Tomaschitz, R. (1993). Classical and quantum dispersion in Robertson-Walker cosmologies, Journal of Mathematical Physics 34, 1022-1042, DOI: 10.1063/1.530235
Abstract (AIP, CDS, SAO/NASA ADS, Zbl 0811.53075)
The instability of world lines in Robertson–Walker universes of negative spatial curvature is investigated. A probabilistic description of this instability, similar to the Liouville equation, is developed, but in a manifestly covariant, non-Hamiltonian form. To achieve this the concept of a horospherical geodesic flow of expanding bundles of parallel world lines is introduced. An invariant measure and a covariant evolution equation for the probability density on which this flow acts is constructed. The orthogonal surfaces to these bundles of trajectories are horospheres, closed surfaces in three-space, touching the boundary at infinity of hyperbolic space, where the flow lines emerge. These horospheres are just the wave fronts of spherical waves, which constitute a complete set of eigenfunctions of the Klein–Gordon equation. This fact suggests that the evolution of the quantum mechanical density with the classical one be compared, and asymptotic identity in the asymptotically flat region is found. This leads, furthermore, to the study of the time behavior of the dispersion of the energy and the coordinates and the energy-time uncertainty relation, and identity in the late stage of the cosmic evolution is again found. In an example it is finally demonstrated that this identity can persist in the early phase of the expansion with a rapidly varying scale factor, provided the fields are conformally coupled to the curvature.
COSMOLOGICAL MODELS, CURVATURE, GEODESICS, KLEIN−GORDON EQUATION, INSTABILITY, MEASURE THEORY, DISTRIBUTION FUNCTIONS, STATISTICAL MECHANICS
98.80.Jk
Mathematical and relativistic aspects of cosmology
03.65.Ta
Foundations of quantum mechanics; measurement theory
05.45.-a
Nonlinear dynamics and chaos
Zbl
0811.53075
Tomaschitz, Roman
Classical and quantum dispersion in Robertson-Walker cosmologies
[J] J.
Math. Phys. 34, No.3, 1022-1042 (1993). ISSN 0022-2488
MSC 2000:
*53Z05
Appl. of differential geometry to physics
83C10 Equations of motion
Keywords: instability of world lines; Robertson-Walker universes; horospherical geodesic flow
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Classical and quantum dispersion in Robertson-Walker cosmologies |
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Tomaschitz, Roman |
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Journal of Mathematical Physics, Volume 34, Issue 3, March 1993, pp.1022-1042 |
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03/1993 |
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description: Roman Tomaschitz (1993) Classical and quantum dispersion in Robertson-Walker cosmologies, Journal of Mathematical Physics 34, 1022.
Keywords: open Robertson–Walker universe, hyperbolic 3-space, constant negative curvature, instability of geodesic world lines, Lyapunov instability, horospherical geodesic flow, semiclassical mechanics, Klein–Gordon equation, cosmic time evolution of classical and quantum probability densities, Hamilton–Jacobi equation, horospherical boundary action, classical and quantum mechanical energy–time uncertainty relations, dispersion, space expansion, cosmic time, Poincaré ball model of hyperbolic geometry, Möbius transformations in the Poincaré half-space, scalar point-pair invariants, Poisson kernel, spectral decomposition of scalar wave fields in hyperbolic space
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