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Hardy’s paradox made simple – what we infer from it?

Authors:
  • retired from Technion Israeli Institute of Technology

Abstract

When the Bohmian mechanics became a serious hope to get rid of the enigmatic postulate of “collapse” of the wavefunction, Hardy’s paradox came and showed that the idea of continuous trajectories for particles – not only Bohmian trajectories, but any continuous trajectories – is in conflict with the relativity. In addition, Hardy’s rationale showed that in experiments with entangled particles, it is a hard problem how to describe the state of the system of particles in the situation that one particle is tested and the other not yet. Admitting “collapse at a distance”, i.e. that the measurement of one particle collapses the description of the other particle to a certain state, is at odds with the relativity theory. It seems that situation of the particle not yet tested remains an entangled state, as long as it does not encounter a macroscopic apparatus of measurement.
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... This interpretation shows a major weakness when applied to entanglements and relativistic situations, as found by Berndl et al. [20] in an analysis of the consequences of the famous Hardy's paradox [21]. A simpler explanation of the contradiction can be found in [22]. It is repeated below in general lines. ...
... In a frame of coordinates + F flying in the direction from the lab where is tested p − to the lab where is tested p + , the time axis would show that when p + is detected, p − didn't yet meet the beam-splitter BS − . The calculi in [22] show that the corresponding state of the system is ( ) ...
... By symmetry, according to the time axis of a frame of coordinates − F flying in opposite direction than + F , by the time when p − is detected, p + didn't yet meet the beam-splitter BS + . The calculi in [22] show that the state reflecting this situation is ( ) ...
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The postulate of the collapse of the wave-function stands between the micro-scopic, quantum world, and the macroscopic world. Because of this interme-diate position, the collapse process cannot be examined with the formalism of the quantum mechanics (QM), neither with that of classical mechanics. This fact makes some physicists propose interpretations of QM, which avoid this postulate. However, the common procedure used in that is making assump-tions incompatible with the QM formalism. The present work discusses the most popular interpretations. It is shown that because of such assumptions those interpretations fail, i.e. predict for some experiments results which differ from the QM predictions. Despite that, special attention is called to a proposal of S. Gao, the only one which addresses and tries to solve an obvious and major contradiction. A couple of theorems are proved for showing that the collapse postulate is necessary in the QM. Although non-explainable with the quantum formalism, this postulate cannot be denied, otherwise one comes to conclusions which disagree with the QM. It is also proved here that the idea of “collapse at a distance” is problematic especially in relativistic cas-es, and is a misunderstanding. Namely, in an entanglement of two quantum systems, assuming that the measurement of one of the systems (accompanied by collapse of that system on one of its states) collapses the other systems, too without the second system being measured, which leads to a contradiction.
... This interpretation shows a major weakness when applied to entanglements and relativistic situations, as found by Berndl et al. [20] in an analysis of the consequences of the famous Hardy's paradox [21]. A simpler explanation of the contradiction can be found in [22]. It is repeated below in general lines. ...
... In a frame of coordinates  F flying in the direction from the lab where is tested  p to the lab where is tested  p , the time axis would show that when  p is detected,  p didn't yet meet the beam-splitter  BS . The calculi in [22] show that the corresponding state of the system is ] [ ...
... By symmetry, according to the time axis of a frame of coordinates  F flying in opposite direction than  F , by the time when  p is detected,  p didn't yet meet the beam-splitter  BS . The calculi in [22] show that the state reflecting this situation is ] [ ...
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Accepted for publication in JQIS, vol. 11, no. 1, March 2021 ***************************************************************************************** The postulate of the collapse of the wave-function stands between the microscopic, quantum world, and the macroscopic world. Because of this intermediate position, the collapse process cannot be examined with the formalism of the quantum mechanics (QM), neither with that of classical mechanics. This fact makes some physicists to propose interpretations of QM, which avoid this postulate. However, the common procedure used in that, is making assumptions incompatible with the QM formalism. The present work discusses the most popular interpretations. It is shown that because of such assumptions those interpretations fail, i.e. predict for some experiments results which differ from the QM predictions. Despite of that, special attention is called to a proposal of S. Gao, the only one which addresses and tries to solve an obvious and major contradiction. A couple of theorems are proved for showing that the collapse postulate is necessary in the QM. Although nonexplainable with the quantum formalism this postulate cannot be denied, otherwise one comes to conclusions which disagree with the QM. It is also proved here that the idea of ‘collapse at a distance’ is problematic especially in relativistic cases, and is a misunderstanding. Namely, in an entanglement of two quantum systems, assuming that the measurement of one of the systems (accompanied by collapse of that system on one of its states) collapses the other system too without the second system being measured, leads to a contradiction.
... It is known from the famous example with Schrödinger's cat, that such a thing is impossible. 9 The position in [17] and the explanation in [18] solve the famous "Hardy's paradox" [19] (see also [20]) in the way proposed by Berndl and Goldstein [21] and in agreement with the famous Peres' dictum "unperformed experiments have no results", [22]. ...
... For the equation (20) to reduce to the Schrödinger equation, the last two terms in (20i) must bring a negligible contribution to the solution, in comparison with t H d i . It will be seen in the end of this section and in the next section that as long as a system consists in a few microscopic components the last two terms have a negligible effect. ...
... Introducing (22) in (20) and projecting on the eigenstate  m |a , ...
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Journal of Quantum Information Science (JQIS) volume 10, number 4, December 2020 ********************************************************************************** Different attempts to solve the measurement problem of the quantum mechanics (QM) by denying the collapse principle, and replacing it with changes in the quantum formalism, failed because the changes in the formalism lead to contradictions with QM predictions. To the difference, Ghirardi, Rimini and Weber took the collapse as a real phenomenon, and proposed a calculus by which the wave-function should undergo a sudden localization. Later on, Ghirardi, Pearle and Rimini came with a change of this calculus into the CSL (continuous spontaneous localization) model of collapse. Both these proposals rely on the experimental fact that the reduction of the wave-function occurs when the microscopic system encounters a macroscopic object and involves a big amount of its particles. Both these proposals also change the quantum formalism by introducing in the Schrödinger equation additional terms with noisy behavior. However, these terms have practically no influence as long as the studied system contains only one or a few components. Only when the amount of components is very big, these term become significant and lead to the reduction of the wave-function to one of its components. The present work has two purposes: 1) proving that the collapse postulate is unavoidable; 2) applying the CSL model to the process in a detector and showing step by step the modification of the wave-function, until reduction. As a side detail, it is argued here that the noise cannot originate in some classical field, contrary to the thought/hope of some physicists, because no classical field is tailored by the wave-functions of entanglements.
... Though, about seventy years from de Broglie's Ph.D. thesis and theory of waves [1,2], and about forty years after the publication of Bohm's famous articles [3], [4], L. Hardy found that the Bohmian trajectories are not relativistically covariant [5]. It cast the first strong doubt on the dB-B mechanics as explained in detail by Berndl et al. [6] (see also [7] for a simpler explanation). The supporters of the dB-B mechanics had to accept that its formalism is valid only in the non-relativistic domain. ...
... The problem found in [5], [6], [7] doesn't stem from the dB-B formulas, contrary to the opinion of the authors of the ESSW thought experiment [8], [9] 1 ; it stems from the basic assumption of continuous trajectories. This is the central idea of the proof in the present work, that quantum objects don't follow continuous paths. ...
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A thought experiment is described and the probability of a particular type of results is predicted according to the quantum formalism. Then, the assumption is made that there exists a particle that travels from the source to one of the detectors, along a continuous trajectory. A contradiction appears: for agreeing with the quantum prediction, the particle has to land at once on two space separated detectors. Therefore, the trajectory of the particle, if it exists, cannot be continuous.
... Though, about seventy years from de Broglie's Ph.D. thesis and theory of waves [1,2], and about forty years after the publication of Bohm's famous articles [3], [4], L. Hardy found that the Bohmian trajectories are not relativistically covariant [5]. It cast the first strong doubt on the dB-B mechanics as explained in detail by Berndl et al. [6] (see also [7] for a simpler explanation). The supporters of the dB-B mechanics had to accept that its formalism is valid only in the non-relativistic domain. ...
... The problem found in [5], [6], [7] doesn't stem from the dB-B formulas, contrary to the opinion of the authors of the ESSW thought experiment [8], [9] 1 ; it stems from the basic assumption of continuous trajectories. This is the central idea of the proof in the present work, that quantum objects don't follow continuous paths. ...
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This article is not a chapter from a book, but a section from a much longer article. I noticed that, while the mathematics is not a problem, people have difficulty (or, lack of patience) to follow the logic which deals with the continuous trajectories. This is why I decided to dedicate a separate article to the issue and explain it in detail. I'd be glad, if something is still not clear, to be notified. ***************************************************************************************** A thought-experiment is described and the probability of a particular type of results is predicted according to the quantum formalism. Then, the assumption is made that there exists a particle that travels from the source to one of the detectors, along a continuous trajectory. A contradiction appears: for agreeing with the quantum prediction, the particle has to land at once on two space-separated detectors. Therefore, the trajectory of the particle – if it exists – cannot be continuous.
... results the velocity of the dBB atom in the region I, according to the formalism in[1] and[2], The ESSW experiment.3 The setup comprises SG apparatuses, two cavities, and a screen sensitive to the atoms. ...
... Today it is known that not the velocity formula is the cause of the incompatibility between the dBB interpretation and QM, but the assumption of continuous trajectory of the substructure object supposed to trigger the detectoras proved in[3] and section 3 of[4]. ...
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The formalism of the de Broglie-Bohm (dBB) mechanics was constructed such as to avoid the collapse principle. However, dBB supplemented the quantum formalism with a hypothesis that seems incompatible with the uncertainty principle, a substructure particle traveling on a definite trajectory under the wave-function guidance, which means that the particle has simultaneously definite position and velocity. They proved that the dBB mechanics should make the same predictions as the quantum mechanics (QM). However, some physicists regarded with suspicion this substructure, especially the velocity formula of the dBB mechanics. One of the earliest trials to exemplify such an incompatibility is due to Englert, Scully, Süssmann, and Walther (ESSW). They proposed a thought-experiment, and analyzed it with the formalism of the dBB mechanics. The conclusion was that the dBB mechanics predicts trajectories for particles, which don’t fit what one would infer from a QM analysis. Their work triggered a whole debate which continues until today. However, at least in all the relevant works found by the present author, the judgements were done on modified configurations of the experiment. In consequence, as shown here, the conclusions of those works are irrelevant. The present article also analyses the ESSW experiment. Though, to the difference from all the other works, the original configuration is examined. Here too, an error is found in the ESSW mathematical treatment, namely, although arguing in base of an entanglement, their mathematical treatment averages over one of the particles eliminating the entanglement.
... However, despite the positive fact of removing the need of the enigmatic postulate of collapse of the wave-function, dBBI was proved wrong. The continuous trajectories of the particles were proved incompatible with the theory of relativity [15,16]. Recently, it was proved that even without invoking the relativity such trajectories are incompatible with the experimentsection 5 in [17]. ...
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For seeing the publication go to the page: https://www.researchgate.net/publication/335229497_The_Wave-Particle_Duality-Does_the_Concept_of_Particle_Make_Sense_in_Quantum_Mechanics_Should_We_Ask_the_Second_Quantization
... However, despite the positive fact of removing the need of the enigmatic postulate of collapse of the wave-function, dBBI was proved wrong. The continuous trajectories of the particles were proved incompatible with the theory of relativity [15,16]. Recently, it was proved that even without invoking the relativity such trajectories are incompatible with the experimentsection 5 in [17]. ...
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