- #1
Someone_physics
- 6
- 1
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, let's 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?
---
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, let's 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?
Last edited: