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The “past” and the “delayed-choice” double-slit experiment

In A. R. Marlow (ed.), Mathematical foundations of quantum theory. New York: Academic Press (1978)

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  1. Bell's theorem: A bridge between the measurement and the mind/body problems.Badis Ydri - manuscript
    In this essay a quantum-dualistic, perspectival and synchronistic interpretation of quantum mechanics is further developed in which the classical world-from-decoherence which is perceived (decoherence) and the perceived world-in-consciousness which is classical (collapse) are not necessarily identified. Thus, Quantum Reality or "{\it unus mundus}" is seen as both i) a physical non-perspectival causal Reality where the quantum-to-classical transition is operated by decoherence, and as ii) a quantum linear superposition of all classical psycho-physical perspectival Realities which are governed by synchronicity as well (...)
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  • (1 other version)Some Observations upon "Realistic" Trajectories in Bohmian Quantum Mechanics.María C. Boscá - 2013 - Theoria 28 (1):45-60.
    _Experimental situations in which we observe quantum effects that deviate from the intuitive expectations of the classical world call for an interdisciplinary discussion, and one fundamental issue to be considered is the compatibility between the description of phenomena and the assumption of an objective reality. This paper discusses the ontological interpretation of Bohmian quantum mechanics, focusing on the use of the term “trajectory” and the difficulties associated with its connection to a “real” (objective) trajectory. __My conclusion is that the intended (...)
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  • No Paradox in Wave–Particle Duality.Andrew Knight - 2020 - Foundations of Physics 50 (11):1723-1727.
    The assertion that an experiment by Afshar et al. demonstrates violation of Bohr’s Principle of Complementarity is based on the faulty assumption that which-way information in a double-slit interference experiment can be retroactively determined from a future measurement.
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  • On Testing the Simulation Theory.Tom Campbell, Houman Owhadi, Joe Savageau & David Watkinson - manuscript
    Can the theory that reality is a simulation be tested? We investigate this question based on the assumption that if the system performing the simulation is nite (i.e. has limited resources), then to achieve low computational complexity, such a system would, as in a video game, render content (reality) only at the moment that information becomes available for observation by a player and not at the moment of detection by a machine (that would be part of the simulation and whose (...)
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  • Some resonances between Eastern thought and Integral Biomathics in the framework of the WLIMES formalism for modelling living systems.Plamen L. Simeonov & Andree C. Ehresmann - forthcoming - Progress in Biophysics and Molecular Biology 131 (Special).
    Forty-two years ago, Capra published “The Tao of Physics” (Capra, 1975). In this book (page 17) he writes: “The exploration of the atomic and subatomic world in the twentieth century has …. necessitated a radical revision of many of our basic concepts” and that, unlike ‘classical’ physics, the sub-atomic and quantum “modern physics” shows resonances with Eastern thoughts and “leads us to a view of the world which is very similar to the views held by mystics of all ages and (...)
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  • Why Delayed Choice Experiments do NOT imply Retrocausality.David Ellerman - 2015 - Quantum Studies: Mathematics and Foundations 2 (2):183-199.
    There is a fallacy that is often involved in the interpretation of quantum experiments involving a certain type of separation such as the: double-slit experiments, which-way interferometer experiments, polarization analyzer experiments, Stern-Gerlach experiments, and quantum eraser experiments. The fallacy leads not only to flawed textbook accounts of these experiments but to flawed inferences about retrocausality in the context of delayed choice versions of separation experiments.
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  • Indivisibility, Complementarity and Ontology: A Bohrian Interpretation of Quantum Mechanics.Jairo Roldán-Charria - 2014 - Foundations of Physics 44 (12):1336-1356.
    The interpretation of quantum mechanics presented in this paper is inspired by two ideas that are fundamental in Bohr’s writings: indivisibility and complementarity. Further basic assumptions of the proposed interpretation are completeness, universality and conceptual economy. In the interpretation, decoherence plays a fundamental role for the understanding of measurement. A general and precise conception of complementarity is proposed. It is fundamental in this interpretation to make a distinction between ontological reality, constituted by everything that does not depend at all on (...)
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  • On EPR-Type Entanglement in the Experiments of Scully et al. I. The Micromaser Case and Delayed-Choice Quantum Erasure.F. Herbut - 2008 - Foundations of Physics 38 (11):1046-1064.
    Delayed-choice erasure is investigated in two-photon two-slit experiments that are generalizations of the micromaser experiment of Scully et al. (Nature 351:111–116, 1991). Applying quantum mechanics to the localization detector, it is shown that erasure with delayed choice in the sense of Scully, has an analogous structure as simple erasure. The description goes beyond probabilities. The EPR-type disentanglement, consisting in two mutually incompatible distant measurements, is used as a general framework in both parts of this study. Two simple coherence cases are (...)
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  • Another proof that the future can influence the present.C. W. Rietdijk - 1981 - Foundations of Physics 11 (9-10):783-790.
    A modified Young double-slit experiment proposed by Wootters and Zurek is considered in which a system P of parallel plates covered with a photographic emulsion has been set up in the region where we would normally expect the central interference fringes. Because under certain conditions P makes it possible to conclude with much more than50% certainty through which of the two slits each particular photon passed, the relevant interference pattern becomes blurred. It is proved that this implies a retroactive effect (...)
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  • (1 other version)Some Observations upon “Realistic” Trajectories in Bohmian Quantum Mechanics.María C. Boscá - 2013 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 28 (1):45-60.
    Experimental situations in which we observe quantum effects that deviate from the intuitive expectations of the classical world call for an interdisciplinary discussion, and one fundamental issue to be considered is the compatibility between the description of phenomena and the assumption of an objective reality. This paper discusses the ontological interpretation of Bohmian quantum mechanics, focusing on the use of the term “trajectory” and the difficulties associated with its connection to a “real” (objective) trajectory. My conclusion is that the intended (...)
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  • A Symmetrical Interpretation of the Klein-Gordon Equation.Michael B. Heaney - 2013 - Foundations of Physics 43 (6):733-746.
    This paper presents a new Symmetrical Interpretation (SI) of relativistic quantum mechanics which postulates: quantum mechanics is a theory about complete experiments, not particles; a complete experiment is maximally described by a complex transition amplitude density; and this transition amplitude density never collapses. This SI is compared to the Copenhagen Interpretation (CI) for the analysis of Einstein’s bubble experiment. This SI makes several experimentally testable predictions that differ from the CI, solves one part of the measurement problem, resolves some inconsistencies (...)
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  • Information, physics, quantum: the search for links.John Archibald Wheeler - 1989 - In Wheeler John Archibald (ed.), Proceedings III International Symposium on Foundations of Quantum Mechanics. pp. 354-358.
    This report reviews what quantum physics and information theory have to tell us about the age-old question, How come existence? No escape is evident from four conclusions: (1) The world cannot be a giant machine, ruled by any preestablished continuum physical law. (2) There is no such thing at the microscopic level as space or time or spacetime continuum. (3) The familiar probability function or functional, and wave equation or functional wave equation, of standard quantum theory provide mere continuum idealizations (...)
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  • Can mind affect matter via active information?Basil J. Hiley & Paavo Pylkkanen - 2005 - Mind and Matter 3 (2):8-27.
    Mainstream cognitive neuroscience typically ignores the role of quantum physical effects in the neural processes underlying cogni¬tion and consciousness. However, many unsolved problems remain, suggesting the need to consider new approaches. We propose that quantum theory, especially through an ontological interpretation due to Bohm and Hiley, provides a fruitful framework for addressing the neural correlates of cognition and consciousness. In particular, the ontological interpretation suggests that a novel type of 'active information', connected with a novel type of 'quantum potential energy', (...)
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  • The wave function as a true ensemble.Jonte Hance & Sabine Hossenfelder - 2022 - Proceedings of the Royal Society 478 (2262).
    In quantum mechanics, the wavefunction predicts probabilities of possible measurement outcomes, but not which individual outcome is realised in each run of an experiment. This suggests that it describes an ensemble of states with different values of a hidden variable. Here, we analyse this idea with reference to currently known theorems and experiments. We argue that the ψ-ontic/epistemic distinction fails to properly identify ensemble interpretations and propose a more useful definition. We then show that all local ψ-ensemble interpretations which reproduce (...)
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  • Non-Locality as a Fundamental Principle of Reality: Bell's Theorem and Space-Like Interconnectedness.A. Rauscher Elizabeth - 2017 - Cosmos and History 13 (2):204-216.
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  • Surrealistic Bohmian trajectories appraised.Albert Solé - 2017 - European Journal for Philosophy of Science 7 (3):467-492.
    Englert et al. claim that, in certain circumstances, the Bohmian trajectory of a test particle does not match the reports of which-path detectors, concluding that the Bohmian trajectories are not real, but “surrealistic.” However, Hiley and Callaghan argue that, if Bohm’s interpretation is correctly applied, no such mismatch is ever sanctioned. Unfortunately, the debate was never settled since nobody showed where the source of disagreement resided. In this paper, I reassess the debate over such “surrealistic” trajectories and I derive both (...)
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  • Consistent quantum measurements.Robert B. Griffiths - 2015 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 52 (Part B):188-197.
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  • (1 other version)Delayed-Choice Experiments and the Metaphysics of Entanglement.Matthias Egg - 2013 - Foundations of Physics 43 (9):1124-1135.
    Delayed-choice experiments in quantum mechanics are often taken to undermine a realistic interpretation of the quantum state. More specifically, Healey has recently argued that the phenomenon of delayed-choice entanglement swapping is incompatible with the view that entanglement is a physical relation between quantum systems. This paper argues against these claims. It first reviews two paradigmatic delayed-choice experiments and analyzes their metaphysical implications. It then applies the results of this analysis to the case of entanglement swapping, showing that such experiments pose (...)
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  • The photon exhibits dualism, constant velocity and nonlocality: What do they have in common?Paul Klevgard - 2021 - Optik 248 (168).
    The photon is typically regarded as a unitary object that is both particle-discrete and wave-continuous. This is a paradoxical position and we live with it by making dualism a fundamental feature of radiation. It is argued here that the photon is not unitary; rather it has two identities, one supporting discrete behavior and the other supporting continuous (wave) behavior. There is photon kinetic energy that is always discrete/localized on arrival; it never splits (on half-silvered mirrors) or diffracts (in pinholes or (...)
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  • A theory of presentism.Craig Bourne - 2006 - Canadian Journal of Philosophy 36 (1):1-23.
    Most of us would want to say that it is true that Socrates taught Plato. According to realists about past facts,1 this is made true by the fact that there is, located in the past, i.e., earlier than now, at least one real event that is the teaching of Plato by Socrates. Presentists, however, in denying that past events and facts exist2 cannot appeal to such facts to make their past-tensed statements true. So what is a presentist to do?
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  • (1 other version)On the Conceptuality Interpretation of Quantum and Relativity Theories.Tomas Veloz, Sandro Sozzo, Massimiliano Sassoli de Bianchi & Diederik Aerts - 2020 - Foundations of Science 25 (1):5-54.
    How can we explain the strange behavior of quantum and relativistic entities? Why do they behave in ways that defy our intuition about how physical entities should behave, considering our ordinary experience of the world around us? In this article, we address these questions by showing that the comportment of quantum and relativistic entities is not that strange after all, if we only consider what their nature might possibly be: not an objectual one, but a conceptual one. This not in (...)
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  • Is quantum indeterminism real? Theological implications.Claudia E. Vanney - 2015 - Zygon 50 (3):736-756.
    Quantum mechanics studies physical phenomena on a microscopic scale. These phenomena are far beyond the reach of our observation, and the connection between QM's mathematical formalism and the experimental results is very indirect. Furthermore, quantum indeterminism defies common sense. Microphysical experiments have shown that, according to the empirical context, electrons and quanta of light behave as waves and other times as particles, even though it is impossible to design an experiment that manifests both behaviors at the same time. Unlike Newtonian (...)
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  • De Broglian probabilities in the double-slit experiment.M. Božić, Z. Marić & J. P. Vigier - 1992 - Foundations of Physics 22 (11):1325-1344.
    A new probability interpretation of interference phenomena in the double-slit experiment is proposed. It differs from the standard interpretation (based on elementary events happening in complementary, mutually exclusive setups—arrivals of waves to the screen when one of the slits is closed) which encounters the “paradox” that the law of total probability is violated. This new interpretation is free of such difficulties and paradoxes since it is based on compatible elementary events (events happening in the same setup in which happenall events (...)
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  • Refinements of the interpretation of complementarity from Afshar's experiment.Osvaldo Pessoa Júnior - 2013 - Scientiae Studia 11 (1):119-139.
    O experimento de Afshar foi proposto recentemente como sendo uma violação do princípio de complementaridade. Reconhecendo a novidade trazida pelo experimento, argumentamos que ele permite um refinamento de tal princípio, a partir do estabelecimento de dois pontos: (1) a possibilidade de modificar o "tipo" de fenômeno (onda ou partícula) sem alterar o estado quântico, e (2) a constatação de que o tipo de fenômeno, associado a um quantum detectado, refere-se a um trecho determinado percorrido pelo objeto quântico. O primeiro ponto (...)
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  • On EPR-type Entanglement in the Experiments of Scully et al. II. Insight in the Real Random Delayed-choice Erasure Experiment.F. Herbut - 2010 - Foundations of Physics 40 (3):301-312.
    It was pointed out in the first part of this study (Herbut in Found. Phys. 38:1046–1064, 2008) that EPR-type entanglement is defined by the possibility of performing any of two mutually incompatible distant, i.e., direct-interaction-free, measurements. They go together under the term ‘EPR-type disentanglement’. In this second part, quantum-mechanical insight is gained in the real random delayed-choice erasure experiment of Kim et al. (Phys. Rev. Lett. 84:1–5, 2000) by a relative-reality-of-unitarily-evolving-state (RRUES) approach (explained in the first part). Finally, it is (...)
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  • The Meaning of the Interaction-Free Measurements.Lev Vaidman - 2003 - Foundations of Physics 33 (3):491-510.
    Interaction-free measurements introduced by Elitzur and Vaidman [Found. Phys. 23, 987 (1993)] allow finding infinitely fragile objects without destroying them. Many experiments have been successfully performed showing that indeed, the original scheme and its modifications lead to reduction of the disturbance of the observed systems. However, there is a controversy about the validity of the term “interaction-free” for these experiments. Broad variety of such experiments are reviewed and the meaning of the interaction-free measurements is clarified.
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  • Investigating Puzzling Aspects of the Quantum Theory by Means of Its Hydrodynamic Formulation.A. S. Sanz - 2015 - Foundations of Physics 45 (10):1153-1165.
    Bohmian mechanics, a hydrodynamic formulation of the quantum theory, constitutes a useful tool to understand the role of the phase as the mechanism responsible for the dynamical evolution displayed by quantum systems. This role is analyzed and discussed here in the context of quantum interference, considering to this end two well-known scenarios, namely Young’s two-slit experiment and Wheeler’s delayed choice experiment. A numerical implementation of the first scenario is used to show how interference in a coherent superposition of two counter-propagating (...)
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  • The New Quantum Logic.Robert B. Griffiths - 2014 - Foundations of Physics 44 (6):610-640.
    It is shown how all the major conceptual difficulties of standard (textbook) quantum mechanics, including the two measurement problems and the (supposed) nonlocality that conflicts with special relativity, are resolved in the consistent or decoherent histories interpretation of quantum mechanics by using a modified form of quantum logic to discuss quantum properties (subspaces of the quantum Hilbert space), and treating quantum time development as a stochastic process. The histories approach in turn gives rise to some conceptual difficulties, in particular the (...)
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  • The paradox of two bottles in quantum mechanics.Bogdan Mielnik - 1990 - Foundations of Physics 20 (6):745-755.
    A class of retrospective measurements analogous to the “delayed choice experiments” of Wheeler and Greenberger is considered. A new argument shows that the reduction of the wave packet must affect the past states of the system. As a side product, our argument implies that the axiom about the reduction of the wave packet in relativistic space-time cannot be consistently introduced.
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  • Observing a Quantum Measurement.Jay Lawrence - 2021 - Foundations of Physics 52 (1):1-17.
    With the example of a Stern–Gerlach measurement on a spin-1/2 atom, we show that a superposition of both paths may be observed compatibly with properties attributed to state collapse—for example, the singleness (or mutual exclusivity) of outcomes. This is done by inserting a quantum two-state system (an ancilla) in each path, capable of responding to the passage of the atom, and thus acting as a virtual detector. We then consider real measurements on the compound system of atomic spin and two (...)
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  • Exploratory Evidence for Correlations between Entrained Mental Coherence and Random Physical Systems.Dean Radin & F. Holmes Atwater - 2010 - Journal of Scientific Exploration 23 (3).
    An experiment tested whether mental coherence entrained in groups would affect sequences of data generated by truly random number generators (RNGs) in the vicinity of those groups. Coherence was entrained by having groups listen to a prescribed series of binaural beat rhythms during a 6-day workshop. Two RNGs based on electronic noise and one on radioactive decay latencies were located in the building where the workshops took place. Random data were continually collected from these RNGs during 14 workshops. As controls, (...)
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  • Critique of Quantum Optical Experimental Refutations of Bohr’s Principle of Complementarity, of the Wootters–Zurek Principle of Complementarity, and of the Particle–Wave Duality Relation.P. N. Kaloyerou - 2016 - Foundations of Physics 46 (2):138-175.
    I argue that quantum optical experiments that purport to refute Bohr’s principle of complementarity fail in their aim. Some of these experiments try to refute complementarity by refuting the so called particle–wave duality relations, which evolved from the Wootters–Zurek reformulation of BPC. I therefore consider it important for my forgoing arguments to first recall the essential tenets of BPC, and to clearly separate BPC from WZPC, which I will argue is a direct contradiction of BPC. This leads to a need (...)
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  • Wave–Particle Duality: An Information-Based Approach.R. M. Angelo & A. D. Ribeiro - 2015 - Foundations of Physics 45 (11):1407-1420.
    Recently, Bohr’s complementarity principle was assessed in setups involving delayed choices. These works argued in favor of a reformulation of the aforementioned principle so as to account for situations in which a quantum system would simultaneously behave as wave and particle. Here we defend a framework that, supported by well-known experimental results and consistent with the decoherence paradigm, allows us to interpret complementarity in terms of correlations between the system and an informer. Our proposal offers formal definition and operational interpretation (...)
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  • (1 other version)On the Conceptuality Interpretation of Quantum and Relativity Theories.Diederik Aerts, Massimiliano Sassoli de Bianchi, Sandro Sozzo & Tomas Veloz - 2020 - Foundations of Science 25 (1):5-54.
    How can we explain the strange behavior of quantum and relativistic entities? Why do they behave in ways that defy our intuition about how physical entities should behave, considering our ordinary experience of the world around us? In this article, we address these questions by showing that the comportment of quantum and relativistic entities is not that strange after all, if we only consider what their nature might possibly be: not an objectual one, but a conceptual one. This not in (...)
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  • The Merli-missiroli-Pozzi two-slit electron interference experiment.Rodolfo Rosa - unknown
    In 2002 the readers of the scientific magazine 'Physics World' voted Young's double-slit experiment applied to the interference of single electrons to be 'the most beautiful experiment in physics'; this experiment, in truth, had already been carried out 30 years beforehand. The present article aims to re-examine the latter real experiment and put it into its proper historical perspective. Even though the experiment was not afforded the importance it perhaps deserved among philosophers, its philosophical mplications add new arguments to the (...)
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  • (1 other version)8 Delayed-Choice Experiments and the Metaphysics of Entanglement.Matthias Egg - 2014 - In Scientific Realism in Particle Physics: A Causal Approach. De Gruyter. pp. 137-148.
    Delayed-choice experiments in quantum mechanics are often taken to undermine a realistic interpretation of the quantum state. More specifically, Healey has recently argued that the phenomenon of delayed-choice entanglement swapping is incompatible with the view that entanglement is a physical relation between quantum systems. This paper argues against these claims. It first reviews two paradigmatic delayed-choice experiments and analyzes their metaphysical implications. It then applies the results of this analysis to the case of entanglement swapping, showing that such experiments pose (...)
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  • Retroactive effects from measurements.C. W. Rietdijk - 1987 - Foundations of Physics 17 (3):297-311.
    We consider several thought and practical experiments, and variations thereof, from which the existence can be inferred of retroactive effects on the assumptions of conservation of linear and angular momentum and of realism defined in a wide sense. Such conclusion is made less counterintuitive by research into the proper physical background of the relativistic length contraction of a moving arrow, viz. the fact that the universe is four-dimensional indeed. In one of the experiments considered, the evidence of retroactivity is more (...)
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  • “Haunted” measurements in quantum theory.Daniel M. Greenberger & Alaine YaSin - 1989 - Foundations of Physics 19 (6):679-704.
    Sometimes it is possible in quantum theory for a system to interact with another system in such a way that the information contained in the wave function becomes very scrambled and apparently incoherent. We produce an example which is exactly calculable, in which a macroscopic change is induced in the environment, and all phase information for the system is apparently lost, so that a measurement has seemingly been made. But actually, although the wave function has been badly scrambled, all the (...)
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  • Quantum Information and Inferential Reasoning.Gennaro Auletta - 2005 - Foundations of Physics 35 (1):155-156.
    Information may be extracted from a quantum–mechanical system only by means of inference. For this reason, the observer, although not required as such for obtaining an eigenstate of the measured observable on a given system, is necessary for obtaining information.
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  • Empirical reality, empirical causality, and the measurement problem.B. D'Espagnat - 1987 - Foundations of Physics 17 (5):507-529.
    Does physics describe anything that can meaningfully be called “independent reality,” or is it merely operational? Most physicists implicitly favor an intermediate standpoint, which takes quantum physics into account, but which nevertheless strongly holds fast to quite strictly realistic ideas about apparently “obvious facts” concerning the macro-objects. Part 1 of this article, which is a survey of recent measurement theories, shows that, when made explicit, the standpoint in question cannot be upheld. Part 2 brings forward a proposal for making minimal (...)
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  • Absorbers in the Transactional Interpretation of Quantum Mechanics.Jean-Sébastien Boisvert & Louis Marchildon - 2013 - Foundations of Physics 43 (3):294-309.
    The transactional interpretation of quantum mechanics, following the time-symmetric formulation of electrodynamics, uses retarded and advanced solutions of the Schrödinger equation and its complex conjugate to understand quantum phenomena by means of transactions. A transaction occurs between an emitter and a specific absorber when the emitter has received advanced waves from all possible absorbers. Advanced causation always raises the specter of paradoxes, and it must be addressed carefully. In particular, different devices involving contingent absorbers or various types of interaction-free measurements (...)
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  • Realismo e Interpretación en mecánica bohmiana.Albert Solé - 2010 - Dissertation, Universidad Complutense de Madrid
    En esta tesis hacemos un análisis comparativo de las distintas interpretaciones de la mecánica bohmiana en relación con el realismo científico. En primer lugar discutimos si cabe encontrar una interpretación de la teoría que satisfaga el requisito de que toda entidad real existe en el espacio físico tridimensional. Luego, discutimos el desempeño de las distintas interpretaciones de la teoría en relación con el principio de fiabilidad de la medida. Finalmente, analizamos el argumento de las trayectorias surrealistas. De acuerdo con este (...)
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  • What is Erased in the Quantum Erasure?B. J. Hiley & R. E. Callaghan - 2006 - Foundations of Physics 36 (12):1869-1883.
    In this paper, we re-examine a series of gedanken welcher Weg (WW) experiments introduced by Scully, Englert and Walther that contain the essential ideas underlying the quantum eraser. For this purpose we use the Bohm model which gives a sharp picture of the behaviour of the atoms involved in these experiments. This model supports the thesis that interference disappears in such WW experiments, even though the centre of mass wave function remains coherent throughout the experiment. It also shows exactly what (...)
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  • Postponing the Past: An Operational Analysis of Delayed-Choice Experiments. [REVIEW]M. Bahrami & A. Shafiee - 2009 - Foundations of Physics 40 (1):55-92.
    The prominent characteristic of a delayed-choice effect is to make the choice between complementary types of phenomena after the relevant interaction between the system and measuring instrument has already come to an end. In this paper, we first represent a detailed comparative analysis of some early delayed-choice propositions and also most of the experimentally performed delayed-choice proposals in a coherent and unified quantum mechanical formulation. Taking into the account the represented quantum mechanical descriptions and also the rules of probability theory, (...)
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  • Relativistic Quantum Mechanics as a Telegraph.O. Costa de Beauregard - 2001 - Foundations of Physics 31 (5):837-848.
    A derivation by Fröhner of non-relativistic quantum mechanics via Fourier analysis applied to probability theory is not extendable to relativistic quantum mechanics because Schrödinger's positive definite probability density ψ*ψ is lost (Dirac's spin 1/2 case being the exception). The nature of the Fourier link then changes; it points to a redefinition of the probability scheme as an information carrying telegraph, the code of which is Born's as extended by Dirac and by Feynman. Hermitian symmetry of the transition amplitude 〈ϕ∣ψ〉 between (...)
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