Path Integrals in Noncommutative Quantum Mechanics

We consider an extension of the Feynman path integral to the quantum mechanics of noncommuting spatial coordinates and formulate the corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians). The basis of our approach is that a quantum mechanical system with a noncommutative configuration space can be regarded as another effective system with commuting spatial coordinates. Because the path integral for quadratic Lagrangians is exactly solvable and a general formula for the probability amplitude exists, we restrict our research to this class of Lagrangians. We find a general relation between quadratic Lagrangians in their commutative and noncommutative regimes and present the corresponding noncommutative path integral. This method is illustrated with two quantum mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.     

Noncommutative Integrability,Moment Map and Geodesic Flows

The purpose of this paper is to discuss the relationship betweencommutative and noncommutative integrability of Hamiltonian systemsand to construct new examples of integrable geodesic flows onRiemannian manifolds. In particular, we prove that the geodesic flowof the bi-invariant metric on any bi-quotient of a compact Lie group isintegrable in the noncommutative sense by means of polynomial integrals, andtherefore, in the classical commutative sense by means ofC -smooth integrals.     

Superintegrable Systems with Third-Order Integrals in Classical and Quantum Mechanics

We review systems in E(2) that are separable in Cartesian coordinates and admit a third-order integral both in quantum mechanics and in classical mechanics. Differences and similarities between those two cases are illustrated by numerous examples. Many of these superintegrable systems are new, and a relation is seen between superintegrable potentials and Painlevé transcendents.     

Topological Strings from Quantum Mechanics

We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov–Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local \({{\mathbb{P}}^2}\), local \({{\mathbb{P}}^1 \times {\mathbb{P}}^1}\) and local \({{\mathbb{F}}_1}\). In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi–Yau manifolds, in which the genus expansion emerges as a ’t Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background-independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry.     

Statistical Algebraic Approach to Quantum Mechanics

A scheme for constructing quantum mechanics not based on the Hilbert space and linear operators as primary elements of the theory is proposed. A particular variant of the algebraic approach is discussed. The elements of a noncommutative algebra (i.e., the observables) and the nonlinear functionals on this algebra (i.e., the physical states) serve as the primary components of the theory. The functionals are associated with the results of a single measurement. The ensembles of physical states are suggested for the role of quantum states in the standard quantum mechanics. It is shown that the mathematical formalism of the standard quantum mechanics can be fully recovered within this scheme.     

Quantum Inequalities in Quantum Mechanics

We study a phenomenon occurring in various areas of quantum physics, in which an observable density (such as an energy density) which is classically pointwise non-negative may assume arbitrarily negative expectation values after quantization, even though the spatially integrated density remains non-negative. Two prominent examples which have previously been studied are the energy density (in quantum field theory) and the probability flux of rightwards-moving particles (in quantum mechanics). However, in the quantum field context, it has been shown that the magnitude and space-time extension of negative energy densities are not arbitrary, but restricted by relations which have come to be known as quantum inequalities. In the present work, we explore the extent to which such quantum inequalities hold for typical quantum mechanical systems. We derive quantum inequalities of two types. The first are kinematical quantum inequalities where spatially averaged densities are shown to be bounded below. Specifically, we obtain such kinematical quantum inequalities for the current density in one spatial dimension (imposing constraints on the backflow phenomenon) and for the densities arising in Weyl–Wigner quantization. The latter quantum inequalities are direct consequences of sharp Gårding inequalities. The second type are dynamical quantum inequalities where one obtains bounds from below on temporally averaged densities. We derive such quantum inequalities in the case of the energy density in general quantum mechanical systems having suitable decay properties on the negative spectral axis of the total energy.Furthermore, we obtain explicit numerical values for the quantum inequalities on the one-dimensional current density, using various spatial averaging weight functions. We also improve the numerical value of the related backflow constant previously investigated by Bracken and Melloy. In many cases our numerical results are controlled by rigorous error estimates.submitted 27/01/04, accepted 05/05/04     

Schwinger Functions in Noncommutative Quantum Field Theory

It is shown that the n-point functions of scalar massive free fields on the noncommutative Minkowski space are distributions which are boundary values of analytic functions. Contrary to what one might expect, this construction does not provide a connection to the popular traditional Euclidean approach to noncommutative field theory (unless the time variable is assumed to commute). Instead, one finds Schwinger functions with twistings involving only momenta that are on the mass-shell. This explains why renormalization in the traditional Euclidean noncommutative framework crudely differs from renormalization in the Minkowskian regime.     

Noncommutative Dynamical Models with Quantum Symmetries

We define dynamical models on the q-Minkowski space algebra (which is a particular case of the Reflection Equation Algebra) as deformations (quantizations) of dynamical models with rotational symmetries, and we find their integrals. In particular, we introduce a q-analog of the Runge-Lenz vector and a q-analog of the dynamics in space-time with a spherically symmetric metric.     

Bundles over Quantum Sphere and Noncommutative Index Theorem

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podle quantum sphere equal the winding numbers of the representations defining these bundles. This result gives an estimate of the positive cone of the algebraic K₀ of the standard quantum sphere.     

Quantum Mechanics on Manifolds

In this paper, we investigate the relationships between quantum mechanics and the theory of partial differential equations. We closely follow the De Broglie and Schrödinger picture. Namely, we consider the well-known wave-particle duality as a relation between solutions of partial differential equations, describing waves, and singularities of solutions, that is particles. Our analysis of these relations shows that the necessary ingredients of any quantum mechanical picture are two connections. The first one is a connection in the tangent bundle of the configuration manifold and the second one is a connection in the trivial linear bundle.We also consider mechanical systems equipped with an inner structure and show that quantization of these systems requires a linear connection in the corresponding vector bundle.These are gravity and electromagnetic fields, or Yang–Mills fields if the configuration space is the Minkowski space. In the case of general mechanical systems, they should be considered as natural generalizations of these fields.Explicit formulas for quantizations of some mechanical systems and the corresponding star-products are given.     

Analyticity for Some Operator Functions from Statistical Quantum Mechanics

For rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schr?dinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain boundedly maps the space of square integrable functions to the space of essentially bounded functions. Dedicated to Günter Albinus Submitted: November 21, 2008. Accepted: March 31, 2009.     

微极连续体力学比经典连续体力学更深入一层次

本文通过刚删架(动静态响应与屈曲)的微极梁板模型,血液的牛顿-微极分层流模型以及人骨微极特性的实验论证等阐明微极连续体理论的本质特点,从应用侧面阐述微极连续体力学比经典连续体力学更深一层次的观点,并介绍该理论及其近期应用的部分进展情况。     

Passage from Quantum to Classical Molecular Dynamics in the Presence of Coulomb Interactions

We present a rigorous derivation of classical molecular dynamics (MD) from quantum molecular dynamics (QMD) that applies to the standard Hamiltonians of molecular physics with Coulomb interactions. The derivation is valid away from possible electronic eigenvalue crossings.     

Representations of a Noncommutative Associative Algebra Related to Quantum Torus of Rank Three

In this paper, we present some modules over the rank-three quantized Weyl algebra, which are closely related to modules over some vertex algebras. The isomorphism classes among these modules are also determined.     

Quantum Mechanics and Brownian Motions

From the point of view that the present formulation of quantum mechanics is very close to the theory of Brownian motions, we search for possible origins of the quantum fluctuation within the framework of new quantization schemes, such as stochastic and/or microcanonical quantizations, for increasing additional fictitious time other than the ordinary one. On the same basis we also show that a D-dimensional quantum system is equivalent to a (D+1)-dimensional classical system.     

Three-body Problem in Quantum Mechanics

No Abstract. . Lecture held in the Seminario Matematico e Fisico on May 26, 2004 Received: August 2004     

Absence of Quantum States Corresponding to Unstable Classical Channels

We develop a general theory of absence of quantum states corresponding to unstable classical scattering channels. We treat in detail Hamiltonians arising from symbols of degree zero in x and outline a generalization in an Appendix. E. Skibsted is (partially) supported by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation. Submitted: September 18, 2007. Accepted: January 14, 2008.