Thought experiment in relativistic quantum mechanics?

+ 1 like - 0 dislike
95 views

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:

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

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?

@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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.