Remarks on the Statistical Origin of the Geometrical Formulation of Quantum Mechanics

Molitor, Mathieu

doi:10.1142/s0219887812200010

Cited by 10 publications

(13 citation statements)

References 3 publications

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“…It is on the basis of the above facts (together with others that are collected in [Mol12a, Mol12b,Mol13]), that we arrived at the conclusion that the quantum formalism might have an information-theoretical origin. Now there are two possibilities:…”

Section: Motivation: the Quantum Formalismsupporting

confidence: 56%

Gaussian distributions, Jacobi group, and Siegel-Jacobi space

Molitor

2014

Journal of Mathematical Physics

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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.

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Section: Motivation: the Quantum Formalismsupporting

confidence: 56%

Gaussian distributions, Jacobi group, and Siegel-Jacobi space

Molitor

2014

Journal of Mathematical Physics

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“…For instance, in [53], A. Caticha models the e-phase space as a cotangent bundle, which comes naturally equipped with a symplectic structure and thus all the usual accoutrements of the canonical framework, including Hamiltonian generators, Poisson brackets, etc. However, while such an approach provides an interesting way forward in ED, there are still some lingering issues needing to be resolved; not least of which is whether the cotangent bundle is a geometric structure that is rich enough to model all physically relevant quantum states (see e.g., [107]).…”

Section: Entropic Dynamicsmentioning

confidence: 99%

Entropic dynamics: reconstructing quantum field theory in curved space-time

Ipek

Abedi

Caticha

2019

Class. Quantum Grav.

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The Entropic Dynamics reconstruction of quantum mechanics is extended to the quantum theory of scalar fields in curved space-time. The Entropic Dynamics framework, which derives quantum theory as an application of the method of maximum entropy, is combined with the covariant methods of Dirac, Hojman, Kuchař, and Teitelboim, which they used to develop a framework for classical covariant Hamiltonian theories. The goal is to formulate an information-based alternative to current approaches based on algebraic quantum field theory. One key ingredient is the adoption of a local notion of entropic time in which instants are defined on curved three-dimensional surfaces and time evolution consists of the accumulation of changes induced by local deformations of these surfaces. The resulting dynamics is a non-dissipative diffusion that is constrained by the requirements of foliation invariance and incorporates the necessary local quantum potentials. As applications of the formalism we derive the Ehrenfest relations for fields in curved-spacetime and briefly discuss the nature of divergences in quantum field theory.

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“…In [19], Molitor notes that for the statistical manifold of all positive probability density functions P × n on the set of n points (with counting measure), we can define a map τ : T P × n → P(C n ) that preserves the almost Hermitian structure of T P × n , where we consider P(C n ) as a Kähler manifold equipped with the Fubini-Study metric. Here we consider a simple generalization of the map for the tangent bundle T M, and show that the almost Hermitian structure is preserved under this map.…”

Section: The Tangent Bundle Of a Statistical Manifoldmentioning

confidence: 99%

“…Most of the computation in this section is a simple generalization of Molitor's computation for probability measures on finite point sets in e.g. [19,20] with some modifications.…”

Section: The Tangent Bundle Of a Statistical Manifoldmentioning

confidence: 99%

The Complex Geometry and Representation Theory of Statistical Transformation Models

2021

Preprint

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Given a measure space X , we can construct a number of induced structures: eg. its L 2 space, the space P(X ) of probability distributions on X . If, in addition, X admits a transitive measure-preserving Lie group action, natural actions are induced on those structures. We expect relationships between these induced structures and actions. We study, in particular, the relations between L 2 (X ) and exponential transformation models on X , which are special "submanifolds" of P(X ) closed under the induced action, whose tangent bundles are Kähler manifolds (given by Molitor). Geometrically, we show the tangent bundle has, locally, the "same" Kähler metric with the Fubini-Study metric on the projectivization of L 2 (X ). Moreover we show the action on the tangent bundle is equivariant with that on L 2 (X ), which is a unitary representation. Finally, in some cases, when the symplectic action on the tangent bundle is Hamiltonian, we show that any coadjoint orbit in the image of its moment map induces, via Kirillov's correspondence from orbit method, irreducible unitary representations that are sub-representations of the aforementioned representation in L 2 (X ).

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Remarks on the Statistical Origin of the Geometrical Formulation of Quantum Mechanics

Cited by 10 publications

References 3 publications

Gaussian distributions, Jacobi group, and Siegel-Jacobi space

Gaussian distributions, Jacobi group, and Siegel-Jacobi space

Entropic dynamics: reconstructing quantum field theory in curved space-time

The Complex Geometry and Representation Theory of Statistical Transformation Models

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