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Summary All approaches to quantum theory need to make sense of the objective probabilities which apparently correspond to the square of the amplitudes of components of the quantum state.
Key works The Born rule connecting probabilities and squared-amplitudes was first formulated in Born 1926. The interpretation of probabilities varies widely across different approaches to quantum mechanics: as ever, Bell 2004 is indispensable in setting out the main options. Wallace 2012 provides an authoritative treatment of probability in Everettian QM.
Introductions Dickson 2011
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  1. Diedrik Aerts & Sven Aerts (1995). Applications of Quantum Statistics in Psychological Studies of Decision Processes. Foundations of Science 1 (1):85-97.
    We present a new approach to the old problem of how to incorporate the role of the observer in statistics. We show classical probability theory to be inadequate for this task and take refuge in the epsilon-model, which is the only model known to us caapble of handling situations between quantum and classical statistics. An example is worked out and some problems are discussed as to the new viewpoint that emanates from our approach.
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  2. James Aken (1986). Analysis of Quantum Probability Theory. II. Journal of Philosophical Logic 15 (3):333 - 367.
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  3. James Aken (1985). Analysis of Quantum Probability Theory. I. Journal of Philosophical Logic 14 (3):267 - 296.
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  4. Sergio Albeverio, Philippe Combe & M. Sirugue-Collin (eds.) (1982). Stochastic Processes in Quantum Theory and Statistical Physics: Proceedings of the International Workshop Held in Marseille, France, June 29-July 4, 1981. [REVIEW] Springer-Verlag.
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  5. Charis Anastopoulos (2006). Classical Versus Quantum Probability in Sequential Measurements. Foundations of Physics 36 (11):1601-1661.
    We demonstrate in this paper that the probabilities for sequential measurements have features very different from those of single-time measurements. First, they cannot be modelled by a classical stochastic process. Second, they are contextual, namely they depend strongly on the specific measurement scheme through which they are determined. We construct Positive-Operator-Valued measures (POVM) that provide such probabilities. For observables with continuous spectrum, the constructed POVMs depend strongly on the resolution of the measurement device, a conclusion that persists even if we (...)
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  6. D. M. Appleby (2005). Facts, Values and Quanta. Foundations of Physics 35 (4):627-668.
    Quantum mechanics is a fundamentally probabilistic theory (at least so far as the empirical predictions are concerned). It follows that, if one wants to properly understand quantum mechanics, it is essential to clearly understand the meaning of probability statements. The interpretation of probability has excited nearly as much philosophical controversy as the interpretation of quantum mechanics. 20th century physicists have mostly adopted a frequentist conception. In this paper it is argued that we ought, instead, to adopt a logical or Bayesian (...)
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  7. D. M. Appleby, Åsa Ericsson & Christopher A. Fuchs (2011). Properties of QBist State Spaces. Foundations of Physics 41 (3):564-579.
    Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum state space may thus be thought of as a restricted subset of all potentially available probabilities. A recent publication (Fuchs and Schack, arXiv:0906.2187v1, 2009) advocates such a representation using symmetric informationally complete (SIC) measurements. Building upon this work we study how this subset—quantum-state space—might be characterized. Our leading characteristic is that the inner (...)
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  8. István Aranyosi (2012). Should We Fear Quantum Torment? Ratio 25 (3):249-259.
    The prospect, in terms of subjective expectations, of immortality under the no-collapse interpretation of quantum mechanics is certain, as pointed out by several authors, both physicists and, more recently, philosophers. The argument, known as quantum suicide, or quantum immortality, has received some critical discussion, but there hasn't been any questioning of David Lewis's point that there is a terrifying corollary to the argument, namely, that we should expect to live forever in a crippled, more and more damaged state, that barely (...)
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  9. David Baker (2007). Measurement Outcomes and Probability in Everettian Quantum Mechanics. Studies in History and Philosophy of Science Part B 38 (1):153-169.
    The decision-theoretic account of probability in the Everett or many-worlds interpretation, advanced by David Deutsch and David Wallace, is shown to be circular. Talk of probability in Everett presumes the existence of a preferred basis to identify measurement outcomes for the probabilities to range over. But the existence of a preferred basis can only be established by the process of decoherence, which is itself probabilistic.
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  10. L. E. Ballentine (1990). Limitations of the Projection Postulate. Foundations of Physics 20 (11):1329-1343.
    The projection postulate, which prescribes “collapse of the state vector” upon measurement, is not an essential part of quantum mechanics. Rather it is only an optional discarding of certain branches of the state vector that are expected to be irrelevant for the purpose at hand. However, its use is hazardous, and there are examples of repeated measurements for which the conventional application of the projection postulate leads to incorrect results.
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  11. L. E. Ballentine (1973). Can the Statistical Postulate of Quantum Theory Be Derived?—A Critique of the Many-Universes Interpretation. Foundations of Physics 3 (2):229-240.
    The attempt to derive (rather than assume) the statistical postulate of quantum theory from the many-universes interpretation of Everett and De Witt is analyzed The many-universes interpretation is found to be neither necessary nor sufficient for the task.
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  12. E. G. Beltrametti & S. Bugajski (2000). Remarks on Two-Slit Probabilities. Foundations of Physics 30 (9):1415-1429.
    The probability pattern emerging in two-slit experiments is a typical quantum feature whose essential ingredients are examined by translating them into the spin- $ \frac{1}{2} $ formalism. In view of the existence of extensions of quantum theory preserving some classical structure, we discuss how the two-slit probabilities behave under such extensions. We consider a generalization of the standard classical probability theory, to be called operational probability theory, that turns out to host the so called quantum probabilities.
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  13. Max Born (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 37 (12):863-867.
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  14. A. J. Bracken & R. J. B. Fawcett (1993). Compact Quantum Systems and the Pauli Data Problem. Foundations of Physics 23 (2):277-289.
    Compact quantum systems have underlying compact kinematical Lie algebras, in contrast to familiar noncompact quantum systems built on the Weyl-Heisenberg algebra. Pauli asked in the latter case: to what extent does knowledge of the probability distributions in coordinate and momentum space determine the state vector? The analogous question for compact quantum systems is raised, and some preliminary results are obtained.
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  15. Paul Busch & Gregg Jaeger (2010). Unsharp Quantum Reality. Foundations of Physics 40 (9-10):1341-1367.
    The positive operator (valued) measures (POMs) allow one to generalize the notion of observable beyond the traditional one based on projection valued measures (PVMs). Here, we argue that this generalized conception of observable enables a consistent notion of unsharp reality and with it an adequate concept of joint properties. A sharp or unsharp property manifests itself as an element of sharp or unsharp reality by its tendency to become actual or to actualize a specific measurement outcome. This actualization tendency—or potentiality—of (...)
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  16. Nancy Cartwright (1978). The Only Real Probabilities in Quantum Mechanics. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1978:54 - 59.
    Position probabilities play a privileged role in the interpretation of quantum mechanics. The standard interpretation has it that |Ψ (r)| 2 represents the probability that the system is at (or will be found at) the location r. Use of these probabilities, however, creates tremendous conceptual difficulties. It forces us either to adopt a non-standard logic, or to be saddled with an intractable measurement problem. This paper proposes that we try to eliminate position probabilities, and instead to interpret quantum mechanics (...)
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  17. Carlton M. Caves, Christopher A. Fuchs, Kiran K. Manne & Joseph M. Renes (2004). Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements. Foundations of Physics 34 (2):193-209.
  18. C. T. K. Chari (1971). Towards Generalized Probabilities in Quantum Mechanics. Synthese 22 (3-4):438 - 447.
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  19. Leon Cohen (1988). Rules of Probability in Quantum Mechanics. Foundations of Physics 18 (10):983-998.
    We show that the quantum mechanical rules for manipulating probabilities follow naturally from standard probability theory. We do this by generalizing a result of Khinchin regarding characteristic functions. From standard probability theory we obtain the methods usually associated with quantum theory; that is, the operator method, eigenvalues, the Born rule, and the fact that only the eigenvalues of the operator have nonzero probability. We discuss the general question as to why quantum mechanics seemingly necessitates different methods than standard probability theory (...)
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  20. Leon Cohen (1966). Can Quantum Mechanics Be Formulated as a Classical Probability Theory? Philosophy of Science 33 (4):317-322.
    It is shown that quantum mechanics cannot be formulated as a stochastic theory involving a probability distribution function of position and momentum. This is done by showing that the most general distribution function which yields the proper quantum mechanical marginal distributions cannot consistently be used to predict the expectations of observables if phase space integration is used. Implications relating to the possibility of establishing a "hidden" variable theory of quantum mechanics are discussed.
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  21. R. Eugene Collins (1977). Quantum Theory: A Hilbert Space Formalism for Probability Theory. [REVIEW] Foundations of Physics 7 (7-8):475-494.
    It is shown that the Hilbert space formalism of quantum mechanics can be derived as a corrected form of probability theory. These constructions yield the Schrödinger equation for a particle in an electromagnetic field and exhibit a relationship of this equation to Markov processes. The operator formalism for expectation values is shown to be related to anL 2 representation of marginal distributions and a relationship of the commutation rules for canonically conjugate observables to a topological relationship of two manifolds is (...)
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  22. Nicola Cufaro-Petroni (1992). On the Structure of the Quantum-Mechanical Probability Models. Foundations of Physics 22 (11):1379-1401.
    In this paper the role of the mathematical probability models in the classical and quantum physics is shortly analyzed. In particular the formal structure of the quantum probability spaces (QPS) is contrasted with the usual Kolmogorovian models of probability by putting in evidence the connections between this structure and the fundamental principles of the quantum mechanics. The fact that there is no unique Kolmogorovian model reproducing a QPS is recognized as one of the main reasons of the paradoxical behaviors pointed (...)
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  23. Michael Dickson (2011). Aspects of Probability in Quantum Theory. In Claus Beisbart & Stephan Hartmann (eds.), Probabilities in Physics. Oxford University Press. 171.
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  24. William Michael Dickson (1998). Quantum Chance and Non-Locality: Probability and Non-Locality in the Interpretations of Quantum Mechanics. Cambridge University Press.
    This book examines in detail two of the fundamental questions raised by quantum mechanics. First, is the world indeterministic? Second, are there connections between spatially separated objects? In the first part, the author examines several interpretations, focusing on how each proposes to solve the measurement problem and on how each treats probability. In the second part, the relationship between probability (specifically determinism and indeterminism) and non-locality is examined, and it is argued that there is a non-trivial relationship between probability and (...)
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  25. Zoltan Domotor (1974). The Probability Structure of Quantum-Mechanical Systems. Synthese 29 (1-4):155 - 185.
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  26. Igor Douven & Jos Uffink (2012). Quantum Probabilities and the Conjunction Principle. Synthese 184 (1):109-114.
    A recent argument by Hawthorne and Lasonen-Aarnio purports to show that we can uphold the principle that competently forming conjunctions is a knowledge-preserving operation only at the cost of a rampant skepticism about the future. A key premise of their argument is that, in light of quantum-mechanical considerations, future contingents never quite have chance 1 of being true. We argue, by drawing attention to the order of magnitude of the relevant quantum probabilities, that the skeptical threat of Hawthorne and Lasonen-Aarnio’s (...)
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  27. Ehtibar N. Dzhafarov & Janne V. Kujala (2014). No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics. Foundations of Physics 44 (3):248-265.
    Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov’s probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice–Bob EPR paradigm, non-contextuality means that the identity of Alice’s spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis $\alpha _{i}$ chosen by Alice, irrespective of Bob’s axis $\beta _{j}$ (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice’s (...)
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  28. C. Martin Edwards & Gottfried T. Rüttimann (1990). On Conditional Probability in GL Spaces. Foundations of Physics 20 (7):859-872.
    We investigate the notion of conditional probability and the quantum mechanical concept of state reduction in the context of GL spaces satisfying the Alfsen-Shultz condition.
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  29. Laura Felline & Guido Bacciagaluppi (forthcoming). Locality and Mentality in Everett Interpretations: Albert and Loewer’s Many Minds. Mind and Matter.
    This is the first of two papers reviewing and analysing the approach to locality and to mind-body dualism proposed in Everett interpreta- tions of quantum mechanics. The planned companion paper will focus on the contemporary decoherence-based approaches to Everett. This paper instead treats the explicitly mentalistic Many Minds Interpreta- tion proposed by David Albert and Barry Loewer (Albert and Loewer 1988). In particular, we investigate what kind of supervenience of the mind on the body is implied by Albert and Loewer’s (...)
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  30. Arthur Fine (1973). Probability and the Interpretation of Quantum Mechanics. British Journal for the Philosophy of Science 24 (1):1-37.
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  31. Alan Forrester (2007). Decision Theory and Information Propagation in Quantum Physics. Studies in History and Philosophy of Science Part B 38 (4):815-831.
    In recent papers, Zurek [(2005). Probabilities from entanglement, Born's rule pk=|ψk|2 from entanglement. Physical Review A, 71, 052105] has objected to the decision-theoretic approach of Deutsch [(1999) Quantum theory of probability and decisions. Proceedings of the Royal Society of London A, 455, 3129–3137] and Wallace [(2003). Everettian rationality: defending Deutsch's approach to probability in the Everett interpretation. Studies in History and Philosophy of Modern Physics, 34, 415–438] to deriving the Born rule for quantum probabilities on the grounds that it courts (...)
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  32. Malcolm R. Forster (2010). Miraculous Consilience of Quantum Mechanics. In. In Ellery Eells & James Fetzer (eds.), The Place of Probability in Science. Springer. 201--228.
  33. Roman Frigg & Carl Hoefer (2007). Probability in GRW Theory. Studies in History and Philosophy of Science Part B 38 (2):371-389.
    GRW Theory postulates a stochastic mechanism assuring that every so often the wave function of a quantum system is `hit', which leaves it in a localised state. How are we to interpret the probabilities built into this mechanism? GRW theory is a firmly realist proposal and it is therefore clear that these probabilities are objective probabilities (i.e. chances). A discussion of the major theories of chance leads us to the conclusion that GRW probabilities can be understood only as either single (...)
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  34. Maria Carla Galavotti (1995). Operationism, Probability and Quantum Mechanics. Foundations of Science 1 (1):99-118.
    This paper investigates the kind of empiricism combined with an operationalist perspective that, in the first decades of our Century, gave rise to a turning point in theoretical physics and in probability theory. While quantum mechanics was taking shape, the classical (Laplacian) interpretation of probability gave way to two divergent perspectives: frequentism and subjectivism. Frequentism gained wide acceptance among theoretical physicists. Subjectivism, on the other hand, was never held to be a serious candidate for application to physical theories, despite the (...)
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  35. G. Gerlich (1981). Some Remarks on Classical Probability Theory in Quantum Mechanics. Erkenntnis 16 (3):335 - 338.
  36. GianCarlo Ghirardi (2005). Quantum Theory as an Emergent Phenomenon: The Statistical. Philosophy of Science 72 (4):642-645.
  37. Robin Giles (1979). The Concept of a Proposition in Classical and Quantum Physics. Studia Logica 38 (4):337 - 353.
    A proposition is associated in classical mechanics with a subset of phase space, in quantum logic with a projection in Hilbert space, and in both cases with a 2-valued observable or test. A theoretical statement typically assigns a probability to such a pure test. However, since a pure test is an idealization not realizable experimentally, it is necessary — to give such a statement a practical meaning — to describe how it can be approximated by feasible tests. This gives rise (...)
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  38. Daniel T. Gillespie (1995). Incompatibility of the Schrödinger Equation with Langevin and Fokker-Planck Equations. Foundations of Physics 25 (7):1041-1053.
    Quantum mechanics posits that the wave function of a one-particle system evolves with time according to the Schrödinger equation, and furthermore has a square modulus that serves as a probability density function for the position of the particle. It is natural to wonder if this stochastic characterization of the particle's position can be framed as a univariate continuous Markov process, sometimes also called a classical diffusion process, whose temporal evolution is governed by the classically transparent equations of Langevin and Fokker-Planck. (...)
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  39. Clark Glymour, 5. Markov Properties and Quantum Experiments.
    Few people have thought so hard about the nature of the quantum theory as has Jeff Bub,· and so it seems appropriate to offer in his honor some reflections on that theory. My topic is an old one, the consistency of our microscopic theories with our macroscopic theories, my example, the Aspect experiments (Aspect et al., 1981, 1982, 1982a; Clauser and Shimony, l978;_Duncan and Kleinpoppen, 199,8) is familiar, and my sirnplrcation of it is borrowed. All that is new here is (...)
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  40. Clark Glymour (1971). Determinism, Ignorance, and Quantum Mechanics. Journal of Philosophy 68 (21):744-751.
    is every bit as intelligible and philosophically respectable as many other doctrines currently in favor, e.g., the doctrine that mental events are identical with brain events; the attempt to give a linguistic construal of this latter doctrine meets many of the same sorts of difficulties encountered above (see Hempel, op. cit.). Secondly, I think that evidence for universal determinism may not, as a matter of fact, be so hard to come by as one might imagine. It is a striking fact (...)
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  41. Sheldon Goldstein, Bell-Type Quantum Field Theories.
    In [3] John S. Bell proposed how to associate particle trajectories with a lattice quantum field theory, yielding what can be regarded as a |Ψ|2-distributed Markov process on the appropriate configuration space. A similar process can be defined in the continuum, for more or less any regularized quantum field theory; such processes we call Bell-type quantum field theories. We describe methods for explicitly constructing these processes. These concern, in addition to the definition of the Markov processes, the efficient calculation of (...)
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  42. Ravi Gomatam, Book Review. [REVIEW]
    In this book, Mara Beller, a historian and philosopher of science, undertakes to examine why and how the elusive Copenhagen interpretation came to acquire the status it has. The book appears under the series ‘Science and Its Conceptual Foundations’. The first part traces in seven chapters the early major developmental phases of QT such as matrix theory, Born’s probabilistic interpretation, Heisenberg’s uncertainty principle and Bohr’s complementarity framework. Although the historical and scientific details are authentic, the author’s presentation in this part (...)
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  43. Ravi V. Gomatam, Popper's Propensity Interpretation and Heisenberg's Potentia Interpretation.
    In other words, classically, probabilities add; quantum mechanically, the probability amplitudes add, leading to the presence of the extra product terms in the quantum case. What this means is that in quantum theory, even though always only one of the various outcomes is obtained in any given observation, some aspect of the non -occurring events, represented by the corresponding complex-valued quantum amplitudes, plays a role in determining the overall probabilities. Indeed, the observed quantum interference effects are correctly captured by the (...)
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  44. Hilary Greaves, Probability in Everettian Quantum Mechanics.
    (a) How to design a nuclear power plant 3. Deutsch/Wallace solution to the practical problem (a) Argue that the rational Everettian agent makes decisions by maximizing expected utility, where the expectation value is an average over branches 4. The semantics of branching - two options..
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  45. Hilary Greaves (2007). Probability in the Everett Interpretation. Philosophy Compass 2 (1):109–128.
    The Everett (many-worlds) interpretation of quantum mechanics faces a prima facie problem concerning quantum probabilities. Research in this area has been fast-paced over the last few years, following a controversial suggestion by David Deutsch that decision theory can solve the problem. This article provides a non-technical introduction to the decision-theoretic program, and a sketch of the current state of the debate.
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  46. Hilary Greaves (2004). Understanding Deutsch's Probability in a Deterministic Universe. Studies in History and Philosophy of Modern Physics 35 (3):423-456.
    Difficulties over probability have often been considered fatal to the Everett interpretation of quantum mechanics. Here I argue that the Everettian can have everything she needs from `probability' without recourse to indeterminism, ignorance, primitive identity over time or subjective uncertainty: all she needs is a particular *rationality principle*. The decision-theoretic approach recently developed by Deutsch and Wallace claims to provide just such a principle. But, according to Wallace, decision theory is itself applicable only if the correct attitude to a future (...)
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  47. Neal Grossman (1972). Quantum Mechanics and Interpretations of Probability Theory. Philosophy of Science 39 (4):451-460.
    Several philosophers of science have claimed that the conceptual difficulties of quantum mechanics can be resolved by appealing to a particular interpretation of probability theory. For example, Popper bases his treatment of quantum mechanics on the propensity interpretation of probability, and Margenau bases his treatment of quantum mechanics on the frequency interpretation of probability. The purpose of this paper is (i) to consider and reject such claims, and (ii) to discuss the question of whether the ψ -function refers to an (...)
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  48. B. J. H. (1962). From Dualism to Unity in Quantum Physics. [REVIEW] Review of Metaphysics 15 (4):676-676.
  49. Amit Hagar, Thou Shalt Not Commute!
    For many among the scientifically informed public, and even among physicists, Heisenberg's uncertainty principle epitomizes quantum mechanics. Nevertheless, more than 86 years after its inception, there is no consensus over the interpretation, scope, and validity of this principle. The aim of this chapter is to offer one such interpretation, the traces of which may be found already in Heisenberg's letters to Pauli from 1926, and in Dirac's anticipation of Heisenberg's uncertainty relations from 1927, that stems form the hypothesis of finite (...)
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  50. Amit Hagar, Does Protective Measurement Tell Us Anything About Quantum Reality?
    An analysis of the two routes through which one may disentangle a quantum system from a measuring apparatus, hence protect the state vector of a single quantum system from being disturbed by the measurement, reveals several loopholes in the argument from protective measurement to the reality of the state vector of a single quantum system.
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