2,819 research outputs found
Vector coherent state representations, induced representations, and geometric quantization: II. Vector coherent state representations
It is shown here and in the preceeding paper (quant-ph/0201129) that vector
coherent state theory, the theory of induced representations, and geometric
quantization provide alternative but equivalent quantizations of an algebraic
model. The relationships are useful because some constructions are simpler and
more natural from one perspective than another. More importantly, each approach
suggests ways of generalizing its counterparts. In this paper, we focus on the
construction of quantum models for algebraic systems with intrinsic degrees of
freedom. Semi-classical partial quantizations, for which only the intrinsic
degrees of freedom are quantized, arise naturally out of this construction. The
quantization of the SU(3) and rigid rotor models are considered as examples.Comment: 31 pages, part 2 of two papers, published versio
Quantization of multidimensional cat maps
In this work we study cat maps with many degrees of freedom. Classical cat
maps are classified using the Cayley parametrization of symplectic matrices and
the closely associated center and chord generating functions. Particular
attention is dedicated to loxodromic behavior, which is a new feature of
two-dimensional maps. The maps are then quantized using a recently developed
Weyl representation on the torus and the general condition on the Floquet
angles is derived for a particular map to be quantizable. The semiclassical
approximation is exact, regardless of the dimensionality or of the nature of
the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit
A reliable order-statistics-based approximate nearest neighbor search algorithm
We propose a new algorithm for fast approximate nearest neighbor search based
on the properties of ordered vectors. Data vectors are classified based on the
index and sign of their largest components, thereby partitioning the space in a
number of cones centered in the origin. The query is itself classified, and the
search starts from the selected cone and proceeds to neighboring ones. Overall,
the proposed algorithm corresponds to locality sensitive hashing in the space
of directions, with hashing based on the order of components. Thanks to the
statistical features emerging through ordering, it deals very well with the
challenging case of unstructured data, and is a valuable building block for
more complex techniques dealing with structured data. Experiments on both
simulated and real-world data prove the proposed algorithm to provide a
state-of-the-art performance
Non-hermitean hamiltonians with unitary and antiunitary symmetry
We analyze several non-Hermitian Hamiltonians with antiunitary symmetry from
the point of view of their point-group symmetry. It enables us to predict the
degeneracy of the energy levels and to reduce the dimension of the matrices
necessary for the diagonalization of the Hamiltonian in a given basis set. We
can also classify the solutions according to the irreducible representations of
the point group and thus analyze their properties separately. One of the main
results of this paper is that some PT-symmetric Hamiltonians with point-group
symmetry exhibit complex eigenvalues for all values of a potential
parameter. In such cases the PT phase transition takes place at the trivial
Hermitian limit which suggests that the phenomenon is not robust. Point-group
symmetry enables us to explain such anomalous behaviour and to choose a
suitable antiunitary operator for the PT symmetry
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