7,149 research outputs found
Moduli of quantum Riemannian geometries on <= 4 points
We classify parallelizable noncommutative manifold structures on finite sets
of small size in the general formalism of framed quantum manifolds and
vielbeins introduced previously. The full moduli space is found for
points, and a restricted moduli space for 4 points. The topological part of the
moduli space is found for points based on the known atlas of regular
graphs. We also discuss aspects of the quantum theory defined by functional
integration.Comment: 34 pages ams-latex, 4 figure
The noncommutative Kubo Formula: Applications to Transport in Disordered Topological Insulators with and without Magnetic Fields
The non-commutative theory of charge transport in mesoscopic aperiodic
systems under magnetic fields, developed by Bellissard, Shulz-Baldes and
collaborators in the 90's, is complemented with a practical numerical
implementation. The scheme, which is developed within a -algebraic
framework, enable efficient evaluations of the non-commutative Kubo formula,
with errors that vanish exponentially fast in the thermodynamic limit.
Applications to a model of a 2-dimensional Quantum spin-Hall insulator are
given. The conductivity tensor is mapped as function of Fermi level, disorder
strength and temperature and the phase diagram in the plane of Fermi level and
disorder strength is quantitatively derived from the transport simulations.
Simulations at finite magnetic field strength are also presented.Comment: 10 figure
Infrared spectroscopy of diatomic molecules - a fractional calculus approach
The eigenvalue spectrum of the fractional quantum harmonic oscillator is
calculated numerically solving the fractional Schr\"odinger equation based on
the Riemann and Caputo definition of a fractional derivative. The fractional
approach allows a smooth transition between vibrational and rotational type
spectra, which is shown to be an appropriate tool to analyze IR spectra of
diatomic molecules.Comment: revised + extended version, 9 pages, 6 figure
Discrete approaches to quantum gravity in four dimensions
The construction of a consistent theory of quantum gravity is a problem in
theoretical physics that has so far defied all attempts at resolution. One
ansatz to try to obtain a non-trivial quantum theory proceeds via a
discretization of space-time and the Einstein action. I review here three major
areas of research: gauge-theoretic approaches, both in a path-integral and a
Hamiltonian formulation, quantum Regge calculus, and the method of dynamical
triangulations, confining attention to work that is strictly four-dimensional,
strictly discrete, and strictly quantum in nature.Comment: 33 pages, invited contribution to Living Reviews in Relativity; the
author welcomes any comments and suggestion
On the area expectation values in area tensor Regge calculus in the Lorentzian domain
Wick rotation in area tensor Regge calculus is considered. The heuristical
expectation is confirmed that the Lorentzian quantum measure on a spacelike
area should coincide with the Euclidean measure at the same argument. The
consequence is validity of probabilistic interpretation of the Lorentzian
measure as well (on the real, i.e. spacelike areas).Comment: LaTeX, 7 pages, introduction and discussion given in more detail,
references adde
Slowness and Sparseness Lead to Place, Head-Direction, and Spatial-View Cells
We present a model for the self-organized formation of place cells, head-direction cells, and spatial-view cells in the hippocampal formation based on unsupervised learning on quasi-natural visual stimuli. The model comprises a hierarchy of Slow Feature Analysis (SFA) nodes, which were recently shown to reproduce many properties of complex cells in the early visual system. The system extracts a distributed grid-like representation of position and orientation, which is transcoded into a localized place-field, head-direction, or view representation, by sparse coding. The type of cells that develops depends solely on the relevant input statistics, i.e., the movement pattern of the simulated animal. The numerical simulations are complemented by a mathematical analysis that allows us to accurately predict the output of the top SFA laye
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