Background
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Consider the following thought experiment in the setting of relativistic quantum mechanics (not QFT). I have a particle in superposition of the position basis:
H|ψ⟩=E|ψ⟩
Now I suddenly turn on an interaction potential Hint localized at ro=(xo,yo,zo) at time to:
Hint(r)={kr≤r′r0r>r′
where r is the radial coordinate and r′ is the radius of the interaction of the potential with origin (xo,yo,zo)
By the logic of the sudden approximation out state has not had enough time to react. Thus the increase in average energy is:
⟨ΔE⟩=4πk∫r′0|ψ(r,θ,ϕ)|2dr
(assuming radial symmetry).
Now, lets say while the potential is turned on at t0 I also perform a measurement of energy at time t1 outside a region of space with a measuring apparatus at some other region (x1,y1,z1). Using some geometry it can be shown I choose t1>t0+r′/c such that:
c2(t1−t0−r′/c)2−(x1−x0)2−(y1−y0)2−(z1−z0)2<0
Hence, they are space-like separated. This means I could have one observer who first sees me turn on the potential Hint and measure a bump in energy ⟨ΔE⟩ but I could also have an observer who sees me first measure energy and then turn on the interaction potential.
Obviously the second observer will observe something different.
Question
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How does relativistic quantum mechanics deal with this paradox?