11 research outputs found
Linear Connections on Graphs
In recent years, discrete spaces such as graphs attract much attention as
models for physical spacetime or as models for testing the spirit of
non-commutative geometry. In this work, we construct the differential algebras
for graphs by extending the work of Dimakis et al and discuss linear
connections and curvatures on graphs. Especially, we calculate connections and
curvatures explicitly for the general nonzero torsion case. There is a metric,
but no metric-compatible connection in general except the complete symmetric
graph with two vertices.Comment: 22pages, Latex file, Some errors corrected, To appear in J. Math.
Phy
Quantum Mechanics on the h-deformed Quantum Plane
We find the covariant deformed Heisenberg algebra and the Laplace-Beltrami
operator on the extended -deformed quantum plane and solve the Schr\"odinger
equations explicitly for some physical systems on the quantum plane. In the
commutative limit the behaviour of a quantum particle on the quantum plane
becomes that of the quantum particle on the Poincar\'e half-plane, a surface of
constant negative Gaussian curvature. We show the bound state energy spectra
for particles under specific potentials depend explicitly on the deformation
parameter . Moreover, it is shown that bound states can survive on the
quantum plane in a limiting case where bound states on the Poincar\'e
half-plane disappear.Comment: 16pages, Latex2e, Abstract and section 4 have been revise