(\vec{x})|f(\cdot)\rangle$. I expect that the amplitude for propagation from field configuration $|f(\cdot)\rangle$ to $|g(\cdot)\rangle$ in time $t$ is $$\langle g(\cdot)| e^{-iHt}|f(\cdot)\rangle = \int_{\phi(0, \vec{x}) = f(\vec{x}),\phi(t, \vec{x}) = g(\vec{x})} D\phi \, e^{iS},$$ where $$S = \int d^n x \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m^2 \phi$$ is the usual Klein-Gordon action.
In classical mechanics, some field configuration with finite support about the origin cannot, at the very next instant in time, have support far away from the origin. However, that is not clear in QFT because the integral is over all paths, not just classically allowed paths.
Can $\langle g(\cdot)| e^{-iHt}|f(\cdot)\rangle$ be nonzero for widely separated field configurations?
If yes, an example calculation showing a nonzero amplitude would be appreciated.
For an example of "separated" field configurations, I have in mind something like $f(\vec{x}) = e^{-L^2/(r^2-|\vec{x}|^2)}\theta(r^2-|\vec{x}|^2)$ and $g(\vec{x}) = e^{-L^2/(r^2-|\vec{x}-\vec{s}|^2)}\theta(r^2-|\vec{x}-\vec{s}|^2)$ for some $|\vec{s}| > 2r$. At very short times, can the amplitude for propagation between such configurations be nonzero?
I'll add a few comments on causality. A typical QFT with Lorentz symmetry enjoys causality. For example, the existence of a unitary representation of the proper, orthochronous Lorentz group and Lorentz invariance of the vacuum implies that the commutator (or anticommutator, for a fermion field) of two fields vanishes at spacelike separation. This property of the commutators, along with a few more assumptions, generally rules out faster-than-light communication (e.g. the open-access Eberhard and Ross).
In this question, I identify a quantity that I believe may "move" faster than light/evolve faster than allowed classically. Even if this quantity does evolve more quickly than naively expected, this still does not mean the end of causality; consider this toy example. However, if it does turn out that we have non-zero amplitudes for propagation between widely separated field configurations at arbitrarily short times, I would still appreciate a comment on resolving anxieties about causality.
