# Thought experiment in relativistic quantum mechanics?

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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:

$$H | \psi \rangle = E | \psi \rangle$$

Now I suddenly turn on an interaction potential $H_{int}$ localized at $r_o = (x_o,y_o,z_o)$ at time $t_o$:

$$H_{int}(r) = \begin{cases} k & r \leq r_r' \\ 0 & r > r' \end{cases}$$

where $r$ is the radial coordinate and $r'$ is the radius of the interaction of the potential with origin $(x_o,y_o,z_o)$

By the logic of the sudden approximation out state has not had enough time to react. Thus the increase in average energy is:

$$\langle \Delta E \rangle = 4 \pi k \int_0^{r'} |\psi(r,\theta,\phi)|^2 d r$$

(assuming radial symmetry).

Now, lets say while the potential is turned on at $t_0$ I also perform a measurement of energy at time $t_1$ outside a region of space with a measuring apparatus at some other region $(x_1,y_1,z_1)$. Using some geometry it can be shown I choose $t_1 > t_0 + r'/c$ such that:

$$c^2(t_1 - t_0 - r'/c)^2 -(x_1 - x_0)^2 - (y_1 - y_0)^2 - (z_1 - z_0)^2 < 0$$

Hence, they are space-like separated. This means  I could have one observer who first sees me turn on the potential $H_{int}$ and measure a bump in energy $\langle \Delta E \rangle$ 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?

asked Jan 16

QM has nothing to do with your behavior.

Direct (retarded) observation results in the relativistic mechanics are recalculated to the proper reference frame. For example, even in Classical mechanics, when you compare the lengths of two equal rods, one of which is next to you and the other is far away (and thus looks "small"), you conclude that the distant rod is of the same length - by recalculating its apparent length to your proper RF.

@Vladimir in the paradox since they are spacelike separated. Observer 2 will see the  measurement first and then the change in potential and conclude that the experimenter will observe $\langle H \rangle$ whereas the first Observer will see the change in potential first and then the measurement and conclude the experimenter has observed $\langle H + H_{int} \rangle$. Both can obviously not be correct.

Two spacelike separated events may be observed as simultaneous in one RF ($\Delta t=0$) and time-separated in another; the sequence of events depending on particular RF position.

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