Showing papers on "Consistent histories published in 2002"

Quantum probabilities as Bayesian probabilities

Abstract: In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. In this paper, we show that, despite being prescribed by a fundamental law, probabilities for individual quantum systems can be understood within the Bayesian approach. We argue that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode. In the classical world, maximal information about a physical system is complete in the sense of providing definite answers for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed. Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule, that maximal information about a quantum system leads to a unique quantum-state assignment, and that quantum theory provides a stronger connection between probability and measured frequency than can be justified classically. Finally, we give a Bayesian formulation of quantum-state tomography.

Time in quantum mechanics

Abstract: Time is often said to play an essentially different role from position in quantum mechanics: whereas position is represented by a Hermitian operator, time is represented by a c-number. This difference is puzzling and has given rise to a vast literature and many efforts at a solution. It is argued that the problem is only apparent and that there is nothing in the formalism of quantum mechanics that forces us to treat position and time differently. The apparent problem is caused by the dominant role point particles play in physics and can be traced back to classical mechanics.

Foundations of Quantum Mechanics, an Empiricist Approach

Abstract: Preface. 1. Standard and generalized formalisms of quantum mechanics. 2. Empiricist and realist interpretations of quantum mechanics. 3. Quantum mechanical description of measurement, and the 'measurement problem'. 4. The Copenhagen interpretation. 5. The Einstein-Podolsky-Rosen problem. 6. Individual-particle and ensemble interpretations of quantum mechanics. 7. Generalized quantum mechanics. 8. Applications of generalized quantum mechanics. 9. The Bell inequality in quantum mechanics. 10. Subquantum or hidden-variables theories. A: Mathematical appendix. Bibliography. Index.

Modal Interpretations of Quantum Mechanics

Abstract: Fil: Lombardi, Olimpia Iris Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina Universidad de Buenos Aires Facultad de Ciencias Exactas y Naturales; Argentina

Worlds in the Everett interpretation

Abstract: This is a discussion of how we can understand the world-view given to us by the Everett interpretation of quantum mechanics, and in particular the role played by the concept of `world'. The view presented is that we are entitled to use `many-worlds' terminology even if the theory does not specify the worlds in the formalism; this is defended by means of an extensive analogy with the concept of an `instant' or moment of time in relativity, with the lack of a preferred foliation of spacetime being compared with the lack of a preferred basis in quantum theory. Implications for identity of worlds over time, and for relativistic quantum mechanics, are discussed.

Contextual objectivity: a realistic interpretation of quantum mechanics

Abstract: An attempt is made to formulate quantum mechanics (QM) in physical rather than in mathematical terms. It is argued that the appropriate conceptual framework for QM is `contextual objectivity', which includes an objective definition of the quantum state. This point of view sheds new light on topics such as the reduction postulate and the quantum measurement process.

The Anti-Vaxjo Interpretation of Quantum Mechanics

Abstract: In this note, I try to accomplish two things First, I fulfill Andrei Khrennikov's request that I comment on his "Vaxjo Interpretation of Quantum Mechanics," contrasting it with my own present view of the subject matter Second, I try to paint an image of the hopeful vistas an information-based conception of quantum mechanics indicates

A Simple Model for an Objective Interpretation of Quantum Mechanics

Abstract: An SR model is presented that shows how an objective (noncontextual and local) interpretation of quantum mechanics can be constructed, which contradicts some well-established beliefs following from the standard interpretation of the theory and from known no-go theorems. The SR model is not a hidden variables theory in the standard sense, but it can be considered a hidden parameters theory which satisfies constraints that are weaker than those usually imposed on standard hidden variables theories. The SR model is also extended in a natural way that shows how a broader theory embodying quantum mechanics can be envisaged which is realistic in a semantic sense, hence compatible with various “realistic” perspectives.

Vaxjo Interpretation of Quantum Mechanics

Abstract: We present critical arguments against individual interpretation of Bohr’s complementarity and Heisenberg’s uncertainty principles. Statistical interpretation of these principles is discussed in the contextual framework. We support the possibility to use Statistical Contextual Realist Interpretation of quantum formalism. In spite of all no-go theorems (e.g., von Neumann, Kochen and Specker,..., Bell,...), recently (quant-ph/0306003 and 0306069) we constructed a realist basis of quantum mechanics. In our model both classical and quantum spaces are rough images of the fundamental prespace. Quantum mechanics cannot be reduced to classical one. Both classical and quantum representations induce reductions of prespace information.

Consistent resolution of some relativistic quantum paradoxes

Abstract: A relativistic version of the (consistent or decoherent) histories approach to quantum theory is developed on the basis of earlier work by Hartle, and used to discuss relativistic forms of the paradoxes of spherical wave packet collapse, Bohm's formulation of the Einstein-Podolsky-Rosen paradox, and Hardy's paradox. It is argued that wave function collapse is not needed for introducing probabilities into relativistic quantum mechanics, and in any case should never be thought of as a physical process. Alternative approaches to stochastic time dependence can be used to construct a physical picture of the measurement process that is less misleading than collapse models. In particular, one can employ a coarse-grained but fully quantum-mechanical description in which particles move along trajectories, with behavior under Lorentz transformations the same as in classical relativistic physics, and detectors are triggered by particles reaching them along such trajectories. States entangled between spacelike separate regions are also legitimate quantum descriptions, and can be consistently handled by the formalism presented here. The paradoxes in question arise because of using modes of reasoning which, while correct for classical physics, are inconsistent with the mathematical structure of quantum theory, and are resolved (or tamed) by using a proper quantum analysis. Inmore » particular, there is no need to invoke, nor any evidence for, mysterious long-range superluminal influences, and thus no incompatibility, at least from this source, between relativity theory and quantum mechanics.« less

Quantum causality, stochastics, trajectories and information

Abstract: A history of the discovery of `new' quantum mechanics and the paradoxes of its probabilistic interpretation are briefly reviewed from the modern point of view of quantum probability and information. Modern quantum theory, which has been developed during the last 20 years for the treatment of quantum open systems including quantum noise, decoherence, quantum diffusions and spontaneous jumps occurring under continuous in time observation, is not yet a part of the standard curriculum of quantum physics. It is argued that the conventional formalism of quantum mechanics is insufficient for the description of quantum events, such as spontaneous decays say, and the new experimental phenomena related to individual quantum measurements, but they have all received an adequate mathematical treatment in quantum stochastics of open systems. Moreover, the only reasonable probabilistic interpretation of quantum mechanics put forward by Max Born was, in fact, in irreconcilable contradiction with traditional mechanical reality and causality. This led to numerous quantum paradoxes, some of them due to the great inventors of quantum theory such as Einstein and Schrodinger. They are reconsidered in this paper from the modern point of view of quantum stochastics and information. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing the algebra of the dynamical generators and the algebra of the actual observables, or Bell's beables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the actual histories with the potential future. This rule, together with the self-compatibility of the measurements ensuring the consistency of the histories, is called the nondemolition, or causality principle in modern quantum theory. The application of this rule in the form of dynamical commutation relations leads to the derivation of the von Neumann projection postulate, and also to the more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a dynamical solution, in the form of the quantum stochastic filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schrodinger and Bohr. It has been recently proved that the quantum stochastic model for the continuous in time measurements is equivalent to a Dirac type boundary-value problem for the secondary quantized input `offer waves from future' in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the `trajectories of the output particles'. This supports the corpuscular-wave duality in the form of the thesis that everything in the future are quantized waves, while everything in the past are trajectories of the recorded particles.

The Observer in the Quantum Experiment

Abstract: A goal of most interpretations of quantum mechanics is to avoid the apparent intrusion of the observer into the measurement process. Such intrusion is usually seen to arise because observation somehow selects a single actuality from among the many possibilities represented by the wavefunction. The issue is typically treated in terms of the mathematical formulation of the quantum theory. We attempt to address a different manifestation of the quantum measurement problem in a theory-neutral manner. With a version of the two-slit experiment, we demonstrate that an enigma arises directly from the results of experiments. Assuming that no observable physical phenomena exist beyond those predicted by the theory, we argue that no interpretation of the quantum theory can avoid a measurement problem involving the observer.

Seven Principles of Quantum Mechanics

Abstract: The list of basic axioms of quantum mechanics as it was formulated by von Neumann includes only the mathematical formalism of the Hilbert space and its statistical interpretation. We point out that such an approach is too general to be considered as the foundation of quantum mechanics. In particular in this approach any finite-dimensional Hilbert space describes a quantum system. Though such a treatment might be a convenient approximation it can not be considered as a fundamental description of a quantum system and moreover it leads to some paradoxes like Bell's theorem. I present a list from seven basic postulates of axiomatic quantum mechanics. In particular the list includes the axiom describing spatial properties of quantum system. These axioms do not admit a nontrivial realization in the finite-dimensional Hilbert space. One suggests that the axiomatic quantum mechanics is consistent with local realism.

A Spontaneous collapse model on a lattice

Abstract: We present a spontaneous collapse model of a field theory on a 1+1 null lattice, in which the causal structure of the lattice plays a central role. Issues such as ``locality,'' ``non-locality'' and superluminal signaling are addressed in the context of the model which has the virtue of extreme simplicity. The formalism of the model is related to that of the consistent histories approach to quantum mechanics.

Causality, symmetries and quantum mechanics

Abstract: It is argued that there is no evidence for causality as a metaphysical relation in quantum phenomena. The assumptions that there are no causal laws, but only probabilities for physical processes constrained by symmetries, leads naturally to quantum mechanics. In particular, an argument is made for why there are probability amplitudes that are complex numbers. This argument generalizes the Feynman path integral formulation of quantum mechanics to include all possible terms in the action that are allowed by the symmetries, but only the lowest order terms are observable at the presently accessible energy scales, which is consistent with observation. The notion of relational reality is introduced in order to give physical meaning to probabilities. This appears to give rise to a new interpretation of quantum mechanics.

Quantum Mechanical Histories and the Berry Phase

Abstract: We elaborate on the distinction between geometric and dynamical phase in quantum theory and we show that the former is intrinsically linked to the quantum mechanical probabilistic structure. In particular, we examine the appearance of the Berry phase in the consistent histories scheme and establish that it is the basic building block of the decoherence functional. These results are consequences of the novel temporal structure of histories-based theories.

The Linearity of Quantum Mechanics at Stake:. the Description of Separated Quantum Entities

Abstract: We consider the situation of a physical entity that is the compound entity consisting of two ‘separated’ quantum entities. In earlier work it has been proved by one of the authors that such a physical entity cannot be described by standard quantum mechanics. More precisely, it was shown that two of the axioms of traditional quantum axiomatics are at the origin of the impossibility for standard quantum mechanics to describe this type of compound entity. One of these axioms is equivalent with the superposition principle, which means that separated quantum entities put the linearity of quantum mechanics at stake. We analyze the conceptual steps that are involved in this proof, and expose the necessary material of quantum axiomatics to be able to understand the argument.

V\"axj\"o Interpretation of Quantum Mechanics

Abstract: We present critical arguments against individual interpretation of Bohr's complementarity and Heisenberg's uncertainty principles. Statistical interpretation of these principles is discussed in the contextual framework. We support the possibility to use Statistical Contextual Realist Interpretation of quantum formalism. In spite of all {\bf no-go} theorems (e.g., von Neumann, Kochen and Specker,..., Bell,...), recently (quant-ph/0306003 and 0306069) we constructed a realist basis of quantum mechanics. In our model both classical and quantum spaces are rough images of the fundamental {\bf prespace.} Quantum mechanics cannot be reduced to classical one. Both classical and quantum representations induce reductions of prespace information.

The Emergence of Classical Dynamics in a Quantum World

Abstract: Ever since the advent of quantum mechanics, it has been clear that the atoms composing matter do not obey Newton's laws. Instead, their behavior is described by the Schrodinger equation. Surprisingly though, until recently, no clear explanation was given for why every- day objects, which are merely collections of atoms, are observed to obey Newton's laws. It would seem that, if quantum mechanics explains all the properties of atoms accurately, they, too, should obey quantum mechanics. This reasoning led some scientists to believe in a dis- tinct macroscopic, or "big and complicated," world in which quantum mechanics fails and classical mechanics takes over, although there has never been experimental evidence for such a failure. Even those who insisted that Newtonian mechanics would somehow emerge from the underlying quantum mechanics as the system became increasingly macroscopic were hindered by the lack of adequate experimental and theoretical tools. In the last decade, however, this quantum-to-classical transition has become accessible to experimental study and quantitative description, and the resulting insights are the subject of this article 1 .

The Interpretation of Quantum Cosmology and the Problem of Time

Abstract: The development of quantum cosmology, in which Stephen Hawking played a crucial role, has frequently encountered substantial conceptual and technical difficulties related to the problem of time in quantum gravity and to general issues concerning the foundations of quantum theory. In this contribution to Stephen's 60th Birthday Conference, I describe some recent work in which the decoherent histories approach to quantum theory is used to quantize simple cosmological models and perhaps shed some light on some of these difficulties.

The Nature of Information in Quantum Mechanics

Abstract: A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process.

Holography, Time and Quantum Mechanics

Abstract: In this talk we entertain the possibility that the synthesis of general covariance and quantum mechanics requires an extension of the basic kinematical setup of quantum mechanics. According to the holographic principle, regions of spacetime bounded by a finite area carry finite entropy. When we in addition assume that the origin of the entropy is a finite dimensional Hilbert space, and apply this to cosmological solutions using a suitable notion of complementarity, we find as a consequence that gravitational effects can lead to dynamical variation in the dimensionality of such Hilbert spaces. This happens generally in cosmological settings like our own universe.

Events and Covariance in the Interpretation of Quantum Field Theory

Abstract: In relativistic quantum field theory the notion of a local operation is regarded as basic: each open space-time region is associated with an algebra of observables representing possible measurements performed within this region. It is much more difficult to accommodate the notions of events taking place in such regions or of localized objects. But how can the notion of a local operation be basic in the theory if this same theory would not be able to represent localized measuring devices and localized events? After briefly reviewing these difficulties we discuss a strategy for eliminating the tension, namely by interpreting quantum theory in a realist way. To implement this strategy we use the ideas of the modal interpretation of quantum mechanics. We then consider the question of whether the resulting scheme can be made Lorentz invariant.

Decoherent Histories for Space–Time Domains

Abstract: The decoherent histories approach is a natural medium in which to address problems in quantum theory which involve time in a non-trivial way. This chapter reviews the various attempts and difficulties involved in using the decoherent histories approach to calculate the probability for crossing the surface x = 0 during a finite interval of time. The commonly encountered difficulties in assigning crossing times arise here as difficulties in satisfying the consistency (no-interference) condition. This can be overcome by introducing an environment to produce decoherence, and probabilities exhibiting the expected classical limit are obtained. The probabilities are, however, dependent to some degree on the decohering environment. The results are compared with a recently proposed irreversible detector model. A third method is introduced, involving continuous quantum measurement theory. Some closely related work on the interpretation of the wave function in quantum cosmology is described.

Bohmian mechanics as a heuristic device: Wave packets in the harmonic oscillator

Abstract: Although Bohmian mechanics has attracted considerable interest as a causal interpretation of quantum mechanics, it also possesses intrinsic heuristic value, arising from calculational tools and physical insights that are unavailable in “standard” quantum mechanics. We illustrate by examining the behavior of Gaussian harmonic oscillator wave packets from the Bohmian perspective. By utilizing familiar classical concepts and techniques, we obtain a physically transparent picture of packet behavior. This example provides, at a level accessible to students, a concrete illustration of Bohmian mechanics as a heuristic device that can enhance both understanding and discovery.

Quantum Computation and Many Worlds

Abstract: An Everett (`Many Worlds') interpretation of quantum mechanics due to Saunders and Zurek is presented in detail This is used to give a physical description of the process of a quantum computation Objections to such an understanding are discussed

The maze of quantum mechanics

Abstract: Students can explore quantum mechanics with a concept map that uses only the most elementary solutions to Schrodinger's equation. The content has been slightly modified from the traditional introduction to the subject because the issue of interpretation is postponed until Parseval's theorem is reached and used to postulate two fundamental equations of probability, simultaneously. A set of canonical but approximate equations can describe wavepacket motion for most linear waves. If restricted to certain special cases, these equations of wavepacket motion are easily derived and can serve as a temporary substitute for Ehrenfest's theorem. A number of exercises can be incorporated into this maze of quantum mechanics.

Critical note on the role of the quantum mechanical "collapse" in quantum modeling of consciousness.

Abstract: We give a brief account on the existing strategies in quantum-mechanical approaching the problem of consciousness. To a list of distinguished approaches we add the next plausible notion: when treated quantum- mechanically, consciousness should be modeled as an open quantum system. This notion is tightly connected to the von Neumann's "collapse" ("wave packet reduction"). Here we strongly emphasize that the problem of the "collapse" cannot be resolved within the quantum mechanics of open quantum systems (or: decoherence theory). Some clues in this regard are briefly outlined.

Quantum vs stochastic processes and the role of complex numbers

Abstract: We argue that the complex numbers are an irreducible object of quantum probability. This can be seen in the measurements of geometric phases that have no classical probabilistic analogue. Having complex phases as primitive ingredient implies that we need to accept non-additive probabilities. This has the desirable consequence of removing constraints of standard theorems about the possibility of describing quantum theory with commutative variables. Motivated by the formalism of consistent histories and keeping an analogy with the theory of stochastic processes, we develop a (statistical) theory of quantum processes. They are characterised by the introduction of a "density matrix" on phase space paths -thus including phase information- and fully reproduce quantum mechanical predictions. In this framework wecan write quantum differential equations, that could be interpreted as referring to a single system (in analogy to Langevin's equation). We describe a reconstruction theorem by which a quantum process can yield the standard Hilbert space structure if the Markov property is imposed. Finally, we discuss the relevance of our iresults for the interpretation of quantum theory (a sample space if possible if probabilities are non-additive) and quantum gravity (the Hilbert space arises after the consideration of a background causal structure).