Natural star-products on symplectic manifolds and related quantum mechanical operators

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.     

Probability and complex quantum trajectories: Finding the missing links

It is shown that a normalisable probability density can be defined for the entire complex plane in the modified de Broglie-Bohm quantum mechanics, which gives complex quantum trajectories. This work is in continuation of a previous one that defined a conserved probability for most of the regions in the complex space in terms of a trajectory integral, indicating a dynamical origin of quantum probability. There it was also shown that the quantum trajectories obtained are the same characteristic curves that propagate information about the conserved probability density. Though the probability density we now adopt for those regions left out in the previous work is not conserved locally, the net source of probability for such regions is seen to be zero in the example considered, allowing to make the total probability conserved. The new combined probability density agrees with the Born’s probability everywhere on the real line, as required. A major fall out of the present scheme is that it explains why in the classical limit the imaginary parts of trajectories are not observed even indirectly and particles are confined close to the real line.     

Unitarity as Preservation of Entropy and Entanglement in Quantum Systems

The logical structure of Quantum Mechanics (QM) and its relation to other fundamental principles of Nature has been for decades a subject of intensive research. In particular, the question whether the dynamical axiom of QM can be derived from other principles has been often considered. In this contribution, we show that unitary evolutions arise as a consequences of demanding preservation of entropy in the evolution of a single pure quantum system, and preservation of entanglement in the evolution of composite quantum systems. ⁶ We would also like to dedicate this work to the memory of Asher Peres, whose contributions and sharp comments guided the first steps of the present article.     

Quantum Law of Motion: Analysis and Extension to Higher Dimensions

In this paper, we review the recently formulated quantum laws of motion and provide new observations. We also extend these laws to higher dimensions. By applying in two dimensions the obtained relations to charge submitted to an electric central potential, we decide between these laws. Furthermore, we extend the selected law to the relativistic case in higher dimensions.     

Logical reformulation of quantum mechanics. II. Interferences and the Einstein-Podolsky-Rosen experiment

The consistent quantum interpretations of logic that were introduced in a previous paper are applied to four experiments: (1) ordinary interferences, (2) the Badurek-Rauch-Tuppinger neutron interferometry experiment, (3) the Einstein-Podolsky-Rosen experiment, and (4) the detection far away of the origin of a nonrelativistic particle initially near the origin. In the first two cases, the proposition calculus excludes the possibility of observing interferences and of asserting together through which path the particles went. It is used to provide a somewhat complete discussion of the Badurek-Rauch-Tuppinger experiment. The possibility of using logical implication allows a rather complete discussion of the EPR experiment, including the question of causality, although the lack of a relativistic version of the theory does not allow a complete discussion of causality. The last experiment leads to the following result: Detecting the position of a particle at timet sometimes allows one to determine with a finite uncertainty what its momentum was just before the position measurement, even when it is infinitely precise.     

Stochastic differential equations in quantum statistical mechanics. Observables and multiple wiener integrals

The physical and mathematical framework for quantum mechanical stochastic differential equations is discussed as the quantization ofc-number equations that typically describe Brownian motion in polynomial potentials.     

Canonical conjugate momentum of discrete label operators in quantum mechanics II: formalism

The formalism developed in Part I of the present paper for the conjugate momentum of label operators is applied to standard physical examples, such as the regular one-dimensional lattice, the one-dimensional box, and the angular momentum of the plane rotator. A non-standard application is also considered, i.e., the timeoperator, conjugate to the one-dimensional harmonic Hamiltonian.     

The infinite well and Dirac delta function potentials as pedagogical,mathematical and physical models in quantum mechanics

The infinite square well and the attractive Dirac delta function potentials are arguably two of the most widely used models of one-dimensional bound-state systems in quantum mechanics. These models frequently appear in the research literature and are staples in the teaching of quantum theory on all levels. We review the history, mathematical properties, and visualization of these models, their many variations, and their applications to physical systems.     

Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum

In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.     

A comparative study of iterative solutions to linear systems arising in quantum mechanics

This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed Lanczos biconjugate A-orthonormalization methods to other classic and popular iterative methods. By the way, experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence.     

Distribution of local entropy in the Hilbert space of bi-partite quantum systems: origin of Jaynes' principle

For a closed bi-partite quantum system partitioned into system proper and environment we interpret the microcanonical and the canonical condition as constraints for the interaction between those two subsystems. In both cases the possible pure-state trajectories are confined to certain regions in Hilbert space. We show that in a properly defined thermodynamical limit almost all states within those accessible regions represent states of some maximum local entropy. For the microcanonical condition this dominant state still depends on the initial state; for the canonical condition it coincides with that defined by Jaynes' principle. It is these states which thermodynamical systems should generically evolve into. Received 13 June 2002 / Received in final form 14 November 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: jochen@theol.physik.uni-stuttgart.de     

Role of environment and confinement in quantum dissipative dynamics: A pedestrian approach

In this paper, the long time and short time behavior of a free quantum Brownian particle and of a quantum harmonic oscillator coupled to a quantum heat bath is investigated for different spectral density functions. Not only are the usual Ohmic, Drude or radiation heat bath schemes employed to derive explicit expressions for the generalized susceptibility χ(t)$\chi \left(t\right)$ and the position correlation function σ(t)$\sigma \left(t\right)$ but also the most intriguing frequency dependent heat bath spectra are considered. Emphasis is laid on the dependence on the low-frequency behavior but in some cases also the high-frequency cutoff becomes relevant. The present investigation also sheds light on the role of boundary in long time and short time behavior of quantum dissipative systems.     

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order ħ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.     

Fractional corresponding operator in quantum mechanics and applications: A uniform fractional Schrödinger equation in form and fractional quantization methods

In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same.     

Quantum stochastic differential equations for boson and fermion systems—method of non-equilibrium thermo field dynamics

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the Itô and the Stratonovich types, is presented within the framework of non-equilibrium thermo field dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schrödinger and the Heisenberg representations with fermionic and bosonic Brownian motions. In order to decide the double tilde conjugation rule and the thermal state conditions for fermions, a generalization of the system consisting of a vector field and Faddeev-Popov ghosts to dissipative open situations is carried out within NETFD.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

On the motion of particles and strings,presymplectic mechanics,and the variational bicomplex

Examples of equations of motion in classical relativistic mechanics are studied: the equations of motion of a charged spinning particle moving in a space-time (with or without torsion) in the presence of an electromagnetic field are derived via Souriau presymplectic reduction. Then, the extension of Souriaus ideas to Lagrangian field theory due to Witten, Crnkovi, Zuckerman is reviewed using the variational bicomplex, the basic properties of the Lund–Regge equations describing the motion of a string interacting with a scalar field and moving in Minkowski spacetime are recalled, and a symplectic structure for their space of solutions is found.This revised version was published online in April 2005. The publishing date was inserted.     

Quantum equilibrium and the origin of absolute uncertainty

The quantum formalism is a measurement formalism-a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrödinger's equation for a system of particles when we merely insist that particles means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that anappearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by = ¦¦ ². A crucial ingredient in our analysis of the origin of this randomness is the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.This paper is dedicated to the memory of J. S. Bell.