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 µ and deviation σ , form a 2-dimensional exponential family.In this paper, we show that the tangent bundle of an exponential family is naturally a Kähler manifold. This simple but crucial observation leads to the formalism of quantum mechanics in its geometrical form, i.e. based on the Kähler structure of the complex projective space, but generalizes also to more general Kähler manifolds, providing a natural geometric framework for the description of quantum systems.Many questions related to this "statistical Kähler 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.
Let (M, g) be a compact, connected and oriented Riemannian manifold with volume form dvolg . We denote by D the space of smooth probability density functions on M , i.e. D := {ρ ∈ C ∞ (M, R) | ρ > 0 and M ρ • dvolg = 1} . We regard D as an infinite dimensional manifold.In this paper, we consider the almost Hermitian structure on T D associated, via Dombrowski's construction, to the Wasserstein metric g D and a natural connection ∇ D on D. Using geometric mechanical methods, we show that the corresponding fundamental 2-form on T D leads to the Schrödinger equation for a quantum particle living in M . Geometrically, we exhibit a map which pulls back the Fubini-Study symplectic form to the 2-form on T D. The integrability of the almost complex structure on T D is also discussed.These results echo other papers of the author where it is stressed that the Fisher metric and exponential connection are related (via Dombrowski's construction) to Kähler geometry and the quantum formalism in finite dimension.
Let N be the space of Gaussian distribution functions over R, regarded as a 2-dimensional statistical manifold parameterized by the mean µ and the deviation σ. In this paper we show that the tangent bundle of N , endowed with its natural Kähler structure, is the Siegel-Jacobi space appearing in the context of Number Theory and Jacobi forms. Geometrical aspects of the Siegel-Jacobi space are discussed in detail (completeness, curvature, group of holomorphic isometries, space of Kähler functions, relationship to the Jacobi group), and are related to the quantum formalism in its geometrical form, i.e., based on the Kähler structure of the complex projective space.This paper is a continuation of our previous work [Mol12b, Mol12a, Mol13], where we studied the quantum formalism from a geometric and information-theoretical point of view.
A quantum system can be entirely described by the Kähler structure of the projective space P(H) associated to the Hilbert space H of possible states; this is the socalled geometrical formulation of quantum mechanics. In this paper, we give an explicit link between the geometrical formulation (of finite dimensional quantum systems) and statistics through the natural geometry of the space P × n of non-vanishing probabilities p : En → R * + defined on a finite set En := {x 1 , . . . , xn}. More precisely, we use the Fisher metric g F and the exponential connection ∇ (1) (both being natural statistical objects living on P × n ) to construct, via the Dombrowski splitting theorem, a Kähler structure on T P × n which has the property that it induces the natural Kähler structure of a suitably chosen open dense subset of P(C n ). As a direct physical consequence, a significant part of the quantum mechanical formalism (in finite dimension) is encoded in the triple (P × n , g F , ∇ (1) ).
Let (N, g) be a Riemannian manifold. For a compact, connected and oriented submanifold M of N , we define the space of volume preserving embeddings Embµ(M, N ) as the set of smooth embeddings f : M ֒→ N such that f * µ f = µ , where µ f (resp. µ) is the Riemannian volume form on f (M ) (resp. M ) induced by the ambient metric g (the orientation on f (M ) being induced by f ). In this article, we use the Nash-Moser inverse function Theorem to show that the set of volume preserving embeddings in Embµ(M, N ) whose mean curvature is nowhere vanishing forms a tame Fréchet manifold, and determine explicitly the Euler-Lagrange equations of a natural class of Lagrangians.As an application, we generalize the Euler equations of an incompressible fluid to the case of an "incompressible membrane" of arbitrary dimension moving in N .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.