31 research outputs found
Nilpotent classical mechanics: s-geometry
We introduce specific type of hyperbolic spaces. It is not a general linear
covariant object, but of use in constructing nilpotent systems. In the present
work necessary definitions and relevant properties of configuration and phase
spaces are indicated. As a working example we use a D=2 isotropic harmonic
oscillator.Comment: 8 pages, presented at QGIS, June 2006, Pragu
Lagrangian Models of the Particles with Spin: The First Seventy Years
We briefly review models of relativistic particles with spin. Departing from
the oldest attempts to describe the spin within the lagrangian framework we
pass through various non supersymmetric models. Then the component and
superfield formulations of the spinning particle and superparticle models are
reviewed. Our focus is mainly on the classical side of the problem, but some
quantization questions are mentioned as well.Comment: plain LaTeX, 25 pages. To appear in "From Field Theory to Quantum
Groups" World Scientific, proceedings to honour J.Lukierski in his 60th
birthda
Quantum brachistochrone problem for spin-1 in a magnetic field
We study quantum brachistochrone problem for the spin-1 system in a magnetic
field of a constant absolute value. Such system gives us a possibility to
examine in detail the statement of papers [A. Carlini {\it et al.}, Phys. Rev.
Lett. {\bf 96}, 060503 (2006), D. C. Brody, D. W. Hook, J. Phys. A {\bf 39},
L167, (2006)] that {\it the state vectors realizing the evolution with the
minimal time of passage evolve along the subspace spanned by the initial and
final state vectors.} Using explicit example we show the existence of quantum
brachistochrone with minimal possible time, but the state vector of which,
during the evolution {\em leaves} the subspace spanned by the initial and final
state vectors. This is the result of the choice of more constrained Hamiltonian
then assumed in the general quantum brachistochrone problem, but what is worth
noting, despite that such evolution is more complicated it is still time
optimal. This might be important for experiment, where general Hamiltonian with
the all allowed parameters is difficult to implement, but constrained one
depending on magnetic field can be realized. However for pre-constrained
Hamiltonian not all final states are accessible. Present result does not
contradict general statement of the quantum brachistochrone problem, but gives
new insight how time optimal passage can be realized.Comment: 7 pages, no figure
Nilpotent Classical Mechanics
The formalism of nilpotent mechanics is introduced in the Lagrangian and
Hamiltonian form. Systems are described using nilpotent, commuting coordinates
. Necessary geometrical notions and elements of generalized differential
-calculus are introduced. The so called geometry, in a special case
when it is orthogonally related to a traceless symmetric form, shows some
resemblances to the symplectic geometry. As an example of an -system the
nilpotent oscillator is introduced and its supersymmetrization considered. It
is shown that the -symmetry known for the Graded Superfield Oscillator (GSO)
is present also here for the supersymmetric -system. The generalized
Poisson bracket for -variables satisfies modified Leibniz rule and
has nontrivial Jacobiator.Comment: 23 pages, no figures. Corrected version. 2 references adde
Spinor particle. An indeterminacy in the motion of relativistic dynamical systems with separately fixed mass and spin
We give an argument that a broad class of geometric models of spinning
relativistic particles with Casimir mass and spin being separately fixed
parameters, have indeterminate worldline (while other spinning particles have
definite worldline). This paradox suggests that for a consistent description of
spinning particles something more general than a worldline concept should be
used. As a particular case, we study at the Lagrangian level the Cauchy problem
for a spinor particle and then, at the constrained Hamiltonian level, we
generalize our result to other particles.Comment: 10 pages, 1 figur
SU(2) reductions in N=4 multidimensional supersymmetric mechanics
We perform an su(2) Hamiltonian reduction in the bosonic sector of the
su(2)-invariant action for two free (4, 4, 0) supermultiplets. As a result, we
get the five dimensional N=4 supersymmetric mechanics describing the motion of
an isospin carrying particle interacting with a Yang monopole. We provide the
Lagrangian and Hamiltonian descriptions of this system. Some possible
generalizations of the action to the cases of systems with a more general
bosonic action, a four-dimensional system which still includes eight fermionic
components, and a variant of five-dimensional N=4 mechanics constructed with
the help of the ordinary and twisted N=4 hypermultiplets were also considered.Comment: 11 pages, LaTeX file, no figures; 3 references added, minor
correction
Extension of the Shirafuji model for Massive Particles with Spin
We extend the Shirafuji model for massless particles with primary spacetime
coordinates and composite four-momenta to a model for massive particles with
spin and electric charge. The primary variables in the model are the spacetime
four-vector, four scalars describing spin and charge degrees of freedom as well
as a pair of Weyl spinors. The geometric description proposed in this paper
provides an intermediate step between the free purely twistorial model in
two-twistor space in which both spacetime and four-momenta vectors are
composite, and the standard particle model, where both spacetime and
four-momenta vectors are elementary. We quantize the model and find explicitly
the first-quantized wavefunctions describing relativistic particles with mass,
spin and electric charge. The spacetime coordinates in the model are not
commutative; this leads to a wavefunction that depends only on one covariant
projection of the spacetime four-vector (covariantized time coordinate)
defining plane wave solutions.Comment: Latex, 27 pages, appendix.sty, newlfont.sty (attached