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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. 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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  8. 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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  9. Max Born (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 37 (12):863-867.
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  10. 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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  11. C. T. K. Chari (1971). Towards Generalized Probabilities in Quantum Mechanics. Synthese 22 (3-4):438 - 447.
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  12. 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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  13. 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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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. 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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  21. 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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  22. 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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  23. 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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  24. 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 that the argument from protected measurement to the reality of the state vector of a single quantum system is circular. From this negative result I draw some lessons on the available "interpretations" of quantum theory and on the debate on the quantum measurement problem.
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  25. Amit Hagar (2012). Decoherence: The View From the History and the Philosophy of Science. Phil. Trans. Royal Soc. London A 375 (1975).
    We present a brief history of decoherence, from its roots in the foundations of classical statistical mechanics, to the current spin bath models in condensed matter physics. We analyze the philosophical import of the subject matter in three different foundational problems, and find that, contrary to the received view, decoherence is less instrumental to their solutions than it is commonly believed. What makes decoherence more philosophically interesting, we argue, are the methodological issues it draws attention to, and the question of (...)
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  26. Amit Hagar (2004). Chance and Time. Dissertation, UBC
    One of the recurrent problems in the foundations of physics is to explain why we rarely observe certain phenomena that are allowed by our theories and laws. In thermodynamics, for example, the spontaneous approach towards equilibrium is ubiquitous yet the time-reversal-invariant laws that presumably govern thermal behaviour in the microscopic level equally allow spontaneous departure from equilibrium to occur. Why are the former processes frequently observed while the latter are almost never reported? Another example comes from quantum mechanics where the (...)
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  27. Amit Hagar & Giuseppe Sergioli, Counting Steps: A Finitist Interpretation of Objective Probability in Physics.
    We propose a new interpretation of objective deterministic chances in statistical physics based on physical computational complexity. This notion applies to a single physical system (be it an experimental set--up in the lab, or a subsystem of the universe), and quantifies (1) the difficulty to realize a physical state given another, (2) the 'distance' (in terms of physical resources) from a physical state to another, and (3) the size of the set of time--complexity functions that are compatible with the physical (...)
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  28. Robin Hanson (2003). When Worlds Collide: Quantum Probability From Observer Selection? [REVIEW] Foundations of Physics 33 (7):1129-1150.
    In Everett's many worlds interpretation, quantum measurements are considered to be decoherence events. If so, then inexact decoherence may allow large worlds to mangle the memory of observers in small worlds, creating a cutoff in observable world size. Smaller world are mangled and so not observed. If this cutoff is much closer to the median measure size than to the median world size, the distribution of outcomes seen in unmangled worlds follows the Born rule. Thus deviations from exact decoherence can (...)
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  29. Meir Hemmo (2007). Quantum Probability and Many Worlds. Studies in History and Philosophy of Science Part B 38 (2):333-350.
    We discuss the meaning of probabilities in the many worlds interpretation of quantum mechanics. We start by presenting very briefly the many worlds theory, how the problem of probability arises, and some unsuccessful attempts to solve it in the past. Then we criticize a recent attempt by Deutsch to derive the quantum mechanical probabilities from the nonprobabilistic parts of quantum mechanics and classical decision theory. We further argue that the Born probability does not make sense even as an additional probability (...)
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  30. Meir Hemmo (1996). Possible Worlds in the Modal Interpretation. Philosophy of Science 63 (3):337.
    An outline for a modal interpretation in terms of possible worlds is presented. The so-called Schmidt histories are taken to correspond to the physically possible worlds. The decoherence function defined in the histories formulation of quantum theory is taken to prescribe a non-classical probability measure over the set of the possible worlds. This is shown to yield dynamics in the form of transition probabilities for occurrent events in each world. The role of the consistency condition is discussed.
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  31. R. I. G. Hughes (1989). The Structure and Interpretation of Quantum Mechanics. Harvard University Press.
    R.I.G Hughes offers the first detailed and accessible analysis of the Hilbert-space models used in quantum theory and explains why they are so successful.
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  32. J. T. Ismael (2009). Probability in Deterministic Physics. Journal of Philosophy 106 (2):89-108.
    The role of probability is one of the most contested issues in the interpretation of contemporary physics. In this paper, I’ll be reevaluating some widely held assumptions about where and how probabilities arise. Larry Sklar voices the conventional wisdom about probability in classical physics in a piece in the Stanford Online Encyclopedia of Philosophy, when he writes that “Statistical mechanics was the first foundational physical theory in which probabilistic concepts and probabilistic explanation played a fundamental role.” And the conventional wisdom (...)
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  33. Jenann Ismael (2009). Probability in Deterministic Physics. Journal of Philosophy 106 (2):89-108.
    The role of probability is one of the most contested issues in the interpretation of contemporary physics. In this paper, I’ll be reevaluating some widely held assumptions about where and how probabilities arise. Larry Sklar voices the conventional wisdom about probability in classical physics in a piece in the Stanford Online Encyclopedia of Philosophy, when he writes that “Statistical mechanics was the first foundational physical theory in which probabilistic concepts and probabilistic explanation played a fundamental role.” And the conventional wisdom (...)
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  34. Jenann Ismael (2003). How to Combine Chance and Determinism: Thinking About the Future in an Everett Universe. Philosophy of Science 70 (4):776-790.
    I propose, in the context of Everett interpretations of quantum mechanics, a way of understanding how there can be genuine uncertainty about the future notwithstanding that the universe is governed by known, deterministic dynamical laws, and notwithstanding that there is no ignorance about initial conditions, nor anything in the universe whose evolution is not itself governed by the known dynamical laws. The proposal allows us to draw some lessons about the relationship between chance and determinism, and to dispel one source (...)
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  35. Lars-Göran Johansson (2007). Interpreting Quantum Mechanics. A Realist View in Schrödinger's Vein. Ashgate.
    Presenting a realistic interpretation of quantum mechanics and, in particular, a realistic view of quantum waves, this book defends, with one exception, ...
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  36. Peter J. Lewis (2010). Probability in Everettian Quantum Mechanics. Manuscrito 33:285--306.
    The main difficulty facing no-collapse theories of quantum mechanics in the Everettian tradition concerns the role of probability within a theory in which every possible outcome of a measurement actually occurs. The problem is two-fold: First, what do probability claims mean within such a theory? Second, what ensures that the probabilities attached to measurement outcomes match those of standard quantum mechanics? Deutsch has recently proposed a decision-theoretic solution to the second problem, according to which agents are rationally required to weight (...)
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  37. Nicholas Maxwell (1995). A Philosopher Struggles to Understand Quantum Theory: Particle Creation and Wavepacket Reduction. In M. Ferrero & A. van der Merwe (eds.), Fundamental Problems in Quantum Physics.
    Work on the central problems of the philosophy of science has led the author to attempt to create an intelligible version of quantum theory. The basic idea is that probabilistic transitions occur when new stationary or particle states arise as a result of inelastic collisions.
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  38. Nicholas Maxwell (1994). Particle Creation as the Quantum Condition for Probabilistic Events to Occur. Physics Letters A 187 (2 May 1994):351-355.
    A new version of quantum theory is proposed, according to which probabilistic events occur whenever new statioinary or bound states are created as a result of inelastic collisions. The new theory recovers the experimental success of orthodox quantum theory, but differs form the orthodox theory for as yet unperformed experiments.
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  39. Nicholas Maxwell (1993). Beyond Fapp: Three Approaches to Improving Orthodox Quantum Theory and An Experimental Test. In F. Selleri and G. Tarozzi van der Merwe, F. Selleri & G. Tarozzi (eds.), Bell's Theorem and the Foundations of Modern Physics. World Scientific.
    Because it fails to solve the wave-particle problem, orthodox quantum theory is obliged to be about observables and not quantum beables. As a result the theory is imprecise, ambiguous, ad hoc, lacking in explanatory power, restricted in scope and resistant to unification. A new version of quantum theory is needed that is about quantum beables.
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  40. Nicholas Maxwell (1993). On Relativity Theory and Openness of the Future. Philosophy of Science 60 (2):341-348.
    In a recent paper, Howard Stein makes a number of criticisms of an earlier paper of mine ('Are Probabilism and Special Relativity Incompatible?', Phil. Sci., 1985), which explored the question of whether the idea that the future is genuinely 'open' in a probabilistic universe is compatible with special relativity. I disagree with almost all of Stein's criticisms.
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  41. Nicholas Maxwell (1988). Quantum Propensiton Theory: A Testable Resolution of the Wave/Particle Dilemma. British Journal for the Philosophy of Science 39 (1):1-50.
    In this paper I put forward a new micro realistic, fundamentally probabilistic, propensiton version of quantum theory. According to this theory, the entities of the quantum domain - electrons, photons, atoms - are neither particles nor fields, but a new kind of fundamentally probabilistic entity, the propensiton - entities which interact with one another probabilistically. This version of quantum theory leaves the Schroedinger equation unchanged, but reinterprets it to specify how propensitons evolve when no probabilistic transitions occur. Probabilisitic transitions occur (...)
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  42. Nicholas Maxwell (1988). Are Probabilism and Special Relativity Compatible? Philosophy of Science 55 (4):640-645.
    Are special relativity and probabilism compatible? Dieks argues that they are. But the possible universe he specifies, designed to exemplify both probabilism and special relativity, either incorporates a universal "now" (and is thus incompatible with special relativity), or amounts to a many world universe (which I have discussed, and rejected as too ad hoc to be taken seriously), or fails to have any one definite overall Minkowskian-type space-time structure (and thus differs drastically from special relativity as ordinarily understood). Probabilism and (...)
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  43. Nicholas Maxwell (1985). Are Probabilism and Special Relativity Incompatible? Philosophy of Science 52 (1):23-43.
    In this paper I expound an argument which seems to establish that probabilism and special relativity are incompatible. I examine the argument critically, and consider its implications for interpretative problems of quantum theory, and for theoretical physics as a whole.
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  44. Nicholas Maxwell (1982). Instead of Particles and Fields: A Micro Realistic Quantum "Smearon" Theory. [REVIEW] Foundatioins of Physics 12 (6):607-631.
    A fully micro realistic, propensity version of quantum theory is proposed, according to which fundamental physical entities - neither particles nor fields - have physical characteristics which determine probabilistically how they interact with one another (rather than with measuring instruments). The version of quantum "smearon" theory proposed here does not modify the equations of orthodox quantum theory: rather, it gives a radically new interpretation to these equations. It is argued that (i) there are strong general reasons for preferring quantum "smearon" (...)
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  45. Nicholas Maxwell (1976). Towards a Micro Realistic Version of Quantum Mechanics, Part I. Foundations of Physics 6 (3):275-292.
    This paper investigates the possibiity of developing a fully micro realistic version of elementary quantum mechanics. I argue that it is highly desirable to develop such a version of quantum mechanics, and that the failure of all current versions and interpretations of quantum mechanics to constitute micro realistic theories is at the root of many of the interpretative problems associated with quantum mechanics, in particular the problem of measurement. I put forward a propensity micro realistic version of quantum mechanics, and (...)
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  46. Nicholas Maxwell (1976). Towards a Micro Realistic Version of Quantum Mechanics, Part II. Foundations of Physics 6 (6):661-676.
    In this paper, possible objections to the propensity microrealistic version of quantum mechanics proposed in Part I are answered. This version of quantum mechanics is compared with the statistical, particle microrealistic viewpoint, and a crucial experiment is proposed designed to distinguish between these to microrealistic versions of quantum mechanics.
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  47. Nicholas Maxwell (1972). A New Look at the Quantum Mechanical Problem of Measurement. American Journal of Physics 40:1431-5..
    According to orthodox quantum mechanics, state vectors change in two incompatible ways: "deterministically" in accordance with Schroedinger's time-dependent equation, and probabilistically if and only if a measurement is made. It is argued here that the problem of measurement arises because the precise mutually exclusive conditions for these two types of transitions to occur are not specified within orthodox quantum mechanics. Fundamentally, this is due to an inevitable ambiguity in the notion of "meawurement" itself. Hence, if the problem of measurement is (...)
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  48. S. I. Melnyk & I. G. Tuluzov, Fundamental Measurements in Economics and in the Theory of Consciousness.
    A new constructivist approach to modeling in economics and theory of consciousness is proposed. The state of elementary object is defined as a set of its measurable consumer properties. A proprietor's refusal or consent for the offered transaction is considered as a result of elementary economic measurement. Elementary (indivisible) technology, in which the object's consumer values are variable, in this case can be formalized as a generalized economic measurement. The algebra of such measurements has been constructed. It has been shown (...)
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  49. Huw Price (2010). Decisions, Decisions, Decisions: Can Savage Salvage Everettian Probability? In Simon Saunders, Jonathan Barrett, Adrian Kent & David Wallace (eds.), Many Worlds? Everett, Quantum Theory, and Reality. Oxford University Press.
    [Abstract and PDF at the Pittsburgh PhilSci Archive] A slightly shorter version of this paper is to appear in a volume edited by Jonathan Barrett, Adrian Kent, David Wallace and Simon Saunders, containing papers presented at the Everett@50 conference in Oxford in July 2007, and the Many Worlds@50 meeting at the Perimeter Institute in September 2007. The paper is based on my talk at the latter meeting (audio, video and slides of which are accessible here).
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  50. Simon Saunders (forthcoming). What is Probability? Arxiv Preprint Quant-Ph/0412194.
    Probabilities may be subjective or objective; we are concerned with both kinds of probability, and the relationship between them. The fundamental theory of objective probability is quantum mechanics: it is argued that neither Bohr's Copenhagen interpretation, nor the pilot-wave theory, nor stochastic state-reduction theories, give a satisfactory answer to the question of what objective probabilities are in quantum mechanics, or why they should satisfy the Born rule; nor do they give any reason why subjective probabilities should track objective ones. But (...)
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