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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.

Content uploaded by Sofia D. Wechsler

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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 ( ) ...

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 ] [ ...

Accepted for publication in JQIS, vol. 11, no. 1, March 2021
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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 , ...

Journal of Quantum Information Science (JQIS) volume 10, number 4, December 2020
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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. ...

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. ...

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.
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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.

... 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]. ...

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]. ...

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]. ...

The quantum object is in general considered as displaying both wave and particle nature. By particle is understood an item localized in a very small volume of the space, and which cannot be simultaneously in two disjoint regions of the space. By wave, to the contrary, is understood a distributed item, occupying in some cases two or more disjoint regions of the space. The quantum formalism did not explain until today the so-called "collapse" of the wave-function, i.e. the shrinking of the wave-function to one small region of the space, when a macroscopic object is encountered. This seems to happen in "which-way" experiments. A very appealing explanation for this behavior is the idea of a particle, localized in some limited part of the wave-function. The present article challenges the concept of particle. It proves in base of a variant of the Tan, Walls and Collett experiment, that this concept leads to a situation in which the particle has to be simultaneously in two places distant from one another-situation that contradicts the very definition of a particle. Another argument is based on a modified version of the Afshar experiment, showing that the concept of particle is problematic. The concept of particle makes additional difficulties when the wave-function passes through fields. An unexpected possibility to solve these difficulties seems to arise from the cavity quantum electrodynamics studies done recently by S. Savasta and his collaborators. It involves virtual particles. One of these studies is briefly described here. Though, experimental results are expected, so that it is too soon to draw conclusions whether it speaks in favor, or against the concept of particle.

Interpretations of quantum mechanics (QM), or proposals for underlying theories, that attempt to present a definite realist picture, such as Bohmian mechanics, require strong non-local effects. Naively, these effects would violate causality and contradict special relativity. However if the theory agrees with QM the violation cannot be observed directly. Here, we demonstrate experimentally such an effect: we steer the velocity and trajectory of a Bohmian particle using a remote measurement. We use a pair of photons and entangle the spatial transverse position of one with the polarization of the other. The first photon is sent to a double-slit-like apparatus, where its trajectory is measured using the technique of Weak Measurements. The other photon is projected to a linear polarization state. The choice of polarization state, and the result, steer the first photon in the most intuitive sense of the word. The effect is indeed shown to be dramatic, while being easy to visualize. We discuss its strength and what are the conditions for it to occur.

We consider two separate atoms interacting with a single-mode optical or microwave resonator. When the frequency of the resonator field is twice the atomic transition frequency, we show that there exists a resonant coupling between one photon and two atoms, via intermediate virtual states connected by counterrotating processes. If the resonator is prepared in its one-photon state, the photon can be jointly absorbed by the two atoms in their ground state which will both reach their excited state with a probability close to one. Like ordinary quantum Rabi oscillations, this process is coherent and reversible, so that two atoms in their excited state will undergo a downward transition jointly emitting a single cavity photon. This joint absorption and emission process can also occur with three atoms. The parameters used to investigate this process correspond to experimentally demonstrated values in circuit quantum electrodynamics systems.

A study of interferometers with one-bit which-way detectors demonstrates that the trajectories, which David Bohm invented in his attempt at a realistic interpretation of quantum mechanics, are in fact surrealistic, because they may be macroscopically at variance with the observed track of the particle. We consider a two-slit interferometer and an incomplete Stern-Gerlach interferometer, and propose an experimentum crucis based on the latter.

It has been proposed that the ability to perform joint weak measurements on postselected systems would allow us to study quantum paradoxes. These measurements can investigate the history of those particles that contribute to the paradoxical outcome. Here we experimentally perform weak measurements of joint (i.e., nonlocal) observables. In an implementation of Hardy's paradox, we weakly measure the locations of two photons, the subject of the conflicting statements behind the paradox. Remarkably, the resulting weak probabilities verify all of these statements but, at the same time, resolve the paradox.

We have implemented a novel double-slit "which-way" experiment which raises interesting questions of interpretation. Coherent laser light is passed through a converging lens and then through a dual pinhole producing two beams crossing over at the focal point of the lens, and fully separating further downstream providing which-way information. A thin wire is then placed at a minimum of the interference pattern formed at the cross-over region. No significant reduction in the total flux or resolution of the separated beams is found, providing evidence for coexistence of perfect interference and which-way information in the same experiment, contrary to the common readings of Bohr's principle of complementarity. This result further supports the conclusions of the original experiment by the author in which an imaging lens was employed to obtain which-way information. Finally, a short discussion of the novel non-perturbative measurement technique for ensemble properties is offered.

The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of 10-13 cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.

First, we demonstrate Bell’s theorem, without using inequalities, for an experiment with two particles. Then we show that, if we assume realism and we assume that the “elements of reality” corresponding to Lorentz-invariant observables are themselves Lorentz invariant, we can derive a contradiction with quantum mechanics.

A Comment on the Letter by Lucien Hardy, Phys. Rev. Lett. 73, 2279 (1994). The authors of the Letter offer a Reply.

A Comment on the Letter by L. Hardy, Phys. Rev. Lett. 68, 2981 (1992).