581 research outputs found
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
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
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
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
Auto-parallel equation as Euler-Lagrange's equation in spaces with affine connections and metrics
The auto-parallel equation over spaces with affine connections and metrics is
considered as a result of the application of the method of Lagrangians with
covariant derivatives (MLCD) on a given Lagrangian density.Comment: 19 pages, LaTe
Normal frames and the validity of the equivalence principle
We investigate the validity of the equivalence principle along paths in
gravitational theories based on derivations of the tensor algebra over a
differentiable manifold. We prove the existence of local bases, called normal,
in which the components of the derivations vanish along arbitrary paths. All
such bases are explicitly described. The holonomicity of the normal bases is
considered. The results obtained are applied to the important case of linear
connections and their relationship with the equivalence principle is described.
In particular, any gravitational theory based on tensor derivations which obeys
the equivalence principle along all paths, must be based on a linear
connection.Comment: 14 pages, LaTeX 2e, the package amsfonts is neede
Flows and particles with shear-free and expansion-free velocities in (L^-_n,g)- and Weyl's spaces
Conditions for the existence of flows with non-null shear-free and
expansion-free velocities in spaces with affine connections and metrics are
found. On their basis, generalized Weyl's spaces with shear-free and
expansion-free conformal Killing vectors as velocity's vectors of spinless test
particles moving in a Weyl's space are considered. The necessary and sufficient
conditions are found under which a free spinless test particle could move in
spaces with affine connections and metrics on a curve described by means of an
auto-parallel equation. In Weyl's spaces with Weyl's covector, constructed by
the use of a dilaton field, the dilaton field appears as a scaling factor for
the rest mass density of the test particle. PACS numbers: 02.40.Ky, 04.20.Cv,
04.50.+h, 04.90.+eComment: 20 pages, LaTeX, to appear in Classical and Quantum Gravity. arXiv
admin note: substantial text overlap with arXiv:gr-qc/001104
Fibre bundle formulation of nonrelativistic quantum mechanics: I. Introduction. The evolution transport
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. Its 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 first part of this investigation is devoted to the introduction
of basic concepts on which the fibre bundle approach to quantum mechanics
rests. We show that the evolution of pure quantum-mechanical states can be
described as a suitable linear transport along paths, called evolution
transport, of the state sections in the Hilbert fibre bundle of states of a
considered quantum system.Comment: 26 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and
amsfonts are required. Revised: new material, references, and comments are
added. Minor style chages. Continuation of quan-ph/9803083. For continuation
of the this series see http://www.inrne.bas.bg/mathmod/bozhome
Transformation laws of the components of classical and quantum fields and Heisenberg relations
The paper recalls and point to the origin of the transformation laws of the
components of classical and quantum fields. They are considered from the
"standard" and fibre bundle point of view. The results are applied to the
derivation of the Heisenberg relations in quite general setting, in particular,
in the fibre bundle approach. All conclusions are illustrated in a case of
transformations induced by the Poincar\'e group.Comment: 22 LaTeX pages. The packages AMS-LaTeX and amsfonts are required. For
other papers on the same topic, view http://theo.inrne.bas.bg/~bozho/ . arXiv
admin note: significant text overlap with arXiv:0809.017
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