Many worlds and the appearance of probability in quantum mechanics

The interpretation of the squared norm as probability and the apparent stochastic nature of observation in the quantum theory are derived from the law of large numbers and the algebraic properties of infinite sequences of simultaneous quantum observables. It is argued that this result validates the many-worlds view of quantum reality.     

Hidden variables in quantum mechanics: Noncontextual generic models

The method of generic extensions (forcing) in quantum mechanics is applied to create hidden-variables models for certain systems of noncommuting quantum observables.     

Hidden variables in quantum mechanics: Generic models, set-theoretic forcing, and the appearance of probability

The hidden-variables premise is shown to be equivalent to the existence of generic filters for systems of commuting observables of a quantum system. The significance of this equivalence is interpreted in light of the theory of generic filters and boolean-valued models in set theory. The apparent stochastic nature of quantum observation is derived for these hidden-variables models.     

Generalized classical, quantum and intermediate statistics and the Pólya urn model

Generalized probability distributions for Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics, with unequal source (“prior”) probabilities qi for each level i, are obtained by combinatorial reasoning. For equiprobable degenerate sublevels, these reduce to those given by Brillouin in 1930, more commonly given as a statistical weight for each statistic. These distributions and corresponding cross-entropy (divergence) functions are shown to be special cases of the Pólya urn model, involving neither independent nor identically distributed (“ninid”) sampling. The most probable Pólya distribution is shown to contain the Acharya-Swamy intermediate statistic.     

A Bayesian account of quantum histories

We investigate whether quantum history theories can be consistent with Bayesian reasoning and whether such an analysis helps clarify the interpretation of such theories. First, we summarise and extend recent work categorising two different approaches to formalising multi-time measurements in quantum theory. The standard approach consists of describing an ordered series of measurements in terms of history propositions with non-additive ‘probabilities.’ The non-standard approach consists of defining multi-time measurements to consist of sets of exclusive and exhaustive history propositions and recovering the single-time exclusivity of results when discussing single-time history propositions. We analyse whether such history propositions can be consistent with Bayes’ rule. We show that certain class of histories are given a natural Bayesian interpretation, namely, the linearly positive histories originally introduced by Goldstein and Page. Thus, we argue that this gives a certain amount of interpretational clarity to the non-standard approach. We also attempt a justification of our analysis using Cox’s axioms of probability theory.     

Quantum probability calculations using a first-principles quantum Monte Carlo method

A first-principles Monte Carlo code that exactly simulates quantum dynamics is presented which makes no use of amplitude calculations, only noise sources. The subtle question concerning how to map random choices in amplitude interferences is explained. In this formalism negative values of the Wigner function have clear logical meaning.     

Dark energy from quantum wave function collapse of dark matter

Dynamical wave function collapse models entail the continuous liberation of a specified rate of energy arising from the interaction of a fluctuating scalar field with the matter wave function. We consider the wave function collapse process for the constituents of dark matter in our universe. Beginning from a particular early era of the universe chosen from physical considerations, the rate of the associated energy liberation is integrated to yield the requisite magnitude of dark energy around the era of galaxy formation. Further, the equation of state for the liberated energy approaches w→−1$w\to -1$ asymptotically, providing a mechanism to generate the present acceleration of the universe.     

Symplectic quantum mechanics

Using the notion of symplectic structure and Weyl (or star) product of non-commutative geometry, we construct unitary representations for the Galilei group and show how to rewrite the Schrödinger equation in phase space. This approach gives rise to a new procedure to derive Wigner functions without the use of the Liouville-von Neumann equation. Applications are presented by deriving the states of linear and nonlinear oscillators in terms of amplitudes of probability in phase space. The notion of coherent states is also discussed in this context.     

Strategies to measure a quantum state

We consider the problem of determining the mixed quantum state of a large but finite number of identically prepared quantum systems from data obtained in a sequence of ideal (von Neumann) measurements, each performed on an individual copy of the system. In contrast to previous approaches, we do not average over the possible unknown states but work out a “typical” probability distribution on the set of states, as implied by the experimental data. As a consequence, any measure of knowledge about the unknown state and thus any notion of “best strategy” (i.e., the choice of observables to be measured, and the number of times they are measured) depend on the unknown state. By learning from previously obtained data, the experimentalist re-adjusts the observable to be measured in the next step, eventually approaching an optimal strategy. We consider two measures of knowledge and exhibit all “best” strategies for the case of a two-dimensional Hilbert space. Finally, we discuss some features of the problem in higher dimensions and in the infinite dimensional case.     

Quantum mechanics of Klein-Gordon-type fields and quantum cosmology

With a view to address some of the basic problems of quantum cosmology, we formulate the quantum mechanics of the solutions of a Klein-Gordon-type field equation: (∂_t²+D)ψ(t)=0, where and D is a positive-definite operator acting in a Hilbert space . In particular, we determine all the positive-definite inner products on the space of the solutions of such an equation and establish their physical equivalence. This specifies the Hilbert space structure of uniquely. We use a simple realization of the latter to construct the observables of the theory explicitly. The field equation does not fix the choice of a Hamiltonian operator unless it is supplemented by an underlying classical system and a quantization scheme supported by a correspondence principle. In general, there are infinitely many choices for the Hamiltonian each leading to a different notion of time-evolution in . Among these is a particular choice that generates t-translations in and identifies t with time whenever D is t-independent. For a t-dependent D, we show that regardless of the choice of the inner product the t-translations do not correspond to unitary evolutions in , and t cannot be identified with time. We apply these ideas to develop a formulation of quantum cosmology based on the Wheeler-DeWitt equation for a Friedman-Robertson-Walker model coupled to a real scalar field with an arbitrary positive confining potential. In particular, we offer a complete solution of the Hilbert space problem, construct the observables, use a position-like observable to introduce the wave functions of the universe (which differ from the Wheeler-DeWitt fields), reformulate the corresponding quantum theory in terms of the latter, reduce the problem of the identification of time to the determination of a Hamiltonian operator acting in , show that the factor-ordering problem is irrelevant for the kinematics of the quantum theory, and propose a formulation of the dynamics. Our method is based on the central postulates of nonrelativistic quantum mechanics, especially the quest for a genuine probabilistic interpretation and a unitary Schrödinger time-evolution. It generalizes to arbitrary minisuperspace (spatially homogeneous) models and provides a way of unifying the two main approaches to the canonical quantum cosmology based on these models, namely quantization before and after imposing the Hamiltonian constraint.     

Infinite quantum well: A coherent state approach

A new family of 2-component vector-valued coherent states for the quantum particle motion in an infinite square well potential is presented. They allow a consistent quantization of the classical phase space and observables for a particle in this potential. We then study the resulting position and (well-defined) momentum operators. We also consider their mean values in coherent states and their quantum dispersions.     

Singularities of Berry connections inhibit the accuracy of the adiabatic approximation

Adiabatic approximation for quantum evolution is investigated addressing its dependence on the Berry connections that are functions of a slowly-varying parameter R  . When the Berry connections have singularities of type 1/Rσ$1/R^{\sigma }$ with σ<1$\sigma <1$, the adiabatic fidelity converges to unit according to a power-law; When the singularity index σ becomes larger than one, adiabatic approximation breaks down. Two-level models are used to substantiate our theory.     

Feynman-path analysis of Hardy's paradox: Measurements and the uncertainty principle

Hardy's paradox is analysed within Feynman's formulation of quantum mechanics. A transition amplitude is represented as a sum over virtual paths which different intermediate measurements convert into different sets of real pathways. Contradictions arise if conflicting statements are applied to the same statistical ensemble. Usefulness of “strange” weak values for resolving the paradox is disputed.     

Trajectory interpretation of the uncertainty principle in 1D systems using complex Bohmian mechanics

Complex Bohmian mechanics is introduced to investigate the validity of a trajectory interpretation of the uncertainty principles ΔqΔp??/2 and ΔEΔt??/2 by replacing probability mean values with time-averaged mean values. It is found that the ?/2 factor in the uncertainty relation ΔEΔt??/2 stems from a quantum potential whose time-averaged mean value taken along any closed trajectory with a period T=2π/ω is proved to be an integer multiple of ?ω/2 for one-dimensional systems.     

A Solved Model to Show Insufficiency of Quantitative Adiabatic Condition

The adiabatic theorem is a useful tool in processing quantum systems slowly evolving, but its practical application depends on the quantitative condition expressed by Hamiltonian＇s eigenvalues and eigenstates, which is usually taken as a sufficient condition. Recently, the sufficiency of the condition was questioned, and several counterexamples have been reported. Here we present a new solved model to show the insufficiency of the traditional quantitative adiabatic condition.     

Can chaotic quantum energy levels statistics be characterized using information geometry and inference methods?

In this paper, we review our novel information-geometrodynamical approach to chaos (IGAC) on curved statistical manifolds and we emphasize the usefulness of our information-geometrodynamical entropy (IGE) as an indicator of chaoticity in a simple application. Furthermore, knowing that integrable and chaotic quantum antiferromagnetic Ising chains are characterized by asymptotic logarithmic and linear growths of their operator space entanglement entropies, respectively, we apply our IGAC to present an alternative characterization of such systems. Remarkably, we show that in the former case the IGE exhibits asymptotic logarithmic growth while in the latter case the IGE exhibits asymptotic linear growth.At this stage of its development, IGAC remains an ambitious unifying information-geometric theoretical construct for the study of chaotic dynamics with several unsolved problems. However, based on our recent findings, we believe that it could provide an interesting, innovative and potentially powerful way to study and understand the very important and challenging problems of classical and quantum chaos.     

Arbitrary Lagrangian-Eulerian rate equation for the Born probability density in complex space

The arbitrary Lagrangian-Eulerian rate equation is derived for the complex-extended Born probability density. This equation describes the rate of change in the density along an arbitrary path in the complex plane. Several special cases for the rate equation depending on the grid velocity are discussed. This study provides a further understanding of the probability density in the complex plane.