62 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
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
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
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