125 research outputs found
The structure of Gelfand-Levitan-Marchenko type equations for Delsarte transmutation operators of linear multi-dimensional differential operators and operator pencils. Part 1
An analog of Gelfand-Levitan-Marchenko integral equations for multi-
dimensional Delsarte transmutation operators is constructed by means of
studying their differential-geometric structure based on the classical Lagrange
identity for a formally conjugated pair of differential operators. An extension
of the method for the case of affine pencils of differential operators is
suggested.Comment: 12 page
Collineations of a symmetric 2-covariant tensor: Ricci collineations
The infinitesimal transformations that leave invariant a two-covariant symmetric tensor are studied. The interest of these symmetry transformations lays in the fact that this class of tensors includes the energy-momentum and Ricci tensors. We find that in most cases the class of infinitesimal generators of these transformations is a finite dimensional Lie algebra, but in some cases exhibiting a higher degree of degeneracy, this class is infinite dimensional and may fail to be a Lie algebra. As an application, we study the Ricci collineations of a type B warped spacetime
Transversely projective foliations on surfaces: existence of normal forms and prescription of the monodromy
We introduce a notion of normal form for transversely projective structures
of singular foliations on complex manifolds. Our first main result says that
this normal form exists and is unique when ambient space is two-dimensional.
From this result one obtains a natural way to produce invariants for
transversely projective foliations on surfaces. Our second main result says
that on projective surfaces one can construct singular transversely projective
foliations with prescribed monodromy
On the geometry of mixed states and the Fisher information tensor
In this paper, we will review the co-adjoint orbit formulation of finite
dimensional quantum mechanics, and in this framework, we will interpret the
notion of quantum Fisher information index (and metric). Following previous
work of part of the authors, who introduced the definition of Fisher
information tensor, we will show how its antisymmetric part is the pullback of
the natural Kostant-Kirillov-Souriau symplectic form along some natural
diffeomorphism. In order to do this, we will need to understand the symmetric
logarithmic derivative as a proper 1-form, settling the issues about its very
definition and explicit computation. Moreover, the fibration of co-adjoint
orbits, seen as spaces of mixed states, is also discussed.Comment: 27 pages; Accepted Manuscrip
Dimensional Reduction of Dirac Operator
We construct an explicit example of dimensional reduction of the free
massless Dirac operator with an internal SU(3) symmetry, defined on a
twelve-dimensional manifold that is the total space of a principal SU(3)-bundle
over a four-dimensional (nonflat) pseudo-Riemannian manifold. Upon dimensional
reduction the free twelve-dimensional Dirac equation is transformed into a
rather nontrivial four-dimensional one: a pair of massive Lorentz spinor
SU(3)-octets interacting with an SU(3)-gauge field with a source term depending
on the curvature tensor of the gauge field. The SU(3) group is complicated
enough to illustrate features of the general case. It should not be confused
with the color SU}(3) of quantum chromodynamics where the fundamental spinors,
the quark fields, are SU(3) triplets rather than octets.Comment: 11 pages, LATEX
New insights in particle dynamics from group cohomology
The dynamics of a particle moving in background electromagnetic and
gravitational fields is revisited from a Lie group cohomological perspective.
Physical constants characterising the particle appear as central extension
parameters of a group which is obtained from a centrally extended kinematical
group (Poincare or Galilei) by making local some subgroup. The corresponding
dynamics is generated by a vector field inside the kernel of a presymplectic
form which is derived from the canonical left-invariant one-form on the
extended group. A non-relativistic limit is derived from the geodesic motion
via an Inonu-Wigner contraction. A deeper analysis of the cohomological
structure reveals the possibility of a new force associated with a non-trivial
mixing of gravity and electromagnetism leading to in principle testable
predictions.Comment: 8 pages, LaTeX, no figures. To appear in J. Phys. A (Letter to the
editor
Variational principles for involutive systems of vector fields
In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector
field can be characterized by means of a variational principle based on a
one-form. We discuss how a vector field on a manifold can also be characterized
in a similar way by means of an higher order variational principle, and how
this extends to involutive systems of vector fields.Comment: 31 pages. To appear in International Journal of Geometric Methods in
Modern Physics (IJGMMP
A variational principle for volume-preserving dynamics
We provide a variational description of any Liouville (i.e. volume
preserving) autonomous vector fields on a smooth manifold. This is obtained via
a ``maximal degree'' variational principle; critical sections for this are
integral manifolds for the Liouville vector field. We work in coordinates and
provide explicit formulae
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Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings
The aim of this paper is to study the relationship between Hamiltonian
dynamics and constrained variational calculus. We describe both using the
notion of Lagrangian submanifolds of convenient symplectic manifolds and using
the so-called Tulczyjew's triples. The results are also extended to the case of
discrete dynamics and nonholonomic mechanics. Interesting applications to
geometrical integration of Hamiltonian systems are obtained.Comment: 33 page
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