Thought experiment in relativistic quantum mechanics?

[Someone_physics]

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?

 

[vanhees71]

It deals with it by using QFT. A 1st-quantization formalism is doomed to fail, precisely because of the causality considerations you just observed!

 

[Someone_physics]

It deals with it by using QFT. A 1st-quantization formalism is doomed to fail, precisely because of the causality considerations you just observed!

Can I have a reference for this? I've skimmed through a book of RQM (https://www.springer.com/gp/book/9783540674573) which makes not mention of this :/

 

[vanhees71]

It's because it's a book about "relativistic quantum mechanics". A nice heuristic argument is given in the beginning of the well-known textbook by Peskin and Schroeder (though in general I'd rather recommend Schwartz as a relativistic QFT intro book).

 

[Someone_physics]

It's because it's a book about "relativistic quantum mechanics". A nice heuristic argument is given in the beginning of the well-known textbook by Peskin and Schroeder (though in general I'd rather recommend Schwartz as a relativistic QFT intro book).

I'll have a look.

In a similar spirit to page 21 I can modify the last inequality by stating the time taken for the sudden approximation to be valid is

$$\tau = t_{1/2} - t_0 >> \frac{\hbar}{\langle \Delta E \rangle}$$

The time after the sudden approximation is measured is given by $\Delta t_1 = t_1 - t_{1/2}$ then:$$c^2 (\Delta t_1 - \frac{\hbar}{ \langle \Delta E \rangle} - r'/c)^2 < (x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 -z_0)^2$$

I haven't seen this expression before as a breakdown condition for QM