29 research outputs found
Generalization of Hasimoto's transformation
In this paper, we generalize the famous Hasimoto's transformation by showing
that the dynamics of a closed unidimensional vortex filament embedded in a
three-dimensional manifold of constant curvature gives rise under Hasimoto's
transformation to the non-linear Schrodinger equation.
We also give a natural interpretation of the function \psi introduced by
Hasimoto in terms of moving frames associated to a natural complex bundle over
the filament
Exponential families, Kahler geometry and quantum mechanics
Exponential families are a particular class of statistical manifolds which
are particularly important in statistical inference, and which appear very
frequently in statistics. For example, the set of normal distributions, with
mean {\mu} and deviation {\sigma}, form a 2-dimensional exponential family.
In this paper, we show that the tangent bundle of an exponential family is
naturally a Kahler manifold. This simple but crucial observation leads to the
formalism of quantum mechanics in its geometrical form, i.e. based on the
Kahler structure of the complex projective space, but generalizes also to more
general Kahler manifolds, providing a natural geometric framework for the
description of quantum systems. Many questions related to this "statistical
Kahler geometry" are discussed, and a close connection with representation
theory is observed. Examples of physical relevance are treated in details. For
example, it is shown that the spin of a particle can be entirely understood by
means of the usual binomial distribution. This paper centers on the
mathematical foundations of quantum mechanics, and on the question of its
potential generalization through its geometrical formulation
Generalization Of Hasimoto'S Transformation
In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold M of constant curvature, gives rise under Hasimoto's transformation to the nonlinear Schrodinger equation. We also give a natural interpretation of the function. introduced by Hasimoto in terms of moving frames associated to a natural complex bundle over the filament
Weyl group of the group of holomorphic isometries of a K\"ahler toric manifold
We compute the Weyl group (in the sense of Segal) of the group of holomorphic
isometries of a K\"ahler toric manifold with real analytic K\"ahler metric
Geometric spectral theory for K\"ahler functions
We consider K\"ahler toric manifolds that are torifications of
statistical manifolds in the sense of [M. Molitor, "K\"ahler
toric manifolds from dually flat spaces", arXiv:2109.04839], and prove a
geometric analogue of the spectral decomposition theorem in which Hermitian
matrices are replaced by K\"ahler functions on . The notion of "spectrum" of
a K\"ahler function is defined, and examples are presented. This paper is
motivated by the geometrization program of Quantum Mechanics that we pursued in
previous works (see, e.g., [M. Molitor, "Exponential families, K\"ahler
geometry and quantum mechanics", J. Geom. Phys. 70, 54-80 (2013)]).Comment: arXiv admin note: text overlap with arXiv:2305.0937