Even if this quantity does evolve more quickly than naively expected, this still does not mean the end of causality consider this toy example. In this question, I identify a quantity that I believe may "move" faster than light/evolve faster than allowed classically. This property of the commutators, along with a few more assumptions, generally rules out faster-than-light communication (e.g. 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. A typical QFT with Lorentz symmetry enjoys causality. At very short times, can the amplitude for propagation between such configurations be nonzero? I will label these eigenstates by a function $f(\cdot)$: $\phi(0, \vec| > 2r$.
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. 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.
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).
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