64 research outputs found
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. 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
Fibre bundle formulation of nonrelativistic quantum mechanics. V. Theory's interpretation, summary and discussion
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.
This is the fifth closing part of our investigation. We briefly discuss the
observer's role in the theory and different realizations of the space-time
model used as a base space in the bundle approach to quantum mechanics. We
point the exact conditions for the equivalence of Hilbert bundle and Hilbert
space formulations of the theory. A comparison table between the both
description of nonrelativistic quantum mechanics is presented. We discuss some
principal moments of the Hilbert bundle description and show that it is more
general than the Hilbert space one. Different directions for further research
are pointed too.Comment: 17 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and
amsfonts are required. Concluding part of the series of works
quant-ph/9803083, quant-ph/9803084, quant-ph/9804062, quant-ph/9806046, and
quant-ph/9901039. For related papers, view
http://www.inrne.bas.bg/mathmod/bozhome
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
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
A mathematical base for Fibre bundle formulation of Lagrangian Quantum Field Theory
The paper contains a differential-geometric foundations for an attempt to
formulate Lagrangian (canonical) quantum field theory on fibre bundles. In it
the standard Hilbert space of quantum field theory is replace with a Hilbert
bundle; the former playing a role of a (typical) fibre of the letter one.
Suitable sections of that bundle replace the ordinary state vectors and the
operators on the system's Hilbert space are transformed into morphisms of the
same bundle. In particular, the field operators are mapped into corresponding
field morphisms.Comment: 12 LaTeX pages. The packages AMS-LaTeX and amsfonts are required. For
other papers on the same topic, view http://theo.inrne.bas.bg/~bozho
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
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