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

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

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

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