391 research outputs found
On Foundation of the Generalized Nambu Mechanics
We outline the basic principles of canonical formalism for the Nambu
mechanics---a generalization of Hamiltonian mechanics proposed by Yoichiro
Nambu in 1973. It is based on the notion of Nambu bracket which generalizes the
Poisson bracket to the multiple operation of higher order on
classical observables and is described by Hambu-Hamilton equations of motion
given by Hamiltonians. We introduce the fundamental identity for the
Nambu bracket which replaces Jacobi identity as a consistency condition for the
dynamics. We show that Nambu structure of given order defines a family of
subordinated structures of lower order, including the Poisson structure,
satisfying certain matching conditions. We introduce analogs of action from and
principle of the least action for the Nambu mechanics and show how dynamics of
loops (-dimensional objects) naturally appears in this formalism. We
discuss several approaches to the quantization problem and present explicit
representation of Nambu-Heisenberg commutation relation for case. We
emphasize the role higher order algebraic operations and mathematical
structures related with them play in passing from Hamilton's to Nambu's
dynamical picture.Comment: 27 page
Formal symplectic groupoid
The multiplicative structure of the trivial symplectic groupoid over associated to the zero Poisson structure can be expressed in terms of a
generating function. We address the problem of deforming such a generating
function in the direction of a non-trivial Poisson structure so that the
multiplication remains associative. We prove that such a deformation is unique
under some reasonable conditions and we give the explicit formula for it. This
formula turns out to be the semi-classical approximation of Kontsevich's
deformation formula. For the case of a linear Poisson structure, the deformed
generating function reduces exactly to the CBH formula of the associated Lie
algebra. The methods used to prove existence are interesting in their own right
as they come from an at first sight unrelated domain of mathematics: the
Runge--Kutta theory of the numeric integration of ODE's.Comment: 28 pages, 4 figure
Perturbative Construction of Models of Algebraic Quantum Field Theory
We review the construction of models of algebraic quantum field theory by
renormalized perturbation theory.Comment: 38 page
On infinite walls in deformation quantization
We examine the deformation quantization of a single particle moving in one
dimension (i) in the presence of an infinite potential wall, (ii) confined by
an infinite square well, and (iii) bound by a delta function potential energy.
In deformation quantization, considered as an autonomous formulation of quantum
mechanics, the Wigner function of stationary states must be found by solving
the so-called \*-genvalue (``stargenvalue'') equation for the Hamiltonian.
For the cases considered here, this pseudo-differential equation is difficult
to solve directly, without an ad hoc modification of the potential. Here we
treat the infinite wall as the limit of a solvable exponential potential.
Before the limit is taken, the corresponding \*-genvalue equation involves
the Wigner function at momenta translated by imaginary amounts. We show that it
can be converted to a partial differential equation, however, with a
well-defined limit. We demonstrate that the Wigner functions calculated from
the standard Schr\"odinger wave functions satisfy the resulting new equation.
Finally, we show how our results may be adapted to allow for the presence of
another, non-singular part in the potential.Comment: 22 pages, to appear in Annals of Physic
Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!
Nambu's construction of multi-linear brackets for super-integrable systems
can be thought of as degenerate Poisson brackets with a maximal set of Casimirs
in their kernel. By introducing privileged coordinates in phase space these
degenerate Poisson brackets are brought to the form of Heisenberg's equations.
We propose a definition for constructing quantum operators for classical
functions which enables us to turn the maximally degenerate Poisson brackets
into operators. They pose a set of eigenvalue problems for a new state vector.
The requirement of the single valuedness of this eigenfunction leads to
quantization. The example of the harmonic oscillator is used to illustrate this
general procedure for quantizing a class of maximally super-integrable systems
Flatness-based control of open-channel flow in an irrigation canal using SCADA
Open channels are used to distribute water to large irrigated areas. In these systems, ensuring timely water delivery is essential to reduce operational water losses. This article derives a method for open-loop control of open channel flow, based on the Hayami model, a parabolic partial differential equation resulting from a simplification of the Saint-Venant equations. The open-loop control is represented as infinite series using differential flatness. Experimental results show the effectiveness of the approach by applying the open-loop controller to a real irrigation canal located in South of France
A Path Integral Approach To Noncommutative Superspace
A path integral formula for the associative star-product of two superfields
is proposed. It is a generalization of the Kontsevich-Cattaneo-Felder's formula
for the star-product of functions of bosonic coordinates. The associativity of
the star-product imposes certain conditions on the background of our sigma
model. For generic background the action is not supersymmetric. The
supersymmetry invariance of the action constrains the background and leads to a
simple formula for the star-product.Comment: Latex 13 pages. v2: references and footnotes adde
Toeplitz operators on symplectic manifolds
We study the Berezin-Toeplitz quantization on symplectic manifolds making use
of the full off-diagonal asymptotic expansion of the Bergman kernel. We give
also a characterization of Toeplitz operators in terms of their asymptotic
expansion. The semi-classical limit properties of the Berezin-Toeplitz
quantization for non-compact manifolds and orbifolds are also established.Comment: 40 page
Infinitesimal deformations of a formal symplectic groupoid
Given a formal symplectic groupoid over a Poisson manifold ,
we define a new object, an infinitesimal deformation of , which can be
thought of as a formal symplectic groupoid over the manifold equipped with
an infinitesimal deformation of the Poisson bivector
field . The source and target mappings of a deformation of are
deformations of the source and target mappings of . To any pair of natural
star products having the same formal symplectic groupoid
we relate an infinitesimal deformation of . We call it the deformation
groupoid of the pair . We give explicit formulas for the
source and target mappings of the deformation groupoid of a pair of star
products with separation of variables on a Kaehler- Poisson manifold. Finally,
we give an algorithm for calculating the principal symbols of the components of
the logarithm of a formal Berezin transform of a star product with separation
of variables. This algorithm is based upon some deformation groupoid.Comment: 22 pages, the paper is reworked, new proofs are adde
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