3,006 research outputs found
Clifford-Finsler Algebroids and Nonholonomic Einstein-Dirac Structures
We propose a new framework for constructing geometric and physical models on
nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry
and nonlinear connection structure. Explicit parametrizations of generic
off-diagonal metrics and linear and nonlinear connections define different
types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to
spinor fields and Dirac operators on nonholonomic manifolds motivates the
theory of Clifford algebroids defined as Clifford bundles, in general, enabled
with nonintegrable distributions defining the nonlinear connection. In this
work, we elaborate the algebroid spinor differential geometry and formulate the
(scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids.
The paper communicates new developments in geometrical formulation of physical
theories and this approach is grounded on a number of previous examples when
exact solutions with generic off-diagonal metrics and generalized symmetries in
modern gravity define nonholonomic spacetime manifolds with uncompactified
extra dimensions.Comment: The manuscript was substantially modified following recommendations
of JMP referee. The former Chapter 2 and Appendix were elliminated. The
Introduction and Conclusion sections were modifie
The Use of NVivo in the Different Stages of Qualitative Research
When researchers take their first steps in qualitative research, they face the great lack of models to follow regarding the method to use in the analysis of data, on some occasions falling into the temptation of coveting the high level of systematisation employed by researchers working with quantitative data. This article offers a basic theoretical contribution that allows researcher to approach qualitative data analysis and the use of the NVivo software, highlighting its advantages and describing the main functions at each moment of a qualitative investigation, placing particular emphasis on the analysis process
On the geometric quantization of twisted Poisson manifolds
We study the geometric quantization process for twisted Poisson manifolds.
First, we introduce the notion of Lichnerowicz-twisted Poisson cohomology for
twisted Poisson manifolds and we use it in order to characterize their
prequantization bundles and to establish their prequantization condition. Next,
we introduce a polarization and we discuss the quantization problem. In each
step, several examples are presented
Symplectic and Poisson geometry on b-manifolds
Let be a Poisson manifold with Poisson bivector field . We say
that is b-Poisson if the map intersects the
zero section transversally on a codimension one submanifold . This
paper will be a systematic investigation of such Poisson manifolds. In
particular, we will study in detail the structure of in the
neighbourhood of and using symplectic techniques define topological
invariants which determine the structure up to isomorphism. We also investigate
a variant of de Rham theory for these manifolds and its connection with Poisson
cohomology.Comment: 34 pages. Some changes have been implemented mainly in Sections 2 and
6. Minor changes in exposition. References have been adde
Lie algebroid Fibrations
A degree 1 non-negative graded super manifold equipped with a degree 1 vector
field Q satisfying [Q, Q]=1, namely a so-called NQ-1 manifold is, in plain
differential geometry language, a Lie algebroid. We introduce a notion of
fibration for such super manifols, that essentially involves a complete
Ehresmann connection. As it is the case for Lie algebras, such fibrations turn
out not to be just locally trivial products. We also define homotopy groups and
prove the expected long exact sequence associated to a fibration. In
particular, Crainic and Fernandes's obstruction to the integrability of Lie
algebroids is interpreted as the image of a transgression map in this long
exact sequence.Comment: 28 pages, 1 figur
Poisson Geometry in Constrained Systems
Constrained Hamiltonian systems fall into the realm of presymplectic
geometry. We show, however, that also Poisson geometry is of use in this
context.
For the case that the constraints form a closed algebra, there are two
natural Poisson manifolds associated to the system, forming a symplectic dual
pair with respect to the original, unconstrained phase space. We provide
sufficient conditions so that the reduced phase space of the constrained system
may be identified with a symplectic leaf in one of those. In the second class
case the original constrained system may be reformulated equivalently as an
abelian first class system in an extended phase space by these methods.
Inspired by the relation of the Dirac bracket of a general second class
constrained system to the original unconstrained phase space, we address the
question of whether a regular Poisson manifold permits a leafwise symplectic
embedding into a symplectic manifold. Necessary and sufficient for this is the
vanishing of the characteristic form-class of the Poisson tensor, a certain
element of the third relative cohomology.Comment: 41 pages, more detailed abstract in paper; v2: minor corrections and
an additional referenc
(In)finite extensions of algebras from their Inonu-Wigner contractions
The way to obtain massive non-relativistic states from the Poincare algebra
is twofold. First, following Inonu and Wigner the Poincare algebra has to be
contracted to the Galilean one. Second, the Galilean algebra is to be extended
to include the central mass operator. We show that the central extension might
be properly encoded in the non-relativistic contraction. In fact, any
Inonu-Wigner contraction of one algebra to another, corresponds to an infinite
tower of abelian extensions of the latter. The proposed method is
straightforward and holds for both central and non-central extensions. Apart
from the Bargmann (non-zero mass) extension of the Galilean algebra, our list
of examples includes the Weyl algebra obtained from an extension of the
contracted SO(3) algebra, the Carrollian (ultra-relativistic) contraction of
the Poincare algebra, the exotic Newton-Hooke algebra and some others. The
paper is dedicated to the memory of Laurent Houart (1967-2011).Comment: 7 pages, revtex style; v2: Minor corrections, references added; v3:
Typos correcte
From the Toda Lattice to the Volterra lattice and back
We discuss the relationship between the multiple Hamiltonian structures of
the generalized Toda lattices and that of the generalized Volterra lattices. We
use a symmtery approach for Poisson structures that generalizes the Poisson
involution theorem.Comment: 15 pages; Final version to appear in Reports on Math. Phy
Coordinate-Free Quantization of Second-Class Constraints
The conversion of second-class constraints into first-class constraints is
used to extend the coordinate-free path integral quantization, achieved by a
flat-space Brownian motion regularization of the coherent-state path integral
measure, to systems with second-class constraints.Comment: 21 pages, plain LaTeX, no figure
Formal Deformations of Dirac Structures
In this paper we set-up a general framework for a formal deformation theory
of Dirac structures. We give a parameterization of formal deformations in terms
of two-forms obeying a cubic equation. The notion of equivalence is discussed
in detail. We show that the obstruction for the construction of deformations
order by order lies in the third Lie algebroid cohomology of the Dirac
structure. However, the classification of inequivalent first order deformations
is not given by the second Lie algebroid cohomology but turns out to be more
complicated.Comment: LaTeX 2e, 26 pages, no figures. Minor changes and improvement
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