104 research outputs found
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
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
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
Phase-Space Metric for Non-Hamiltonian Systems
We consider an invariant skew-symmetric phase-space metric for
non-Hamiltonian systems. We say that the metric is an invariant if the metric
tensor field is an integral of motion. We derive the time-dependent
skew-symmetric phase-space metric that satisfies the Jacobi identity. The
example of non-Hamiltonian systems with linear friction term is considered.Comment: 12 page
A Tonnetz Model for pentachords
This article deals with the construction of surfaces that are suitable for
representing pentachords or 5-pitch segments that are in the same class.
It is a generalization of the well known \"Ottingen-Riemann torus for triads of
neo-Riemannian theories. Two pentachords are near if they differ by a
particular set of contextual inversions and the whole contextual group of
inversions produces a Tiling (Tessellation) by pentagons on the surfaces. A
description of the surfaces as coverings of a particular Tiling is given in the
twelve-tone enharmonic scale case.Comment: 27 pages, 12 figure
Singular lagrangians: some geometric structures along the Legendre map
New geometric structures that relate the lagrangian and hamiltonian
formalisms defined upon a singular lagrangian are presented. Several vector
fields are constructed in velocity space that give new and precise answers to
several topics like the projectability of a vector field to a hamiltonian
vector field, the computation of the kernel of the presymplectic form of
lagrangian formalism, the construction of the lagrangian dynamical vector
fields, and the characterisation of dynamical symmetries.Comment: 27 pages; minor changes, a reference update
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
The "Symplectic Camel Principle" and Semiclassical Mechanics
Gromov's nonsqueezing theorem, aka the property of the symplectic camel,
leads to a very simple semiclassical quantiuzation scheme by imposing that the
only "physically admissible" semiclassical phase space states are those whose
symplectic capacity (in a sense to be precised) is nh + (1/2)h where h is
Planck's constant. We the construct semiclassical waveforms on Lagrangian
submanifolds using the properties of the Leray-Maslov index, which allows us to
define the argument of the square root of a de Rham form.Comment: no figures. to appear in J. Phys. Math A. (2002
Lie algebroid foliations and -Dirac structures
We prove some general results about the relation between the 1-cocycles of an
arbitrary Lie algebroid over and the leaves of the Lie algebroid
foliation on associated with . Using these results, we show that a
-Dirac structure induces on every leaf of its
characteristic foliation a -Dirac structure , which comes
from a precontact structure or from a locally conformal presymplectic structure
on . In addition, we prove that a Dirac structure on can be obtained from and we discuss the relation between the leaves of
the characteristic foliations of and .Comment: 25 page
The Electromagnetic Lorentz Condition Problem and Symplectic Properties of Maxwell and Yang-Mills Type Dynamical Systems
Symplectic structures associated to connection forms on certain types of
principal fiber bundles are constructed via analysis of reduced geometric
structures on fibered manifolds invariant under naturally related symmetry
groups. This approach is then applied to nonstandard Hamiltonian analysis of of
dynamical systems of Maxwell and Yang-Mills type. A symplectic reduction theory
of the classical Maxwell equations is formulated so as to naturally include the
Lorentz condition (ensuring the existence of electromagnetic waves), thereby
solving the well known Dirac -Fock - Podolsky problem. Symplectically reduced
Poissonian structures and the related classical minimal interaction principle
for the Yang-Mills equations are also considered. 1
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