114 research outputs found
"Falling cat" connections and the momentum map
We consider a standard symplectic dynamics on TM generated by a natural
Lagrangian L. The Lagrangian is assumed to be invariant with respect to the
action TR_g of a Lie group G lifted from the free and proper action R_g of G on
M. It is shown that under these conditions a connection on principal bundle pi:
M \rightarrow M/G can be constructed based on the momentum map corresponding to
the action TR_g. The horizontal motion is shown to be in physical terms the one
with all the momenta corresponding to the symmetry vanishing. A simple explicit
formula for the connection form is given. For the special case of the standard
action of G = SO(3) on M = R^3 x ... x R^3 corresponding to a rigid rotation of
a N-particle system the formula obtained earlier by Guichardet and
Shapere/Wilczek is reproduced.Comment: 10 pages, no figures, AmsTe
Representations of the conformal Lie algebra in the space of tensor densities on the sphere
Let be the space of tensor densities on
of degree . We consider this space as an induced module
of the nonunitary spherical series of the group and
classify -sim{\mathcal F}_\lambda(\mathbb{S}^n)\lambda$.Comment: Published by JNMP at http://www.sm.luth.se/math/JNMP
Gauge-potential approach to the kinematics of a moving car
A kinematics of the motion of a car is reformulated in terms of the theory of
gauge potentials (connection on principal bundle). E(2)-connection originates
in the no-slipping contact of the car with a road.Comment: 13 pages, AmsTe
Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation
The isoholonomic problem in a homogeneous bundle is formulated and solved
exactly. The problem takes a form of a boundary value problem of a variational
equation. The solution is applied to the optimal control problem in holonomic
quantum computer. We provide a prescription to construct an optimal controller
for an arbitrary unitary gate and apply it to a -dimensional unitary gate
which operates on an -dimensional Hilbert space with . Our
construction is applied to several important unitary gates such as the Hadamard
gate, the CNOT gate, and the two-qubit discrete Fourier transformation gate.
Controllers for these gates are explicitly constructed.Comment: 19 pages, no figures, LaTeX2
Local Differential Geometry on the Tempered Dual of a Semisimple Lie Group
AbstractDelorme proved that the Fell topology on the tempered dual of a real semi simple group G is rather simple: roughly speaking, it is identical with the "parameter topology." The aim of this paper is to prove that the "differential geometry" of the tempered dual is very simple, too; by differential geometry, we mean three types of objects: the categories of finite length (g, K)-modules with tempered subquotients, the Extn-groups between such modules, and the deformations of such modules
Hitting time for the continuous quantum walk
We define the hitting (or absorbing) time for the case of continuous quantum
walks by measuring the walk at random times, according to a Poisson process
with measurement rate . From this definition we derive an explicit
formula for the hitting time, and explore its dependence on the measurement
rate. As the measurement rate goes to either 0 or infinity the hitting time
diverges; the first divergence reflects the weakness of the measurement, while
the second limit results from the Quantum Zeno effect. Continuous-time quantum
walks, like discrete-time quantum walks but unlike classical random walks, can
have infinite hitting times. We present several conditions for existence of
infinite hitting times, and discuss the connection between infinite hitting
times and graph symmetry.Comment: 12 pages, 1figur
Isometric group actions on Banach spaces and representations vanishing at infinity
Our main result is that the simple Lie group acts properly
isometrically on if . To prove this, we introduce property
({\BP}_0^V), for be a Banach space: a locally compact group has
property ({\BP}_0^V) if every affine isometric action of on , such
that the linear part is a -representation of , either has a fixed point
or is metrically proper. We prove that solvable groups, connected Lie groups,
and linear algebraic groups over a local field of characteristic zero, have
property ({\BP}_0^V). As a consequence for unitary representations, we
characterize those groups in the latter classes for which the first cohomology
with respect to the left regular representation on is non-zero; and we
characterize uniform lattices in those groups for which the first -Betti
number is non-zero.Comment: 28 page
Ultracoherence and Canonical Transformations
The (in)finite dimensional symplectic group of homogeneous canonical
transformations is represented on the bosonic Fock space by the action of the
group on the ultracoherent vectors, which are generalizations of the coherent
states.Comment: 24 page
Steady state fluctuations of the dissipated heat for a quantum stochastic model
We introduce a quantum stochastic dynamics for heat conduction. A multi-level
subsystem is coupled to reservoirs at different temperatures. Energy quanta are
detected in the reservoirs allowing the study of steady state fluctuations of
the entropy dissipation. Our main result states a symmetry in its large
deviation rate function.Comment: 41 pages, minor changes, published versio
A two-cocycle on the group of symplectic diffeomorphisms
We investigate a two-cocycle on the group of symplectic diffeomorphisms of an
exact symplectic manifolds defined by Ismagilov, Losik, and Michor and
investigate its properties. We provide both vanishing and non-vanishing results
and applications to foliated symplectic bundles and to Hamiltonian actions of
finitely generated groups.Comment: 16 pages, no figure
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