4,993 research outputs found
Fibre bundle formulation of nonrelativistic quantum mechanics. IV. Mixed states and evolution transport's curvature
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.
The present fourth part of this series is devoted mainly to the fibre bundle
description of mixed quantum states. We show that to the conventional density
operator there corresponds a unique density morphism (along paths) for which
the corresponding equations of motion are derived. It is also investigated the
bundle description of mixed quantum states in the different pictures of motion.
We calculate the curvature of the evolution transport and prove that it is
curvature free iff the values of the Hamiltonian operator at different moments
commute.Comment: 14 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and
amsfonts are required. Minor style changes, a problem with the bibliography
is corrected. Continuation of quant-ph/9803083, quant-ph/9803084,
quant-ph/9804062 and quant-ph/9806046. For continuation of the series and
related papers, view http://www.inrne.bas.bg/mathmod/bozhome
Comments on: "Quantum mechanics as a gauge theory of metaplectic spinor fields" by M. Reuter [Int.J.Mod.Phys. A13 (1998), 3835-3884; hep-th/9804036]
We point out how some mathematically incorrect passages in the paper of M.
Reuter can be formulated in a rigorous way. The fibre bundle approach to
quantum mechanics developed in quant-ph/9803083, quant-ph/9803084,
quant-ph/9804062, quant-ph/9806046, quant-ph/9901039, and quant-ph/9902068 is
compared with the one contained in loc. cit.Comment: 11 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and
amsfonts are required. For related papers, 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
Minimal sections of conic bundles
Let the threefold X be a general smooth conic bundle over the projective
plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this
paper we prove the existence of two natural families C(+) and C(-) of curves on
X, such that the Abel-Jacobi map F sends one of these families onto a copy of
the theta divisor (Theta), and the other -- onto the jacobian J(X). The general
curve C of any of these two families is a section of the conic bundle
projection, and our approach relates such C to a maximal subbundle of a rank 2
vector bundle E(C) on C, or -- to a minimal section of the ruled surface
P(E(C)). The families C(+) and C(-) correspond to the two possible types of
versal deformations of ruled surfaces over curves of fixed genus g(C). As an
application, we find parameterizations of J(X) and (Theta) for certain classes
of Fano threefolds, and study the sets Sing(Theta) of the singularities of
(Theta).Comment: Duke preprint, 29 pages. LaTex 2.0
The fiber of the Griffiths map for the non-hyperelliptic Fano threefolds of genus 6
Among smooth non-rational Fano 3-folds, the non-hyperelliptic Fano 3-fold
X(10) of genus 6 has the unique property to admit a non-trivial orbit of
birationally isomorphic 3-folds, inside its moduli space. Here we prove that
these orbits are, in fact, the same as the fibers of the Griffiths period map
for X(10). This leads upto the main result of the paper: The general fiber of
the period map for X(10) is a union of two irreducible families of 3-folds:
F(1) + F(2), each F(i) -- isomorphic to the Fano surface of conics of any of
its elements. As an application, we give a negative answer to a Tjurin's
conjecture: The general X(10) is birational to a quartic double solid.Comment: Duke preprint, 52 pages, LaTex 2.0
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
Heat kernel expansions on the integers and the Toda lattice hierarchy
We consider the heat equation where is a second-order difference
operator in a discrete variable . The fundamental solution has an expansion
in terms of the Bessel functions of imaginary argument. The coefficients
in this expansion are analogs of Hadamard's coefficients for
the (continuous) Schrodinger operator.
We derive an explicit formula for in terms of the wave and the
adjoint wave functions of the Toda lattice hierarchy. As a first application of
this result, we prove that the values of these coefficients on the diagonals
and define a hierarchy of differential-difference equations which
is equivalent to the Toda lattice hierarchy. Using this fact and the
correspondence between commutative rings of difference operators and algebraic
curves we show that the fundamental solution can be summed up, giving a finite
formula involving only two Bessel functions with polynomial coefficients in the
time variable , if and only if the operator belongs to the family of
bispectral operators constructed in [18].Comment: corrected typo
On the heat kernel and the Korteweg-de Vries hierarchy
We give explicit formulas for Hadamard's coefficients in terms of the
tau-function of the KdV hierarchy. We show that some of the basic properties of
these coefficients can be easily derived from these formulas. The first
immediate corollary is the symmetry of Hadamard's coefficients about the
diagonal. Another well known fact, which follows from this approach, is that on
the diagonal Hadamard's coefficients determine the right-hand sides of the
equations of the KdV hierarchy. The proof of the main result uses Sato theory
and simple properties of Gegenbauer polynomials.Comment: 9 page
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