This is a thought-experiment, see "Quantum Mechanics, Local Realistic Theories, and Lorentz-Invariant Realistic Theories", Phys. Rev. Lett., Vol. 68, No. 20, page 2981, year 1992, that rules out local hidden variables. But does it also rule out nonlocal influence of the measurement result of one particle, on the wave-function of another particle, before it was measured? Is this paradox a proof of impossibility of Bohm's interpretation of the quantum mechanics?
The experiment goes as follows: an electron $e^-$ and a positron $e^+$ land, each one, on a beam-splitter, $BS1^-$ respectively $BS1^+$. One gets the following joint wave-function of two independent particles:
$ (1) \ e^+ \to \frac {|v^+> + i|u^+>}{\sqrt (2)} $
$ (2) \ e^- \to \frac {|v^-> + i|u^->}{\sqrt (2)} $
However, the couple $|u^+>|u^->$ annihilates at the point P, with production ofgamma rays, s.t. an entanglement appears
$ (3) \ |\psi> = \frac {1}{2} (|v^+>|v^-> + i|u^+>|v^-> + i|v^+>|u^-> + |2\gamma>). $
The "remainder" of the beams of the positron and electron are gathered by the beam-splitters $BS2^-$ and $BS2^+$, at which the following transformations take place.
$ (4) \ |v^±> → \frac {i|c^± > + |d^± >}{\sqrt (2)}, $
$ (5) \ |u^±> → \frac {|c^± > + i|d^± >}{\sqrt (2)} . $
So, in all, we get after $BS2^-$ and $BS2^+$ one gets the following wave-function:
$ (6) \ |\psi> → \frac {1}{4} (−3|c^+>|c^−> + i|c^+>|d^−> + i|d^+>|c^−> − |d^+>|d^−> + |2\gamma>) $.
There is nothing special until now, unless the experiment is judged by people traveling in opposite frames of coordinates.
In what follows we are interested in the detections in the detectors $D^+$ and $D^-$. Let's accompany an analyst that travels in a frame $I^+$ in which the positron is at rest. From his point of view, the positron reaches $BS2^+$ before the electron reaches $BS2^-$, s.t. after the positron reaches $BS2^+$ the wave-function is
$ (7) \ |\psi> = \frac {1}{4}(−|c^+>|u^−> + 2i|c^+>|v^−> + i|d^+>|u^−> + |2\gamma>) .$
According to the before last term, the detection in $D^+$ leaves the electron on the path $u^-$.
But an analyst that travels in the frame $I^-$ in which the electron is at rest, would hold the opposite, i.e. that the electron reaches $BS2^-$ before that positron reaches $BS2^+$, and the following wave-function is bound to appear
$ (8) \ |\psi> = \frac {1}{4}(−|u^+>|c^-> + 2i|v^+>|c^-> + i|u^+>|d^-> + |2\gamma>) .$
So, he holds that after the detection in $D^-$, the positron should have been left on the path $u^+$.
Here is the problem: the combination $|u^+>|u^->$ does not exist, it was destroyed into gamma rays. And if the gamma rays appeared, the detections in the detectors $D^±$ and $C^±$ wouldn't be obtained.