Showing papers on "Configuration space published in 1972"

Feynman's path integral. definition without limiting procedure

Abstract: Feynman's integral is defined with respect to a pseudomeasure on the space of paths: for instance, letC be the space of pathsq:T⊂ℝ → configuration space of the system, letC be the topological dual ofC; then Feynman's integral for a particle of massm in a potentialV can be written where $$S_{\operatorname{int} } (q) = \mathop \smallint \limits_T V(q(t)) dt$$ and wheredw is a pseudomeasure whose Fourier transform is defined by for μ∈C′. Pseudomeasures are discussed; several integrals with respect to pseudomeasures are computed.

The Einstein Equations of Evolution -- A Geometric Approach

Abstract: In this paper the exterior Einstein equations are explored from a differential geometric point of view. Using methods of global analysis and infinite-dimensional geometry, we answer sharply the question: "In what sense are the Einstein equations, written as equations of evolution, a Lagrangian dynamical system?" By using our global methods, several aspects of the lapse function and shift vector field are clarified. The geometrical significance of the shift becomes apparent when the Einstein evolution equations are written using Lie derivatives. The evolution equations are then interpreted as evolution equations as seen by an observer in space coordinates. Using the notion of body-space transitions, we then find the relationship between solutions with different shifts by finding the flow of a time-dependent vector field. The use of body and space coordinates is shown to be somewhat analogous to the use of such coordinates in Euler's equations for a rigid body and the use of Eulerian and Lagrangian coordinates in hydrodynamics. We also explore the geometry of the lapse function, and show how one can pass from one lapse function to another by integrating ordinary differential equations. This involves integrating what we call the "intrinsic shift vector field." The essence of our method is to extend the usual configuration space [fraktur M]=Riem(M) of Riemannian metrics to [script T]×[script D]×[fraktur M], where [script T]=C[infinity](M,R) is the group of relativistic time translations and [script D]=Diff(M) is the group of spatial coordinate transformations of M. The lapse and shift then enter the dynamical picture naturally as the velocities canonically conjugate to the configuration fields (xit,etat)[is-an-element-of][script T]×[script D]. On this extended configuration space, a degenerate Lagrangian system is constructed which allows precisely for the arbitrary specification of the lapse and shift functions. We reinterpret a metric given by DeWitt for [fraktur M] as a degenerate metric on [script D]×[fraktur M]. On [script D]×[fraktur M], however, the metric is quadratic in the velocity variables. The groups [script T] and [script D] also serve as symmetry groups for our dynamical system. We establish that the associated conserved quantities are just the usual "constraint equations." A precise theorem is given for a remark of Misner that in an empty space-time we must have [script H]=0. We study the relationship between the evolution equations for the time-dependent metric gt and the Ricci flat condition of the reconstructed Lorentz metric gL. Finally, we make some remarks about a possible "superphase space" for general relativity and how our treatment on [script T]×[script D]×[fraktur M] is related to ordinary superspace and superphase space.

Canonical Transformations and the Radial Oscillator and Coulomb Problems

Abstract: In a previous paper a discussion was given of linear canonical transformations and their unitary representation. We wish to extend this analysis to nonlinear canonical transformations, particularly those that are relevant to physically interesting many‐body problems. As a first step in this direction we discuss the nonlinear canonical transformations associated with the radial oscillator and Coulomb problems in which the corresponding Hamiltonian has a centrifugal force of arbitrary strength. By embedding the radial oscillator problem in a higher dimensional configuration space, we obtain its dynamical group of canonical transformations as well as its unitary representation, from the Sp(2) group of linear transformations and its representation in the higher‐dimensional space. The results of the Coulomb problem can be derived from those of the oscillator with the help of the well‐known canonical transformation that maps the first problem on the second in two‐dimensional configuration space. Finally, we make use of these nonlinear canonical transformations, to derive the matrix elements of powers of r in the oscillator and Coulomb problems from a group theoretical standpoint.

Triatomic potential surfaces. A catalogue of intersections

Abstract: Intersections of adiabatic electronic orbital potential energy surfaces for triatomic systems can be classified according to the point group in which the intersection occurs, and the following geometrical properties. The locus of the intersection in the three-dimensional configuration space of the system may be a surface, a curve or a point. When the locus is a curve, it may be of finite or infinite extent. In the neighbourhood of the intersection, the separation of the surfaces may be a linear, quadratic or higher order function of the displacement coordinates. In the asymptotic case, the surfaces intersect only when one atom is infinitely far from the other two. The various types of intersections are described and examples cited.

Geometrical dynamics: A new approach to periodic orbits aroundL 4

Abstract: Geometrical dynamics is the study of the geometry of the orbits in configuration space of a dynamical system without reference to the system's motion in time. Generalized coordinates for the circular restricted problem of three bodies are taken as polar coordinatesr, θ centered at the triangular libration pointL 4. A time-independent nonlinear second order ordinary differential equation forr as a function of θ is derived. Approximations to periodic solutions are obtained by perturbations and Fourier series.

A phenomenological effective two-body interaction for calcium isotopes

Abstract: The 1]~-2pg shell has long been one of the most studied nuclear regions. Recently there has been an extensive effective-interaction calculation on calcium isotopes by F~I)~MAN~ and PITTnL (1). The single set of matrix elements given by them has a good convergence and appears to reproduce the energy spectra, (t,p) strengths and single-neutron spectroscopic factors with reasonable agreement. Moreover, the /~pg matrix elements have the desh'ed feature of repulsion on the average. This is in agreement with the findings obtained by analysing the data from (3He, d) and (d, p) measurements in this shell (~). These matrix elements, derived from Hamada-Johnston potential (a), are, however, attractive on the average. The effective-interaction method in the framework of the shell model has the disadvantage that one cannot use a large configuration space in such calculations. Then the number of variable two-particle matrix elements increases drastically and the method fails. Even in a reasonable configuration space, the two-particle matrix elements obtained from the least-squares fit to the experimental binding energies and level energies of calcium isotopes have large statistical errors (a-e), when all the required two-particle matrix elements are allowed to vary. On the other hand, a large configuration space can be used with a phenomenological effective two-body interaction with a few adjustable parameters. The wave functions thus obtained, in view of greater configurational freedom, are more realistic than those obtained by the effective-interaction method. But it is difficult to get detailed quanti tat ive fit to the binding energies and level energies like the effective-interaction method because there are only a few adjustable parameters in a phenomenological force. So an initial test of a phenomenological residual twobody interaction is to reproduce the matrix elements obtained by the effective-interaction method to get a good description of the experimental data.

Time Evolving Theory of the Mechanics of Single Particles and Single Photons

Abstract: We introduce space-time ensemble methods to formulate definitions of single particles and single photons as local abstractions of constant processes. We find the general form of the corresponding Stueckelberg Lagrangian for Riemannian and Newtonian spacetimes and supply a physical interpretation for the worldline parameter. We develop the corresponding mechanical theories over the extended configuration space and the extended phase space. We suppose that the background can be represented by an ‘external field’ and we study several general examples. Certain phenomenological forms do not describe particles, others do not seem to describe theories in which the representation of the background is process independent (Riemannian case). At the canonical level the elimination of second-class constraints associated with null processes generates restrictions on the domain of definition of photon coordinates which correspond to the absence of zero energies. The requirement that the canonical process-anti-process classification exist leads to a factorization condition on the extended phase space which is satisfied for all the cases studied in which the configuration formalisms entail no difficulties, except one, which is the ‘minimally’ coupled external vector field case over Riemannian space-times. We discuss the observation theoretical significance of our results.