Showing papers on "Consistent histories published in 2010"

QBism, the Perimeter of Quantum Bayesianism

Abstract: This article summarizes the Quantum Bayesian point of view of quantum mechanics, with special emphasis on the view's outer edges---dubbed QBism. QBism has its roots in personalist Bayesian probability theory, is crucially dependent upon the tools of quantum information theory, and most recently, has set out to investigate whether the physical world might be of a type sketched by some false-started philosophies of 100 years ago (pragmatism, pluralism, nonreductionism, and meliorism). Beyond conceptual issues, work at Perimeter Institute is focused on the hard technical problem of finding a good representation of quantum mechanics purely in terms of probabilities, without amplitudes or Hilbert-space operators. The best candidate representation involves a mysterious entity called a symmetric informationally complete quantum measurement. Contemplation of it gives a way of thinking of the Born Rule as an addition to the rules of probability theory, applicable when an agent considers gambling on the consequences of his interactions with a newly recognized universal capacity: dimension (formerly Hilbert-space dimension). (The word "capacity" should conjure up an image of something like gravitational mass---a body's mass measures its capacity to attract other bodies. With hindsight one can say that the founders of quantum mechanics discovered another universal capacity, "dimension.") The article ends by showing that the egocentric elements in QBism represent no impediment to pursuing quantum cosmology and outlining some directions for future work.

Colloquium: An introduction to consistent quantum theory

Abstract: This paper presents an elementary introduction to consistent quantum theory, as developed by Griffiths and others over the past $25\phantom{\rule{0.3em}{0ex}}\text{years}$. The theory is a version of orthodox (Copenhagen) quantum mechanics based on the notion that the unique and mysterious feature of quantum, as opposed to classical systems, is the simultaneous existence of multiple incompatible representations of reality, referred to as ``frameworks.'' A framework is a maximal set of properties of a system for which probabilities can be consistently defined. This notion is expressed by stating that a framework provides an exhaustive set of exclusive alternatives, but no single framework suffices to fully characterize a quantum system. Any prediction of the theory must be confined to a single framework and combining elements from different frameworks leads to quantum mechanically meaningless statements. This ``single-framework rule'' is the precise mathematical statement of Bohr's complementarity. It is shown that if the microscopic description is assumed to incorporate these elements in a local setting, then distant entanglements, macroscopic measurements, wave-function collapse, and other features of quantum behavior follow in a logical manner. The essential elements of the theory are first explained using the simplest quantum system, a single spin-$\frac{1}{2}$ degree of freedom at one time.

Consistent probabilities in Wheeler-DeWitt quantum cosmology

Abstract: We give an explicit, rigorous framework for calculating quantum probabilities in a model theory of quantum gravity. Specifically, we construct the decoherence functional for the Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmology with a free, massless, minimally coupled scalar field, thus providing a complete decoherent histories formulation for this quantum cosmological model. The decoherence functional is applied to study predictions concerning the model's Dirac (relational) observables; the behavior of semiclassical states and superpositions of such states; and to study the singular behavior of quantum Wheeler-DeWitt universes. Within this framework, rigorous formulas are given for calculating the corresponding probabilities from the wave function when those probabilities may be consistently defined, thus replacing earlier heuristics for interpreting the wave function of the universe with explicit constructions. It is shown according to a rigorously formulated standard, and in a quantum-mechanically consistent way, that in this quantization these models are generically singular. Independent of the choice of state we show that the probability for these Wheeler-DeWitt quantum universes to ever encounter a singularity is unity. In addition, the relation between histories formulations of quantum theory and relational Dirac observables is clarified.

Rovelli’s World

Abstract: Carlo Rovelli’s inspiring “Relational Quantum Mechanics” serves several aims at once: it provides a new vision of what the world of quantum mechanics is like, and it offers a program to derive the theory’s formalism from a set of simple postulates pertaining to information processing. I propose here to concentrate entirely on the former, to explore the world of quantum mechanics as Rovelli depicts it. It is a fascinating world in part because of Rovelli’s reliance on the information-theory approach to the foundations of quantum mechanics, and in part because its presentation involves taking sides on a fundamental divide within philosophy itself.

A Potentiality and Conceptuality Interpretation of Quantum Physics

Abstract: We elaborate on a new interpretation of quantum mechanics which we introduced recently The main hypothesis of this new interpretation is that quantum particles are entities interacting with matter conceptually, which means that pieces of matter function as interfaces for the conceptual content carried by the quantum particles We explain how our interpretation was inspired by our earlier analysis of non-locality as non-spatiality and a specific interpretation of quantum potentiality, which we illustrate by means of the example of two interconnected vessels of water We show by means of this example that philosophical realism is not in contradiction with the recent findings with respect to Leggett's inequalities and their violations We explain our recent work on using the quantum formalism to model human concepts and their combinations and how this has given rise to the foundational ideas of our new quantum interpretation We analyze the equivalence of meaning in the realm of human concepts and coherence in the realm of quantum particles, and how the duality of abstract and concrete leads naturally to a Heisenberg uncertainty relation We illustrate the role played by interference and entanglement and show how the new interpretation explains the problems related to identity and individuality in quantum mechanics We put forward a possible scenario for the emergence of the reality of macroscopic objects

On the Possibility That We Think in a Quantum Probabilistic Manner

Abstract: My discussion is articulated under the neurological as well as the psychological profile. I insist in particular on the view that mental events arise in analogy with quantum probability fields. I review some results obtained on quantum cognition discussing in detail those that we obtained on quantum interference in mental states during perception-cognition in ambiguous figures. Frequently, I use the approach to quantum mechanics by Clifford algebra. I insist in particular on two recent results. The first is the justification that I obtain of the von Neumann postulate on quantum measurement and the second relates my Clifford demonstration on the logical origins of quantum mechanics and thus on the arising feature that quantum mechanics relates conceptual entities. The whole discussion aims me to support the conclusion that we think in a quantum probabilistic manner.

The modal-Hamiltonian interpretation and the Galilean covariance of quantum mechanics

Abstract: The aim of this paper is to analyze the modal-Hamiltonian interpretation of quantum mechanics in the light of the Galilean group. In particular, it is shown that the rule of definite-value assignment proposed by that interpretation has the same properties of Galilean covariance and invariance as the Schrodinger equation. Moreover, it is argued that, when the Schrodinger equation is invariant, the rule can be reformulated in an explicitly invariant form in terms of the Casimir operators of the Galilean group. Finally, the possibility of extrapolating the rule to quantum field theory is considered.

Comment on "Quantum solution to the arrow-of-time dilemma".

Abstract: Recently, a substantial amount of debate has grown up around a proposed quantum resolution to the `arrow of time dilemma' that is based on the role of classical memory records of entropy-decreasing events. In this note we show that the argument is incomplete and furthermore, by providing a counter-example, argue that it is incorrect. Instead of quantum mechanics providing a resolution in the manner suggested, it allows enhanced classical memory records of entropy-decreasing events.

Many Worlds in Context

Abstract: Everett's Many-Worlds Interpretation of quantum mechanics is discussed in the context of other physics disputes and other proposed kinds of parallel universes. We find that only a small fraction of the usual objections to Everett's theory are specific to quantum mechanics, and that all of the most controversial issues crop up also in settings that have nothing to do with quantum mechanics.

The Statistical Origins of Quantum Mechanics

Abstract: It is shown that Schrodinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared, and some fundamental differences are identified.

Formalism and Interpretation in Quantum Theory

Abstract: Quantum Mechanics can be viewed as a linear dynamical theory having a familiar mathematical framework but a mysterious probabilistic interpretation, or as a probabilistic theory having a familiar interpretation but a mysterious formal framework. These points of view are usually taken to be somewhat in tension with one another. The first has generated a vast literature aiming at a “realistic” and “collapse-free” interpretation of quantum mechanics that will account for its statistical predictions. The second has generated an at least equally large literature aiming to derive, or at any rate motivate, the formal structure of quantum theory in probabilistically intelligible terms. In this paper I explore, in a preliminary way, the possibility that these two programmes have something to offer one another. In particular, I show that a version of the measurement problem occurs in essentially any non-classical probabilistic theory, and ask to what extent various interpretations of quantum mechanics continue to make sense in such a general setting. I make a start on answering this question in the case of a rudimentary version of the Everett interpretation.

Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics

Abstract: Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular, the relation between the SIC probability representation and other probability representations is established, the connection with mutually unbiased bases is discussed, and comments to the Lie algebraic structure of SIC-POVMs are presented.

An axiomatic formulation of the Montevideo interpretation of quantum mechanics

Abstract: We make a first attempt to axiomatically formulate the Montevideo interpretation of quantum mechanics. In this interpretation environmental decoherence is supplemented with loss of coherence due to the use of realistic clocks to measure time to solve the measurement problem. The resulting formulation is framed entirely in terms of quantum objects without having to invoke the existence of measurable classical quantities like the time in ordinary quantum mechanics. The formulation eliminates any privileged role to the measurement process giving an objective definition of when an event occurs in a system.

Geometrizing Relativistic Quantum Mechanics

Abstract: We propose a new approach to describe quantum mechanics as a manifestation of non-Euclidean geometry. In particular, we construct a new geometrical space that we shall call Qwist. A Qwist space has a extra scalar degree of freedom that ultimately will be identified with quantum effects. The geometrical properties of Qwist allow us to formulate a geometrical version of the uncertainty principle. This relativistic uncertainty relation unifies the position-momentum and time-energy uncertainty principles in a unique relation that recover both of them in the non-relativistic limit.

A Hilbert Space Representation of Generalized Observables and Measurement Processes in the ESR Model

Abstract: The extended semantic realism (ESR) model recently worked out by one of the authors embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here a Hilbert space representation of the generalized observables introduced by the ESR model that satisfy a simple physical condition, propose a generalization of the projection postulate, and suggest a possible mathematical description of the measurement process in terms of evolution of the compound system made up of the measured system and the measuring apparatus.

Conceptual Aspects of PT-Symmetry and Pseudo-Hermiticity: A status report

Abstract: We survey some of the main conceptual developments in the study of PT-symmetric and pseudo-Hermitian Hamiltonian operators that have taken place during the past ten years or so. We offer a precise mathematical description of a quantum system and its representations that allows us to describe the idea of unitarization of a quantum system by modifying the inner product of the Hilbert space. We discuss the role and importance of the quantum-to-classical correspondence principle that provides the physical interpretation of the observables in quantum mechanics. Finally, we address the problem of constructing an underlying classical Hamiltonian for a unitary quantum system defined by an a priori non-Hermitian Hamiltonian.

Quantum Mechanics: From Realism to Intuitionism

Abstract: The interpretation of quantum mechanics has been a problem since its founding days. A large contribution to the discussion of possible interpretations of quantum mechanics is given by the so-called impossibility proofs for hidden variable models; models that allow a realist interpretation. In this thesis some of these proofs are discussed, like von Neumann’s Theorem, the Kochen-Specker Theorem and the Bell-inequalities. Some more recent developments are also investigated, like Meyer’s nullification of the Kochen-Specker Theorem, the MKC-models and Conway and Kochen’s Free Will Theorem. This last one is taken to suggest that the problems that arise for certain interpretations of quantum mechanics are not limited to realist interpretations only, but also affect certain instrumentalist interpretations. It is argued that one may arrive at a more satisfying interpretation of quantum mechanics if one adopts a logic that seems more compatible with the instrumentalist viewpoint namely, intuitionistic logic. The motivations for adopting this form of logic rather than classical logic or quantum logic are linked to some of the philosophical ideas of Bohr. In particular a new interpretation of Bohr’s notion of complementarity is proposed. Finally some possibilities are explored for linking the intuitionistic interpretation of quantum mechanics to the mathematical formalism of the theory.

Quantum singularities in the FRW universe revisited

Abstract: The components of the Riemann tensor in the tetrad basis are quantized and, through the Einstein equation, we find the local expectation value in the ontological interpretation of quantum mechanics of the energy density and pressure of a perfect fluid with equation of state p=(1/3){rho} in the flat Friedmann-Robertson-Walker quantum cosmological model. The quantum behavior of the equation of state and energy conditions are then studied, and it is shown that the energy conditions are violated since the singularity is removed with the introduction of quantum cosmology, but in the classical limit both the equation of state and the energy conditions behave as in the classical model. We also calculate the expectation value of the scale factor for several wave packets in the many-worlds interpretation in order to show the independence of the nonsingular character of the quantum cosmological model with respect to the wave packet representing the wave function of the Universe. It is also shown that, with the introduction of nonnormalizable wave packets, solutions of the Wheeler-DeWitt equation, the singular character of the scale factor, can be recovered in the ontological interpretation.

Quantum probabilities: an information-theoretic interpretation

Abstract: This Chapter develops a realist information-theoretic interpretation of the nonclassical features of quantum probabilities. On this view, what is fundamental in the transition from classical to quantum physics is the recognition that \emph{information in the physical sense has new structural features}, just as the transition from classical to relativistic physics rests on the recognition that space-time is structurally different than we thought. Hilbert space, the event space of quantum systems, is interpreted as a kinematic (i.e., pre-dynamic) framework for an indeterministic physics, in the sense that the geometric structure of Hilbert space imposes objective probabilistic or information-theoretic constraints on correlations between events, just as the geometric structure of Minkowski space in special relativity imposes spatio-temporal kinematic constraints on events. The interpretation of quantum probabilities is more subjectivist in spirit than other discussions in this book (e.g., the chapter by Timpson), insofar as the quantum state is interpreted as a credence function---a bookkeeping device for keeping track of probabilities---but it is also objective (or intersubjective), insofar as the credences specified by the quantum state are understood as uniquely determined, via Gleason's theorem, by objective correlational constraints on events in the nonclassical quantum event space defined by the subspace structure of Hilbert space.

Classes of Copenhagen interpretations: Mechanisms of collapse as typologically determinative

Abstract: The Copenhagen interpretation of quantum mechanics is the dominant view of the theory among working physicists, if not philosophers. There are, however, several strains of Copenhagenism extant, each largely accepting Born's assessment of the wave function as the most complete possible specification of a system and the notion of collapse as a completely random event. This paper outlines three of these sub-interpretations, typing them by what the author of each names as the trigger of quantum-mechanical collapse. Visions of the theory from von Neumann, Heisenberg, and Wheeler offer different mechanisms to break the continuous, deterministic, superposition-laden quantum chain and yield discrete, probabilistic, classical results in response to von Neumann's catastrophe of infinite regress.

A detailed interpretation of probability, and its link with quantum mechanics

Abstract: In the following we revisit the frequency interpretation of probability of Richard von Mises, in order to bring the essential implicit notions in focus Following von Mises, we argue that probability can only be defined for events that can be repeated in similar conditions, and that exhibit 'frequency stabilization' The central idea of the present article is that the mentioned 'conditions' should be well-defined and 'partitioned' More precisely, we will divide probabilistic systems into object, environment, and probing subsystem, and show that such partitioning allows to solve a wide variety of classic paradoxes of probability theory As a corollary, we arrive at the surprising conclusion that at least one central idea of the orthodox interpretation of quantum mechanics is a direct consequence of the meaning of probability More precisely, the idea that the "observer influences the quantum system" is obvious if one realizes that quantum systems are probabilistic systems; it holds for all probabilistic systems, whether quantum or classical

Foundations of quantum mechanics: decoherence and interpretation

Abstract: In this paper we review Castagnino's contributions to the foundations of quantum mechanics. First, we recall his work on quantum decoherence in closed systems, and the proposal of a general framework for decoherence from which the phenomenon acquires a conceptually clear meaning. Then, we introduce his contribution to the hard field of the interpretation of quantum mechanics: the modal-Hamiltonian interpretation solves many of the interpretive problems of the theory, and manifests its physical relevance in its application to many traditional models of the practice of physics. In the third part of this work we describe the ontological picture of the quantum world that emerges from the modal-Hamiltonian interpretation, stressing the philosophical step toward a deep understanding of the reference of the theory.

Measurements in Quantum Mechanics and von NEUMANN's Model

Abstract: Many textbooks on Quantum Mechanics are not very precise as to the meaning of making a measurement: as a consequence, they frequently make assertions which are not based on a dynamical description of the measurement process. A model proposed by von Neumann allows a dynamical description of measurement in Quantum Mechanics, including the measuring instrument in the formalism.In this article we apply von Neumann’s model to illustrate the measurement of an observable by means of a measuring instrument and show how various results, which are sometimens postulated without a dynamical basis, actually emerge.We also investigate the more complex, intriguing and fundamental problem of two successive measurements in Quantum Mechanics, extending von Neumann’s model to two measuring instruments. We present a description which allows obtaining, in a unified way, various results that have been given in the literature.

Complete quantum mechanics: an axiomatic formulation of the Montevideo interpretation

Abstract: We make a first attempt to axiomatically formulate the Montevideo interpretation of quantum mechanics. In this interpretation environmental decoherence is supplemented with loss of coherence due to the use of realistic clocks to measure time to solve the measurement problem. The resulting formulation is framed entirely in terms of quantum objects without having to invoke the existence of measurable classical quantities like the time in ordinary quantum mechanics. The formulation eliminates any privileged role to the measurement process giving an objective definition of when an event occurs in a system.

Information and Interpretation of Quantum Mechanics

Abstract: This work is a discussion on the concept of information. We define here information as an abstraction that is able to be copied. We consider the connection between the process of copying information in quantum systems and the emergence of the so-called classical realism. The problem of interpretation of quantum mechanics in this context is discussed as well.

Quantum Structures of a Model-Universe: Questioning the Everett Interpretation of Quantum Mechanics

Abstract: Our objective is to demonstrate an inconsistency with both the original and modern Everettian Many Worlds Interpretations. We do this by examining two important corollaries of the universally valid quantum mechanics in the context of the Quantum Brownian Motion (QBM) model: "Entanglement Relativity" and the "parallel occurrence of decoherence." We conclude that the highlighted inconsistency demands that either there is a privileged spatial structure of the QBM model universe or that the Everettian Worlds are not physically real.

Survey of an approach to quantum measurement, classical properties and realist interpretation problems

Abstract: The paper gives a systematic review of the basic ideas of (non-relativistic) quantum mechanics including all changes that result from previous work of the authors. This shows that the new theory is self-consistent and (in certain sense) complete. The most important changes are: 1) A new realist interpretation of quantum mechanics based on the observation that there are enough objective properties of quantum systems if one looks for them elsewhere than among values of observables. This enables us to introduce the notion of quantum object. 2) Classical systems are defined as macroscopic quantum objects in states close to maximum entropy. For classical mechanics, new states of such kind are introduced, the so-called maximum-entropy packets, and shown to approximate classical dynamics better than Gaussian wave packets. 3) A new solution of quantum measurement problem is proposed for measurements that are performed on microsystems. First, it is assumed that readings of registration apparatuses are always signals from detectors. This implies restrictions on what is observable. Second, an application of the cluster separability principle leads to the key notion of the paper: the separation status of microsystems. The processes of preparation and registration include changes of separation status. A crucial observation is that standard quantum mechanics does not prescribe the evolution during such changes. This gap can be filled by new rules without contradicting the rest of quantum mechanics. As an example of such a new rule, Beltrametti-Cassinelli-Lahti model of measurement is modified and shown then to satisfy both the probability-reproducibility and the objectification requirements.

An Introduction to Quantum Cosmology

Abstract: We set out to provide an introduction to the area of quantum cosmology. We begin by introducing quantized general relativity or quantum geometrodynamics on which quantum cosmology is based, discussing issues in this construction. A wave function of the universe is constructed whose dynamics are governed by the Wheeler-DeWitt equation, the quantized Hamiltonian constraint of the system. It is found that, due to ambiguities arising from operator ordering issues in the Wheeler-DeWitt equation and issues in the path integral formulation of the wave function, it is often unreliable to work beyond the semiclassical approximation. In this approximation, the WKB wave function may employed to approximate the wave function. We construct a probability measure valid in this regime with a view to making measurable predictions. In order to make such predictions, it is also found to be necessary to prescribe quantum mechanical boundary conditions on the wave function. There have been various proposals for making such prescriptions. These proposals must be viewed as, at best, being supplementary fundamental physical laws. The ‘problem of time’ is analyzed for quantum cosmology or, more precisely, for reparametrization invariant systems. We outline three approaches to the problem: time before quantization, time after quantization and the timeless approach. We conclude that the most likely candidate is the timeless approach and focus on a particular evolving constants of motion form of this approach. This approach is motivated and analyzed. We find that a unique, with an additional physical argument, Hilbert space structure may be constructed using refined algebraic quantization here. We introduce the two most prominent proposals for prescribing such boundary conditions: the noboundary and tunneling wave functions. We give the definitions and interpretations of the quantum cosmological wave function satisfying these conditions. Working in the semiclassical approximation, various predictions of quantum cosmology are studied. We particularly focus on inflation for the case of a closed Friedmann-Robertson-Walker model, finding that both the no-boundary and tunneling wave functions are peaked about classical inflationary solutions. We then ask whether sufficient inflation is predicted by these wave functions, finding that on a history-by-history basis, with a physically motivated upper cut-off for the initial value of the scalar field, the tunneling wave function is found to predict sufficient inflation whereas the no-boundary wave function is not. An argument is outlined for the inaccuracy of this result for the no-boundary wave function, proposing that it should be multiplied by a volume factor. The resulting volume weighted probability is found to predict sufficient inflation for both the tunneling and no-boundary wave functions. We proceed to examine the probability in for the no-boundary universe in such a model to exhibit either an initial singularity or a bounce. Again a distinction is seen between the history-by-history probability, which is seen to predict a bounce, and the fully conditioned probability which is seen to predict an initial singularity. Structure formation may be described in quantum cosmology by introducing quantum perturbations about a semiclassical spatially homogeneous and isotropic background. This provides perhaps the greatest potential in terms of falsifiable predictions. A spectrum for such anisotropies may be predicted for both boundary proposals. A vacuum state is also found to be selected in this analysis, the Bunch-Davies vacuum state. This is often assumed to be the vacuum state in cosmological calculations and the fact that this is selected by both boundary proposals is encouraging. A non-physical analysis studying topology change in Friedmann-Robertson-Walker models with non-trivial topology is outlined. By non-physical, we mean that the boundary conditions were not chosen by any physical argument, rather for convenience. While the system is non-physical and the exact results are, in some senses, meaningless, a characteristic ‘arrow of topology change’ is observed two of the studied cases. This qualitative behaviour may well carry over to physical cases. The final chapter focuses on the decoherent or consistent histories approach to quantum mechanics. We discuss some of the conceptual foundational issues arising for quantum cosmology. It is noted that the Copenhagen interpretation is incompatible with quantum cosmology. Alternatives are found in the form of generalized quantum mechanics.