287 research outputs found
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
Prevalence and associated factors with complex regional pain syndrome type 1 of hemiplegic upper limb
Volume preserving multidimensional integrable systems and Nambu--Poisson geometry
In this paper we study generalized classes of volume preserving
multidimensional integrable systems via Nambu--Poisson mechanics. These
integrable systems belong to the same class of dispersionless KP type equation.
Hence they bear a close resemblance to the self dual Einstein equation. All
these dispersionless KP and dToda type equations can be studied via twistor
geometry, by using the method of Gindikin's pencil of two forms. Following this
approach we study the twistor construction of our volume preserving systems
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.
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
Cohomology Groups of Deformations of Line Bundles on Complex Tori
The cohomology groups of line bundles over complex tori (or abelian
varieties) are classically studied invariants of these spaces. In this article,
we compute the cohomology groups of line bundles over various holomorphic,
non-commutative deformations of complex tori. Our analysis interpolates between
two extreme cases. The first case is a calculation of the space of
(cohomological) theta functions for line bundles over constant, commutative
deformations. The second case is a calculation of the cohomologies of
non-commutative deformations of degree-zero line bundles.Comment: 24 pages, exposition improved, typos fixe
Cohomologies of the Poisson superalgebra
Cohomology spaces of the Poisson superalgebra realized on smooth
Grassmann-valued functions with compact support on ($C^{2n}) are
investigated under suitable continuity restrictions on cochains. The first and
second cohomology spaces in the trivial representation and the zeroth and first
cohomology spaces in the adjoint representation of the Poisson superalgebra are
found for the case of a constant nondegenerate Poisson superbracket for
arbitrary n>0. The third cohomology space in the trivial representation and the
second cohomology space in the adjoint representation of this superalgebra are
found for arbitrary n>1.Comment: Comments: 40 pages, the text to appear in Theor. Math. Phys.
supplemented by computation of the 3-rd trivial cohomolog
Features of Time-independent Wigner Functions
The Wigner phase-space distribution function provides the basis for Moyal's
deformation quantization alternative to the more conventional Hilbert space and
path integral quantizations. General features of time-independent Wigner
functions are explored here, including the functional ("star") eigenvalue
equations they satisfy; their projective orthogonality spectral properties;
their Darboux ("supersymmetric") isospectral potential recursions; and their
canonical transformations. These features are illustrated explicitly through
simple solvable potentials: the harmonic oscillator, the linear potential, the
Poeschl-Teller potential, and the Liouville potential.Comment: 18 pages, plain LaTex, References supplemente
A General Theory of Phase-Space Quasiprobability Distributions
We present a general theory of quasiprobability distributions on phase spaces
of quantum systems whose dynamical symmetry groups are (finite-dimensional) Lie
groups. The family of distributions on a phase space is postulated to satisfy
the Stratonovich-Weyl correspondence with a generalized traciality condition.
The corresponding family of the Stratonovich-Weyl kernels is constructed
explicitly. In the presented theory we use the concept of the generalized
coherent states, that brings physical insight into the mathematical formalism.Comment: REVTeX, 4 pages. More information on
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