**Background**

---

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**

---

How does relativistic quantum mechanics deal with this paradox?