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

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

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

Geometric spectral theory for K\"ahler functions

We consider K\"ahler toric manifolds $N$ that are torifications of statistical manifolds $\mathcal{E}$ 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 $N$. 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