242 research outputs found
The Moyal-Lie Theory of Phase Space Quantum Mechanics
A Lie algebraic approach to the unitary transformations in Weyl quantization
is discussed. This approach, being formally equivalent to the
-quantization, is an extension of the classical Poisson-Lie formalism
which can be used as an efficient tool in the quantum phase space
transformation theory.Comment: 15 pages, no figures, to appear in J. Phys. A (2001
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
Lattice Gauge Theory
We reformulate the Hamiltonian approach to lattice gauge theories such that,
at the classical level, the gauge group does not act canonically, but instead
as a Poisson-Lie group. At the quantum level, it then gets promoted to a
quantum group gauge symmetry. The theory depends on two parameters - the
deformation parameter and the lattice spacing . We show that the
system of Kogut and Susskind is recovered when , while
QCD is recovered in the continuum limit (for any ). We thus have the
possibility of having a two parameter regularization of QCD.Comment: 26 pages, LATEX fil
Wigner Trajectory Characteristics in Phase Space and Field Theory
Exact characteristic trajectories are specified for the time-propagating
Wigner phase-space distribution function. They are especially simple---indeed,
classical---for the quantized simple harmonic oscillator, which serves as the
underpinning of the field theoretic Wigner functional formulation introduced.
Scalar field theory is thus reformulated in terms of distributions in field
phase space. Applications to duality transformations in field theory are
discussed.Comment: 9 pages, LaTex2
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
On quantum deformation of conformal symmetry: Gauge dependence via field redefinitions
The effective action in gauge theories is known to depend on a choice of
gauge fixing conditions. This dependence is such that any change of gauge
conditions is equivalent to a field redefinition in the effective action. In
this sense, the quantum deformation of conformal symmetry in the N = 4 super
Yang-Mills theory, which was computed in 't Hooft gauge in hep-th/9808039 and
hep-th/0203236, is gauge dependent. The deformation is an intrinsic property of
the theory in that it cannot be eliminated by a local choice of gauge (although
we sketch a field redefinition induced by a nonlocal gauge which, on the
Coulomb branch of the theory, converts the one-loop quantum-corrected conformal
transformations to the classical ones). We explicitly compute the deformed
conformal symmetry in R_\xi gauge. The conformal transformation law of the
gauge field turns out to be \xi-independent. We construct the scalar field
redefinition which relates the 't Hooft and R_\xi gauge results. A unique
feature of 't Hooft gauge is that it makes it possible to consistently truncate
the one-loop conformal deformation to the terms of first order in derivatives
of the fields such that the corresponding transformations form a field
realization of the conformal algebra.Comment: 14 pages, latex, no figures; references and comments added, the final
version to appear in PL
Quantum Mechanics Another Way
Deformation quantization (sometimes called phase-space quantization) is a
formulation of quantum mechanics that is not usually taught to undergraduates.
It is formally quite similar to classical mechanics: ordinary functions on
phase space take the place of operators, but the functions are multiplied in an
exotic way, using the star product. Here we attempt a brief, pedagogical
discussion of deformation quantization, that is suitable for inclusion in an
undergraduate course.Comment: 14 pages, 3 figures, to be published in Eur. J. Phy
Mixed Weyl Symbol Calculus and Spectral Line Shape Theory
A new and computationally viable full quantum version of line shape theory is
obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the
collision--broadened line shape theory is the time dependent dipole
autocorrelation function of the radiator-perturber system. The observed
spectral intensity is the Fourier transform of this correlation function. A
modified form of the Wigner--Weyl isomorphism between quantum operators and
phase space functions (Weyl symbols) is introduced in order to describe the
quantum structure of this system. This modification uses a partial Wigner
transform in which the radiator-perturber relative motion degrees of freedom
are transformed into a phase space dependence, while operators associated with
the internal molecular degrees of freedom are kept in their original Hilbert
space form. The result of this partial Wigner transform is called a mixed Weyl
symbol. The star product, Moyal bracket and asymptotic expansions native to the
mixed Weyl symbol calculus are determined. The correlation function is
represented as the phase space integral of the product of two mixed symbols:
one corresponding to the initial configuration of the system, the other being
its time evolving dynamical value. There are, in this approach, two
semiclassical expansions -- one associated with the perturber scattering
process, the other with the mixed symbol star product. These approximations are
used in combination to obtain representations of the autocorrelation that are
sufficiently simple to allow numerical calculation. The leading O(\hbar^0)
approximation recovers the standard classical path approximation for line
shapes. The higher order O(\hbar^1) corrections arise from the noncommutative
nature of the star product.Comment: 26 pages, LaTeX 2.09, 1 eps figure, submitted to 'J. Phys. B.
Fermionic Ghosts in Moyal String Field Theory
We complete the construction of the Moyal star formulation of bosonic open
string field theory (MSFT) by providing a detailed study of the fermionic ghost
sector. In particular, as in the case of the matter sector, (1) we construct a
map from Witten's star product to the Moyal product, (2) we propose a
regularization scheme which is consistent with the matter sector and (3) as a
check of the formalism, we derive the ghost Neumann coefficients algebraically
directly from the Moyal product. The latter satisfy the Gross-Jevicki nonlinear
relations even in the presence of the regulator, and when the regulator is
removed they coincide numerically with the expression derived from conformal
field theory. After this basic construction, we derive a regularized action of
string field theory in the Siegel gauge and define the Feynman rules. We give
explicitly the analytic expression of the off-shell four point function for
tachyons, including the ghost contribution. Some of the results in this paper
have already been used in our previous publications. This paper provides the
technical details of the computations which were omitted there.Comment: 65 pages, typos correcte
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