Fibre bundle formulation of nonrelativistic quantum mechanics. III. Pictures and integrals of motion

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. It's evolution is determined through the bundle (analogue of the) Schr\"odinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The mentioned quantities are connected by a number of relations derived in this work. In this third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, there are found the state sections and bundle morphisms corresponding to state vectors and observables respectively. The equations of motion for these quantities are derived too. Using the results obtained, we consider from the bundle view-point problems concerning the integrals of motion. An invariant (bundle) necessary and sufficient conditions for a dynamical variable to be an integral of motion are found.Comment: 19 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and amsfonts are required. New references and comments are added. Minor style chages. Continuation of quant-ph/9803083, quant-ph/9803084 and quant-ph/9804062. For continuation of the series view http://www.inrne.bas.bg/mathmod/bozhome

The 21-cm Background from the Cosmic Dark Ages: Minihalos and the Intergalactic Medium before Reionization

The H atoms inside minihalos (i.e. halos with virial temperatures T_vir < 10^4 K, in the mass range roughly from 10^4 M_sun to 10^8 M_sun) during the cosmic dark ages in a LCDM universe produce a redshifted background of collisionally-pumped 21-cm line radiation which can be seen in emission relative to the cosmic microwave background (CMB). Previously, we used semi-analytical calculations of the 21-cm signal from individual halos of different mass and redshift and the evolving mass function of minihalos to predict the mean brightness temperature of this 21-cm background and its angular fluctuations. Here we use high-resolution cosmological N-body and hydrodynamic simulations of structure formation at high redshift (z > 8) to compute the mean brightness temperature of this background from both minihalos and the intergalactic medium (IGM) prior to the onset of Ly-alpha radiative pumping. We find that the 21-cm signal from gas in collapsed, virialized minihalos dominates over that from the diffuse shocked gas in the IGM.Comment: 8 pages, 5 figures. To appear in proceedings of UC Irvine May 2005 workshop on "First Light & Reionization", eds. E. Barton & A. Cooray, New Astronomy Reviews, in pres

Fibre bundle formulation of nonrelativistic quantum mechanics. 0. Preliminary considerations: Quantum mechanics from a geometric-observer's viewpoint

We propose a version of the non-relativistic quantum mechanics in which the pure states of a quantum system are described as sections of a Hilbert (generally infinitely-dimensional) fibre bundle over the space-time. There evolution is governed via (a kind of) a parallel transport in this bundle. Some problems concerning observables are considered. There are derived the equations of motion for the state sections and observables. We show that up to a constant the matrix of the coefficients of the evolution operator (transport) coincides with the matrix of the Hamiltonian of the investigated quantum system.Comment: 15 standard LaTeX 2e (11pt, A4) pages. The packages AMS-LaTeX and amsfonts are require

The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14

The Abel-Jacobi maps of the families of elliptic quintics and rational quartics lying on a smooth cubic threefold are studied. It is proved that their generic fiber is the 5-dimensional projective space for quintics, and a smooth 3-dimensional variety birational to the cubic itself for quartics. The paper is a continuation of the recent work of Markushevich-Tikhomirov, who showed that the first Abel-Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers $c_1=0, c_2=2$ obtained by Serre's construction from elliptic quintics, and that the factorizing map from the moduli space to the intermediate Jacobian is \'etale. The above result implies that the degree of the \'etale map is 1, hence the moduli component of vector bundles is birational to the intermediate Jacobian. As an applicaton, it is shown that the generic fiber of the period map of Fano varieties of degree 14 is birational to the intermediate Jacobian of the associated cubic threefold.Comment: Latex, 28 page

Fibre bundle formulation of relativistic quantum mechanics. I. Time-dependent approach

We propose a new fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present first part of our investigation we consider the time-dependent or Hamiltonian approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced by (state) liftings of paths or sections along paths of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the evolution transport along paths in the space-time) of the state liftings/sections in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations.Comment: 16 standard LaTeX pages. The packages AMS-LaTeX and amsfonts are required. The paper continuous the application of fibre bundle formalism to quantum physics began in the series of works quant-ph/9803083, quant-ph/9803084, quant-ph/9804062, quant-ph/9806046, quant-ph/9901039, quant-ph/9902068, and quant-ph/0004041. For related papers, view http://theo.inrne.bas.bg/~bozho

Normal frames and the validity of the equivalence principle. III. The case along smooth maps with separable points of self-intersection

The equivalence principle is treated on a mathematically rigorous base on sufficiently general subsets of a differentiable manifold. This is carried out using the basis of derivations of the tensor algebra over that manifold. Necessary and/or sufficient conditions of existence, uniqueness, and holonomicity of these bases in which the components of the derivations of the tensor algebra over it vanish on these subsets, are studied. The linear connections are considered in this context. It is shown that the equivalence principle is identically valid at any point, and along any path, in every gravitational theory based on linear connections. On higher dimensional submanifolds it may be valid only in certain exceptional cases.Comment: 15 standard LaTeX 2e (11pt, A4) pages. The package amsfonts is require