Showing papers on "Configuration space published in 1977"

On the theory of identical particles

Abstract: The classical configuration space of a system of identical particles is examined. Due to the identification of points which are related by permutations of particle indices, it is essentially different, globally, from the Cartesian product of the one-particle spaces. This fact is explicity taken into account in a quantization of the theory. As a consequence, no symmetry constraints on the wave functions and the observables need to be postulated. The two possibilities, corresponding to symmetric and antisymmetric wave functions, appear in a natural way in the formalism. But this is only the case in which the particles move in three- or higher-dimensional space. In one and two dimensions a continuum of possible intermediate cases connects the boson and fermion cases. The effect of particle spin in the present formalism is discussed.

Information-Theoretical Aspects of Quantum Measurement

Abstract: We present criteria for comparing measurements on a given system from the point of view of the information they provide. These criteria lead to a concept ofinformational completeness of a set of observables, which generalizes the conventional concept of completeness. The entropy of a state with respect to an arbitrary sample space of potential measurement outcomes is defined, and then studied in the context of configuration space and fuzzy stochastic phase space.

Systems of imprimitivity and representations of quantum mechanics on fuzzy phase spaces

Abstract: The problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so‐called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.

WKB wave function for systems with many degrees of freedom: A unified view of solitons and pseudoparticles

Abstract: The WKB method for systems with many degrees of freedom is developed. Using a given imaginary-time (Euclidean) classical solution of the equations of motion, we explicitly construct the WKB wave function in the classically forbidden region of configuration space. Similarly, we construct the wave function for the allowed region using a real-time (Minkowski) solution. For this purpose we use the collective-coordinate method previously developed for solitons in quantum field theory. The present WKB method is an extention of that by Banks, Bender, and Wu to systems with may degrees of freedom and field theories. This paper is intended to present ideas and the general formalism: two applications are briefly discussed: the quantization condition for periodic solutions and vacuum tunneling in field theories.

Analytical equations vs. table look-up for manipulation: A unifying concept

Abstract: By considering parameterization of the equations of motion for a manipulater, divergent procedures for control involving analytical expressions and table look-up can be examined in the same light. Each approach represents at different point on a continuum characterized by the indicator P, the number of parametric variables. As P increases storage requirements increase, but computational complexity decreases. Typically, for an arm having N degrees of freedom ~(N(3-P/N) operations and ~(MP) stroage cells are required for evaluation of the equations of motion. The State Space Model and the Configuration Space Model, characterized by intermediate values of P, are discussed.

Path integration in phase space

Abstract: We develop a method for defining and performing path integrals in phase space. This work generalizes to phase space the method formulated in recent years for path integration in configuration space. We give the relationship between the Wiener measure and the Liouville measure.

Configuration Space Control

Abstract: : Complicated systems with nonlinear time-varying behavior are difficult to control using classical linear feedback methods applied separately to individual degrees of freedom. At the present, mechanical manipulators, for example, are limited in their rate of movement by the inability of traditional feedback systems to deal with time-varying inertia, torque coupling effects between links and Coriolis forces. Analysis of the dynamics of such systems, however, provides the basic information needed to achieve adequate control. Implementation of a control system based on such analysis is not straightforward, however, since impractical amounts of computation or memory may be called for. We propose a new method that balances the trade-off between computation and storage costs. The actuator torques required to move a manipulator along a trajectory are calculated using coefficients found in a look-up table indexed by the configuration of the manipulator. Feedback plays only an indirect role in correcting for small differences between the state of the actual device and that of a dynamic model.

A study of the convergence in iterative natural orbital procedures

Abstract: A series of five different Iterative Natural Orbital (INO) procedures are tested for the ground state of water and are compared on the basis of their respective convergence properties. The choice of configuration space employed in these methods is shown to be a key factor in determining the results of such calculations. If the CI space is generated by taking all single excitations with respect to a series of dominant or reference configurations, it is concluded that the practice of varying such generating species at each iteration is highly desirable. In general the choice of the configuration space is found to be much more important than the attainment of strict NO convergence, whereby experience indicates that inclusion of all singly and doubly excited configurations (or at least a select subset thereof) relative to a series of dominant configurations provides the most efficient means of approximating the true NOS of a given system within the general INO framework.

Surface Feature Interpolation on Two-Dimensional Time-Space Maps

Abstract: A procedure is described for interpolating surface features on a map based on a two-dimensional time-space configuration of points. Initially an objective procedure is used to give a best fit of a previously obtained time-space configuration of points to the physical configuration of the same points. The resultant set of vectors, each of which shows the residual displacement between a point in time space and physical space, is used as the basis of a fast, objective procedure for interpolating any other points or lines in the time space. This enables such things to be shown as the time-space distortion of an urban street network or of a square graticule from physical space. The process can also be reversed so that, for example, the solution points of a facility location problem solved in time space can be converted to their equivalent locations in physical space.

Affine fields and the operator formulation of augmented scalar fields

Abstract: Affine fields, which can be used to replace the usual canonical fields, and which induce strictly homogeneous transformations of the underlying configuration space, are shown to be relevant in the operator formulation of augmented scalar field models. A characterization of the Hamiltonian and other basic generators by means of the expectation functional of the square of the field replaces the standard one based on the expectation functional of the field. Connection with previous work on augmented models is established through the form of the equation of motion for the field.

Geometrization of configuration space

Abstract: Projective geometry provides the means to geometrize configuration space, granted three elementary postulates: (1) It is possible to describe physical systems as collections of a countable number n of particles; (2) the mass‐inertial tensor of such a system is positive definite; (3) the Galilei group acting on elastic bodies can be rationally extended to the whole of GL(4,R). From these it is possible to deduce that any of the three symmetric spaces O(4,n)/O(4) ×O(n), U(2,n)/U(2) ×U(n), and Sp(1,n)/Sp(1) ×Sp(n) represents the configuration space. These three spaces are Einstein manifolds of constant sectional curvature. Several geometrical theorems are given for all three spaces; the metric on each is given in both Cartesian and polar forms; infinitesimal generators of the Lie algebras so(4,n), su(2,n), and sp(1) ×sp(n) are also obtained. Physical quantities are presumed to be obtained from eigenfunctions of the Laplace–Beltrami operator on the three spaces, and partial solutions for such functions are ob...

Characteristic class of a bundle and the existence of a global routh function

Abstract: The possibility of the global Lagrangian reduction of a mechanical system with symmetry is shown to be connected with the characteristic class of a principal fiber bundle of the configuration space over the factor manifold. It is proved that the reduced system is globally Lagrangian if and only if the product of the momentum constant with this characteristic class is zero. In the case of a rigid body rotating about a fixed point in an axially symmetric force field the bundle over a 2-sphere is non-trivial, therefore the reduced system admits a global Routh function if and only if the momentum constant is zero.

One-boson exchange potentials in relativistic configuration space

Abstract: A relativistic system of two fermions is considered. As quasipotentials we have taken one-boson exchange potentials converted to three-dimensional form in Lobachevskii space. A transition to relativistic coordinate space is made. A relativistic generalization of spin-orbit and tensor forces is suggested.

Normal modes of nonlinear discrete symmetric systems by group representation theory

Abstract: Publisher Summary The motion of a spring mass system with n masses in a three-dimensional physical space can be represented by a 3n-dimensional configuration space. The solution yields a set of 3n nonlinear coupled equations of motion. This chapter explains how, for a symmetrical system, the group representation theory can be employed to determine generalized symmetry adapted coordinates. The chapter presents the equations of motion that are expressed in this coordinate system. These equations are solved for nonlinear modes. The displacements and other kinetic and kinematic quantities can be expressed in terms of components of the symmetry adapted basis vectors. The nonlinear equation of motion is independent, and it can be solved by the classical approach.

Conformal potential inN dimensions (N>1)

Abstract: The conformal invariant model of de Alfaro, Fubini and Furlan is studied in the case of potential\(V(Q) = g/2Q^2 (Q^2 = \sum\limits_{i = 1}^N {Q_i^2 ,N > 1} )\). By means of the invariance under projective transformations and the rotation symmetry of the system in configuration space, explicit solutions are obtained.