Consistent Quantum Histories: Towards a Universal Language of Physics

Time in quantum mechanics

We extend previous work on the arrival time problem in quantum mechanics, in the framework of decoherent histories, to the case of a particle coupled to an environment. The usual arrival time probabilities are related to the probability current, so we explore the properties of the current for general open systems that can be written in terms of a master equation of Lindblad form. We specialise to the case of quantum Brownian motion, and show that after a time of order the localisation time the current becomes positive. We show that the arrival time probabilities can then be written in terms of a POVM, which we compute. We perform a decoherent histories analysis including the effects of the environment and show that time of arrival probabilities are decoherent for a generic state after a time much greater than the localisation time, but that there is a fundamental limitation on the accuracy, $\delta t$, with which they can be specified which obeys $E\delta t>>\hbar$. We confirm that the arrival time probabilities computed in this way agree with those computed via the current, provided there is decoherence. We thus find that the decoherent histories formulation of quantum mechanics provides a consistent explanation for the emergence of the probability current as the classical arrival time distribution, and a systematic rule for deciding when probabilities may be assigned.

Consistent quantum measurements

The decoherent histories approach to quantum theory is applied to a class of reparametrization-invariant models whose state is an energy eigenstate. A key step in this approach is the construction of class operators characterizing the questions of physical interest, such as the probability of the system entering a given region of configuration space without regard to time. In nonrelativistic quantum mechanics these class operators are given by time-ordered products of projection operators. But in reparametrization-invariant models, where there is no time, the construction of the class operators is more complicated, the main difficulty being to find operators which commute with the Hamiltonian constraint (and so respect the invariance of the theory). Here, inspired by classical considerations, we put forward a proposal for the construction of such class operators for a class of reparametrization-invariant systems. They consist of continuous infinite temporal products of Heisenberg picture projection operators. We investigate the consequences of this proposal in a number of simple models and also compare with the evolving constants method. The formalism developed here is ultimately aimed at cosmological models described by a Wheeler-DeWitt equation, but the specific features of such models are left to future papers.

In the present work, I discuss the interpretation of quantum mechanics (QM) from the viewpoint of quantum logic (QL). I regard the objects of QL to be the possible (accidental) properties that can be ascribed to a quantum system SYS. The basic idea is that, at any given moment t, SYS actualizes some properties, but not others. The micro-state of SYS at time t is identified with the set of all properties that SYS actualizes at time t.One thing an interpretation of QM is supposed to do, I believe, is delineate the admissible quantum micro-states. Since the characterization of quantum micro-states is intimately related to the characterization of quantum logical consistency, the interpretation of QM is intimately tied to the interpretation of QL.Two kinds of interpretations are discussed. Strict interpretations are based on the assumption that the properties of a system are individuated by the projection operators on the associated Hilbert space.

This paper discusses the fundamental assumptions and background of the consistent histories (CH) approach to quantum mechanics. The focus of the paper is on the concept of frameworks. It is proposed that frameworks should be interpreted objectively as observer-independent realities. Two further options are considered: a hidden-variables variant of the CH approach, and a many-worlds version, which considers each individual history belonging to a given family as describing a separate world. The latter interpretation is subsequently compared and contrasted with the standard many-worlds interpretation. Finally, the solution to the measurement problem offered by the many-worlds variant of CH is analyzed and amended.

We review recent progress in understanding the arrival time problem in quantum mechanics, from the point of view of the decoherent histories approach to quantum theory. We begin by discussing the arrival time problem, focussing in particular on the role of the probability current in the expected classical solution. After a brief introduction to decoherent histories we review the use of complex potentials in the construction of appropriate class operators. We then discuss the arrival time problem for a particle coupled to an environment, and review how the arrival time probability can be expressed in terms of a POVM in this case. We turn finally to the question of decoherence of the corresponding histories, and we show that this can be achieved for simple states in the case of a free particle, and for general states for a particle coupled to an environment.

After more than 70 years of debate about the difficulties that one encounters in working out a coherent view of physical processes based on the standard formulation of quantum mechanics, there is now a widespread belief that such difficulties do not arise from philosophical prejudices (as has been repeatedly asserted by many of the supporters of textbook quantum mechanics) but represent precise mathematical and physical challenges which call for a physical solution. As J.S. Bell appropriately stated1 “the way ahead is unromantic in that it requires mathematical work by theoretical physicists, rather than interpretations by philosophers”. It is also encouraging, for those who share this position, to see that now there are explicit proposals indicating possible ways out in which the process of measurement (and more generally all those measurement-like processes2 “we are obliged to admit ... are going on more or less all the time, more or less everywhere”) is analysed not in terms of vague assertions or resorting to ill-defined and/or contradictory dualistic evolution processes (the linear and deterministic evolution for microsystems and the nonlinear and stochastic wave packet reduction for the macroscopic ones) but in terms of more fundamental physical concepts. In particular, approaches of this kind, which S. Goldstein3 has appropriately denoted as “observer-free formulations of quantum mechanics”, can be grouped in three categories: the pilot-wave theories: the decoherent histories and the spontaneous localization models.

We propose an alternative interpretation for the meaning of noncommutativity of the string-inspired field theories and quantum mechanics. Arguments are presented to show that the noncommutativity generated in the stringy context should be assumed to be only between the particle coordinate observables, and not of the spacetime coordinates. Some implications of this fact for noncomutative field theories and quantum mechanics are discussed. In particular, a consistent interpretation is given for the wavefunction in quantum mechanics. An analysis of the noncommutative theories in the Schr\"odinger formulation is performed employing a generalized quantum Hamilton-Jacobi formalism. A formal structure for noncommutative quantum mechanics, richer than the one of noncommutative quantum field theory, comes out. Conditions for the classical and commutative limits of these theories have also been determined and applied in some examples.

The quantum mechanics of closed systems such as the universe is formulated using an extension of familiar probability theory that incorporates negative probabilities. Probabilities must be positive for sets of alternative histories that are the basis of fair settleable bets. However, in quantum mechanics there are sets of alternative histories that can be described but which cannot be the basis for fair settleable bets. Members of such sets can be assigned extended probabilities that are sometimes negative. A prescription for extended probabilities is introduced that assigns extended probabilities to all histories that can be described, fine grained or coarse grained, members of decoherent sets or not. All probability sum rules are satisfied exactly. Sets of histories that are recorded to sufficient precision are the basis of settleable bets. This formulation is compared with the decoherent (consistent) histories formulation of quantum theory. Prospects are discussed for using this formulation to provide testable alternatives to quantum theory or further generalizations of it.

We study the spontaneous decay rate of a single molecule close to a metallic nanoparticle in the extreme near‐field regime. The electrodynamic response of the metal is modelled using a non‐local dielectric constant, that accounts for the relevant microscopic length scales. We describe quantitatively the crossover between the macroscopic and microscopic regimes. In the case of silver, for a nanoparticle with radius 25 nm, the transition occurs for a distance between the emitter and the metal surface on the order of 10 nm. We show that below this distance, the non‐radiative decay rate and the quenching efficiency are enhanced due to the non‐local interaction.

Mesoscopic systems, which lie between microscopic and macroscopic scales, are pivotal in understanding quantum transport phenomena. These systems, typically comprising a few hundred to thousand atoms, exhibit quantum effects that are not observable in either the macroscopic or microscopic regimes. Theoretical models of quantum transport in mesoscopic systems, such as the Landauer-Büttiker formalism, the tight-binding model, and the Lippmann-Schwinger equation, provide crucial insights into electron behavior, resistance, and conductance in these systems. This article presents an overview of key theoretical models used to describe quantum transport, discusses their applications, and examines the experimental validation of these models in modern mesoscopic systems.

Based on the nonequilibrium Green function technique, a unified theory of quantum transport and quantum diffusion is developed from quantum-kinetic equations of the full carrier distribution function f<(k, κ|T,t). The rigorous two-time approach is equally applicable to quantum transport as well as quantum diffusion that proceeds either via extended states or by hopping between localized carriers. The approach focuses on the entanglement between the microscopic and macroscopic time regimes and on the related non-Markovian evolution in both time channels. For a one-dimensional superlattice model subject to quantizing dc and ac electric fields, both phononless transport and diffusion mechanisms are identified that have a strict double-time origin. The experimental verification of this intricate quantum effect seems to be feasible by studying quantum-box superlattices at low temperatures.

The articles provides a phenomenological reading of the Many-Worlds Interpretation (MWI) of quantum mechanics and its answer to the measurement problem, or the question of “why only one of a wave function’s probable values is observed when the system is measured.” Transcendental-phenomenological and hermeneutic-phenomenological approaches are employed. The project comprises four parts. Parts one and two review MWI and the standard (Copenhagen) interpretation of quantum mechanics. Part three reviews the phenomenologies. Part four deconstructs the hermeneutics of MWI. It agrees with the confidence the theory derives from its (1) unforgiving appropriation of the Schrödinger equation and (2) association of branching universes with the evolution of the wave function insofar as that understanding comes from the formalism itself. Part four also reveals the hermeneutical shortcomings of the standard interpretation.

In general, Albert Einstein as one of "the founding fathers of quantum mechanics" had some problems to accept especially the Copenhagen dominated interpretation of quantum mechanics. Einstein's dissatisfaction with Copenhagen's interpretation of quantum mechanics, the absence of locality and causality within the Copenhagen dominated quantum mechanics lead to the well known Einstein, Podolsky and Rosen thought experiment. According to Einstein et al., the Copenhagen dominated quantum mechanics cannot be regarded as a complete physical theory. The Einstein, Podolsky and Rosen thought experiment was the origin of J. S. Bell's publication in 1964; known as Bell's theorem. Meanwhile, some dramatic violations of Bell's inequality (by so called Bell test experiments) have been reported which is taken as an empirical evidence against local realism and causality at quantum level and as positive evidence in favor of the Copenhagen dominated quantum mechanics. Thus far, Quantum mechanics is still regarded as a "strictly" non-local theory. The purpose of this publication is to refute Bell's original theorem. Thus far, if we accept Bell's theorem as correct, we must accept that +0> = +1. We can derive a logical contradiction out of Bell's theorem, Bell's theorem is refuted.