Historical Hamiltonian Dynamics: symplectic and covariant

This paper presents a " historical " formalism for dynamical systems, in its Hamiltonian version (Lagrangian version was presented in a previous paper). It is universal, in the sense that it applies equally well to time dynamics and to field theories on space-time. It is based on the notion of (Hamiltonian) histories, which are sections of the (extended) phase space bundle. It is developed in the space of sections, in contradistinction with the usual formalism which works in the bundle manifold. In field theories, the formalism remains covariant and does not require a spitting of space-time. It considers space-time exactly in the same manner than time in usual dynamics, both being particular cases of the evolution domain. It applies without modification when the histories (the fields) are forms rather than scalar functions, like in electromagnetism or in tetrad general relativity. We develop a differential calculus in the infinite dimensional space of histories. It admits a (generalized) symplectic form which does not break the covariance. We develop a covariant symplectic formalism, with generalizations of usual notions like current conservation, Hamiltonian vector-fields, evolution vector-field, brackets, ... The usual multisymplectic approach derives form it, as well as the symplectic form introduced by Crnkovic and Witten in the space of solutions

Examples of Berezin-Toeplitz Quantization: Finite sets and Unit Interval

We present a quantization scheme of an arbitrary measure space based on overcomplete families of states and generalizing the Klauder and the Berezin-Toeplitz approaches. This scheme could reveal itself as an efficient tool for quantizing physical systems for which more traditional methods like geometric quantization are uneasy to implement. The procedure is illustrated by (mostly two-dimensional) elementary examples in which the measure space is a $N$-element set and the unit interval. Spaces of states for the $N$-element set and the unit interval are the 2-dimensional euclidean $\R^2$ and hermitian \C^2 planes

The twin paradox in compact spaces

Twins travelling at constant relative velocity will each see the other's time dilate leading to the apparent paradox that each twin believes the other ages more slowly. In a finite space, the twins can both be on inertial, periodic orbits so that they have the opportunity to compare their ages when their paths cross. As we show, they will agree on their respective ages and avoid the paradox. The resolution relies on the selection of a preferred frame singled out by the topology of the space.Comment: to be published in PRA, 3 page

Decaying Dark Energy in Higher-Dimensional Gravity

We use data from observational cosmology to put constraints on higher-dimensional extensions of general relativity in which the effective four-dimensional dark-energy density (or cosmological "constant") decays with time. In particular we study the implications of this decaying dark energy for the age of the universe, large-scale structure formation, big-bang nucleosynthesis and the magnitude-redshift relation for Type Ia supernovae. Two of these tests (age and the magnitude-redshift relation) place modest lower limits on the free parameter of the theory, a cosmological length scale L akin to the de Sitter radius. These limits will improve if experimental uncertainties on supernova magnitudes can be reduced around z=1.Comment: 11 pages, 5 figures, submitted to A&

Statistical isotropy of the Cosmic Microwave Background

The breakdown of statistical homogeneity and isotropy of cosmic perturbations is a generic feature of ultra large scale structure of the cosmos, in particular, of non trivial cosmic topology. The statistical isotropy (SI) of the Cosmic Microwave Background temperature fluctuations (CMB anisotropy) is sensitive to this breakdown on the largest scales comparable to, and even beyond the cosmic horizon. We propose a set of measures, $\kappa_\ell$ ($\ell=1,2,3, ...$) which for non-zero values indicate and quantify statistical isotropy violations in a CMB map. We numerically compute the predicted $\kappa_\ell$ spectra for CMB anisotropy in flat torus universe models. Characteristic signature of different models in the $\kappa_\ell$ spectrum are noted.Comment: Presented at PASCOS'03, January 3-8, 2003, in TIFR, Mumbai; to be published in a special issue of 'Pramana' (4 pages, 1 figure, style files included

About the Malmquist bias in the determination of H0 and of distances of galaxies

We provide the mathematical framework which elucidates the way of using a Tully-Fisher (TF) like relation in the determination of the Hubble constant $H_0$, as well as for distances of galaxies. The methods related to the so-called Direct and Inverse TF Relations (herein DTF and ITF) are interpreted as maximum likelihood statistics. We show that, as long as the same model is used for the calibration of the TF relation and for the determination of $H_0$, we obtain a coherent Hubble's constant. The choice of the model is motivated by reasons of robustness of statistics, it depends on selection effects in observation which are present in the sample. The difference on the distance estimates when using either the ITF or the DTF model is only due to random fluctuations. It is interesting to point out that the DTF estimate does not depend on the luminosity distribution of sources. Both statistics show a correction for a bias, inadequately believed to be of Malmquist type. The repercussion of measurement errors, and additional selection effects are also analyzedComment: 37 pages,cpt-93/p.2808,latex A&A,4fig available on cpt.univ-mrs.fr directory ftp/pub/preprints/93/cosmology/93-P.280