Does Hardy's paradox represent a proof against Bohm's interpretation of the quantum mechanics?

Question

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.

Answer 1

The answer I give here is the result of many discussions with experts, and I heard many opinions in all the directions. So, I write here what is known to me by the time that I give this answer.

1. There are three interpretations known as the most elaborated, most investigated:

• a) The standard (or Copenhagen) interpretation + the von Neumann's reduction postulate, which can act non-locally;
• b) Bohm's interpretation, that rules out the collapse, and recognizes that entanglements work on a non-local base;
• c) GRW (Ghirardi, Rimini and Weber) interpretation that tries to simulate the collapse by modifying the Schrodinger equation, i.e. introducing a stochastic term and non-local potential.

2. As detailed above, the idea that the entanglements resort to some sort of non-local transmission of information between the measured particles, could not be avoided. On top of this problem comes the relativity theory and tells us that space-separated measurements on particles belonging to an entanglement, don't have an absolute time-order. According to some frame of coordinates the particle A is measured first, while according to another frame, in relative movement, the particle B is measured first. This fact opens a question: $$The \ measurement \ of \ one \ of \ particles \ collapses \ the \ entanglement?$$

How can it be possible? There exists frames of coordinates according to which, by the time the above particle is measured, the other particle didn't reach the detector.

$$Then, it \ is \ not \ at \ the \ contact \ with \ a \ macroscopic \ detector \ that \ the \ collapse \ occurs?$$

$$If \ so, \ could \ it \ be \ that \ Bohm's \ interpretation, \ that \ eliminates \ the \ collapse, \ is \ correct?$$

3. Hardy's thought experiment (actually performed by Aephraim Steinberg's group), shows that Bohm's interpretation has difficulties with the relativity, as explained in the question. This thought-experiment belongs to the class of contextual experiments and applies counterfactual reasoning, but Bohm's interpretation is compatible with counter-factual reasoning.

As a consequence, some researchers are inclined to think that the quantum world admits a preferred frame, despite the theory of relativity which refutes such things.