Showing papers on "Configuration space published in 1993"

Classical equations for quantum systems

Abstract: The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.

Quantum Scattering Theory for Several Particle Systems

Abstract: Introduction 1 General Aspects of the Scattering Problem 2 Stationary Approach to Scattering Theory 3 The Method of Integral Equation 4 Configuration Space Neutral Particles 5 Charged Particles in Configuration Space 6 Mathematical Foundation of the Scattering Problem 7 Some Applications 8 Comments on Literature Bibliography Index

Computation of configuration-space obstacles using the fast Fourier transform

Abstract: This paper presents a new method for computing the configuration-space map of obstacles that is used in motion-planning algorithms. The method derives from the observation that, when the robot is a rigid object that can only translate, the configuration space is a convolution of the workspace and the robot. This convolution is computed with the use of the fast Fourier transform (FFT) algorithm. The method is particularly promising for workspaces with many and/or complicated obstacles, or when the shape of the robot is not simple. It is an inherently parallel method that can significantly benefit from existing experience and hardware on the FFT. >

Asymptotic completeness of long-range N-body quantum systems

Abstract: where X is a euclidean space and {lra: a E A} is a finite family of projections onto certain subspaces xa of X. We will always assume that the family {Xa: a E A} is closed with respect to the algebraic sum and contains Xamin : = {0}. The class of operators of the form (0.2) is a generalization of (0.1) up to a linear change of coordinates in the configuration space X. They usually go under the name of generalized N-body Schrbdinger operators and were first considered in [A]. (In what follows we will drop the word "generalized".) We refer the reader to [RSi], vol. III, for a general introduction to N-body Schr6dinger operators defined as in equation (0.1). We will consistently use the definition (0.2) and assume that the reader is familiar with the physical meaning of various objects under study.

A general framework for interpreting time finite element formulations

Abstract: In this work, an attempt is made at filling the apparent gap existing between the two major approaches evolved in the literature towards formulating space-time finite element methods. The first assumes Hamilton's Law as underlying concept, while the second performs a weighted residual approach on the ordinary differential equations emanating from the semidiscretization in the space dimension. A general framework is proposed in the following pages, where the configuration space and the phase space forms of Hamilton's Law provide the general statements of the problem of motion. Within this framework, different families of integration algorithms are derived, according to different interpretations of the boundary terms. The bi-discontinuous form is obtained as the consequence of a consistent impulsive formulation of dynamics, while the discontinuous Galerkin form is obtained when the boundary terms at the end of the time interval are appropriately approximated.

Exact Dirac Quantization of All 2-D Dilaton Gravity Theories

Abstract: The most general dilaton gravity theory in 2 spacetime dimensions is considered. A Hamiltonian analysis is performed and the reduced phase space, which is two dimensional, is explicitly constructed in a suitable parametrization of the fields. The theory is then quantized via the Dirac method in a functional Schrodinger representation. The quantum constraints are solved exactly to yield the (spatial) diffeomorphism invariant quantum wave functional for all theories considered. This wave function depends explicitly on the (single) configuration space coordinate as well as on the imbedding of space into spacetime (i.e. on the choice of time).

Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology.

Abstract: This paper is concerned with the question of the existence of composition laws in the sum-over-histories approach to relativistic quantum mechanics and quantum cosmology, and its connection with the existence of a canonical formulation. In nonrelativistic quantum mechanics, the propagator is represented by a sum over histories in which the paths move forward in time. The composition law of the propagator then follows from the fact that the paths intersect an intermediate surface of constant time once and only once, and a partition of the paths according to their crossing position may be affected. In relativistic quantum mechanics, by contrast, the propagators (or Green functions) may be represented by sums over histories in which the paths move backward and forward in time. They therefore intersect surfaces of constant time more than once, and the relativistic composition law, involving a normal derivative term, is not readily recovered. The principal technical aim of this paper is to show that the relativistic composition law may, in fact, be derived directly from a sum over histories by partitioning the paths according to their first crossing position of an intermediate surface. We review the various Green functions of the Klein-Gordon equation, and derive their composition laws. We obtain path-integral representations for all Green functions except the causal one. We use the proper time representation, in which the path integral has the form of a nonrelativistic sum over histories but is integrated over time.The question of deriving the composition laws therefore reduces to the question of factoring the propagators of nonrelativistic quantum mechanics across an arbitrary surface in configuration space. This may be achieved using a known result called the path decomposition expansion (PDX). We give a proof of the PDX using a spacetime lattice definition of the Euclidean propagator. We use the PDX to derive the composition laws of relativistic quantum mechanics from the sum over histories. We also derive canonical representations of all of the Green functions of relativistic quantum mechanics, i.e., express them in the form 〈x''\ensuremath{\Vert}x'〉, where the {\ensuremath{\Vert}x〉} are a complete set of configuration-space eigenstates. These representations make it clear why the Hadamard Green function ${\mathit{G}}^{(1)}$ does not obey a standard composition law. They also give a hint as to why the causal Green function does not appear to possess a sum-over-histories representation. We discuss the broader implications of our methods and results for quantum cosmology, and parametrized theories generally. We show that there is a close parallel between the existence of a composition law and the existence of a canonical formulation, in that both are dependent on the presence of a timelike Killing vector. We also show why certain naive composition laws that have been proposed in the past for quantum cosmology are incorrect. Our results suggest that the propagation amplitude between three-metrics in quantum cosmology, as constructed from the sum over histories, does not obey a composition law.

Large Deviations and Maximum Entropy Principle for Interacting Random Fields on $\mathbb{Z}^d$

Abstract: We present a new approach to the principle of large deviations for the empirical field of a Gibbsian random field on the integer lattice Zd. This approach has two main features. First, we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities, and thus obtain estimates which are finer and more widely applicable. Second, we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field, the limits being those predicted by the maximum entropy principle. This result implies a general version of the equivalence of Gibbs ensembles, stating that every microcanonical limiting state is a grand canonical equilibrium state. We also prove a converse to the last statement, and discuss some applications. 0. Introduction. As is well known, the study of the asymptotic probabilities of large fluctuations of time averages or space averages away from the mean is based on two fundamental principles: the principle of large deviations, and the maximum entropy principle. The former provides the exact rate of exponential decay of the fluctuation probabilities, whereas the latter predicts the limiting conditional distribution under the condition that the fluctuations are large. It is obvious that these principles are intimately related. In this paper, we investigate these principles in the case of interacting random fields on the integer lattice 7Zd. The setup is the following. First, we let (E, 4) be any measurable space. We shall assume throughout that (E, a) is standard Borel, but we shall avoid making any explicit topological assumptions on E. So we do not assume that E is Polish. Next we let S = Zd be the d-dimensional integer lattice and (Q. F) = (E, gas the associated product space. In lattice models of statistical mechanics, (E, &) is called the state space or single spin space and (Q, F) the configuration space. We let J = = S F) denote the set of all probability

A qualitative test for N-finger force-closure grasps on planar objects with applications to manipulation and finger gaits

Abstract: A force-closure test function for an n-finger grasp on a planar object with friction is presented. All n-finger grasps can be represented by an n-dimensional contact space. The critical conditions of the test functions are used to define force-closure curves which are the boundaries of force-closure sets in the contact configuration space. It is shown that the force-closure sets can be decomposed into subsets in which m (m >

Path planning and the topology of configuration space

Abstract: This work considers the path planning problem for planar revolute manipulators operating in a workspace of polygonal obstacles. This problem is solved by determining the topological characteristics of obstacles in configuration space, thereby determining where feasible paths can be found. A collision-free path is then calculated by using the mathematical description of the boundaries of only those configuration space obstacles with which collisions are possible. The key to this technique is a simple test for determining whether two disjoint obstacles are connected in configuration space. This test allows the path planner to restrict its calculations to regions in which collision-free paths are guaranteed a priori, thus avoiding unnecessary computations and resulting in an efficient implementation. Typical timing results for environments consisting of four polyhedral obstacles comprising a total of 27 vertices are of the order of 22 ms on a SPARC-IPC workstation. >

Self-motion topology for redundant manipulators with joint limits

Abstract: The topology of self-motion manifolds for serial redundant manipulators in the presence of joint limits is investigated. It is known that the pre-images of singular taskpoints divide the configuration space into regions where self-motion manifolds are homotopic. The authors describe how self-motion manifolds are homotopic. The authors describe how self-motion manifolds rupture moving from one region to the next. The influence of joint limits on those topologies is investigated. This analysis leads to the discovery of the semi-singularity, a new type of singularity introduced by the presence of joint limits in redundant manipulators. As opposed to standard singularities, this type is unidirectional in nature. It has a significant impact in the structure of the kinematic map, as it generates the boundary between regions of different self-motion topologies. A 3-degree-of-freedom planar robot is used to illustrate the phenomena. This analysis is of fundamental importance for global redundancy resolution and path planning, as it describes the connectivity among regions in both the work space and the configuration space, as well as the topology of self-motion manifolds. >

Planning planar grasps of smooth contours

Abstract: A theory for identification and classification of two-fingered grasps is described and demonstrated. It is shown that optimal finger positions are obtained from the pairwise intersections of three subsets in the configuration space for the gripper. The sets are the symmetry set, the antisymmetry set, and the critical set of the grasp map. An implementation of the theory is developed for an ADEPT robot with visual sensing. >

Convergence studies of atomic properties from variational methods: total energy, ionization energy, specific mass shift, and hyperfine parameters for Li

Abstract: Systematic variational calculations using the multi-configuration Hartree-Fock method for Li are reported. For 1s22s 2S of Li, an nl-expansion is attempted in configuration space, similar to a partial wave expansion for helium-like systems. The total energy, ionization energy, specific mass shift, and hyperfine contact term are calculated with respect to increasing size of the configuration set. A comparison is made with n-expansion in configuration space, which is found to be the more efficient method. The total energy obtained from our variational procedure is lower than that of a previously reported partial wave study on ground state of Li. Similar n-expansion studies are performed for 1s22p 2P. For the latter, the 2s-2p transition integrals are reported, showing convergence of the transition energy as well as the length and velocity form of the f-value.

11-DIMENSIONAL Supermembrane in a Spinor Moving Repere Formalism

Abstract: Twistor-like formulations for super p-brane theories in the D-dimensional space-time including D=11 supermembrane are suggested in an extended configuration space. This extension is carried out by the addition of auxiliary spinor coordinates of the moving repere (the Newman-Penrose dyades for D=4). The spinor coordinates are realized as the Lorentz harmonics and necessary for the irreducible covariant description of the κ-symmetry. Using this harmonical realization a generalization of the Newman-Penrose dyades for the case D=11 is suggested.

On the configuration space topology in general relativity

Abstract: The configuration-space topology in canonical General Relativity depends on the choice of the initial data 3-manifold. If the latter is represented as a connected sum of prime 3-manifolds, the topology receives contributions from all configuration spaces associated to each individual prime factor. There are by now strong results available concerning the diffeomorphism group of prime 3-manifolds which are exploited to examine the topology of the configuration spaces in terms of their homotopy groups. We explicitly show how to obtain these for the class of homogeneous spherical primes, and communicate the results for all other known primes except the non-sufficiently large ones of infinite fundamental group.

Geometry-induced yang-mills fields in constrained quantum mechanics

Abstract: We derive the effective Hamiltonian for a quantomechanical system constrained to move on a submanifold M of its configuration space Rn by a confining potential V. Besides potential terms proportional to the intrinsic and mean curvature of M the restriction to the constraint produce the minimal interaction with a geometry-induced Yang-Mills field.

On the Possibility of Spinorial Quantization in the Skyrme Model

Abstract: We consider the configuration space of the Skyrme model and give a simple proof that loops generated by 2π-rotations are contractible in the even-, and non-contractible in the odd-winding-number sectors.

Faddeev calculations in configuration space with Cartesian coordinates

Abstract: The Faddeev-Noyes equations are solved in their natural Cartesian Jacobi coordinates for scattering below break-up threshold and for bound states. This approach is particularly well adapted to deal with strongly varying interactions. The method is proved to be successful in the three-nucleon system. First results concerning the4He trimer in configuration space are presented and further generalizations are suggested.

Quantum and classical mechanics of Q deformed systems

Abstract: Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of Uq(m) in a space of functions on a phase space spanned by commutative coordinates is constructed.

Obstacle Avoidance Control of Multi-Degree-of-Freedom Manipulator Using Electrostatic Potential Field and Sliding Mode

Abstract: The approach presented in this paper is based on two ideas : (1) on the planning level an electrostatic potential field in the configuration space of the robot-manipulator is used. The generalized force curves are attractive to the goal point avoiding obstacles without local minima ; (2) on the control level the system follows these force curves using sliding mode control. The system is attracted to the swishing manifold using sliding mode and therefore the system is robust to disturbances and parameter variations.

Nonholonomic motion planning using Stoke's theorem

Abstract: An algorithm for planning admissible trajectories for nonholonomic systems that will take the system from one point in its configuration space to another is developed. In this algorithm, the independent variables are converged to their desired values. Closed trajectories of the independent variables are used to converge the dependent variables. Stoke's theorem is used in the algorithm to convert the problem of finding a closed path into that of finding a surface area in the space of the independent variables such that the dependent variables converge to their desired values as the independent variables traverse along the boundary of this surface area. The salient features of this algorithm are illustrated in the example of a disk rolling without slipping on a flat surface. >

Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment

Abstract: We consider the problem of planning the motion of an arbitrary k-sided polygonal robot B, free to translate and rotate in a polygonal environment V bounded by n edges. We show that the combinatorial complexity of a single connected component of the free configuration space of B is k/sup 3/n/sup 2/2/sup O(log(2/3)/ n). This is a significant improvement of the naive bound O((kn)/sup 3/); when k is constant, which is often the case in practice, this yields a near-quadratic bound on the complexity of such a component, which almost settles (in this special case) a long-standing conjecture regarding the complexity of a single cell in a three-dimensional arrangement of surfaces. We also present an algorithm that constructs a single component of the free configuration space of B in time O(n/sup 2+/spl epsi//), for any /spl epsi/>0, assuming B has a constant number of sides. This algorithm, combined with some standard techniques in motion planning, yields a solution to the underlying motion planning problem, within the same asymptotic running time. >

Quantum chaos in the configurational quantum cat map.

Abstract: The motion of a classical or quantum-mechanical charged particle in the unit square (with periodic boundary conditions) is investigated under the influence of periodic electromagnetic fields. It is shown that the external fields can be chosen in such a way that the configuration space of the particle is mapped periodically to itself according to Arnold’s cat map. The time evolution of the quantum system shows the same degree of irregularity as does the classical time evolution which is completely dominated by the properties of the hyperbolic map. In particular, the eigenfunctions of the Floquet operator are determined analytically, and, as an immediate consequence, the spectrum of quasienergies in this system is seen to be absolutely continuous. Furthermore, spatial correlations decay exponentially. The observed features are in striking similarity to properties of classically chaotic systems; for example, long-time predictions of the future behavior of the system turn out to be extremely sensitive to the specification of the initial state. In other words, the time evolution of the quantum system is algorithmically complex. These phenomena, based on the formation of arbitrarily fine structures in the two-dimensional configuration space, require that the system absorb energy (provided by the external kicks) at an exponential rate.

Configurational entropy of microemulsions: The fundamental length scale

Abstract: Phenomenological models have been quite successful in characterizing both the various complex phases and the corresponding phase diagrams of microemulsions. In some approaches, e.g., the random mixing model (RMM), the lattice parameter is of the order of the dimension of an oil or water domain and has been used as a length scale for computing a configurational entropy, the so‐called entropy of mixing, of the microemulsion. In the central and material section of this paper (Sec. III), we show that the fundamental length scale for the calculation of the entropy of mixing is of the order of the cube root of the volume per molecule—orders of magnitude smaller than the dimension of such a domain. This length scale is specifically the scale for the configurational entropy—not that which measures either the curvature of the interface, the ‘‘granularity’’ of the microemulsion, or the persistence length. Furthermore, we demonstrate, in general, that mixing entropy, evaluated in configuration space as opposed to phase space, will not be physically correct unless it is made to be consistent with the phase space evaluation. Following this core section, we give a one‐dimensional illustration of the problem (Sec. IV), and discuss the consequences of our general result with respect to the RMM (Sec. V). The RMM not only seriously underestimates the entropy of mixing but exhibits a dependence on composition that is qualitatively very different from the correct dependence. Furthermore, for oil or water rich compositions of the microemulsion, the correct mixing entropy reinforces effects that would normally be attributed to bending energy, i.e., it destabilizes the system.

Some Remarks on Almost Gibbs States

Abstract: We investigate possible ways to generalize the concept of Gibbs states. For classical lattice systems we do so by modifying the configuration space and considering the continuity of conditional probabilities thereupon. For quantum systems we are led by the structure that can be inferred from considering the correlation functions of the two-dimensional (ferromagnetic) Ising model as so-called ‘classical’ states on a quantum system. In the latter context our notion of an almost Gibbs state coincides with the notion of a state that does not satisfy the Kubo-Martin-Schwinger boundary condition but instead has only the structure that follows from Tomita’s theorem for so-called separating states on the observables.