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Correction: Naudts, J. Quantum Statistical Manifolds. <i>Entropy</i> 2018, <i>20</i>, 472.

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Related Experiment Video

Updated: May 25, 2025

Self-Assembly of Microtubule Tactoids
08:49

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Published on: June 23, 2022

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A Complex Structure for Two-Typed Tangent Spaces.

Jan Naudts1

  • 1Physics Department, Universiteit Antwerpen, Universiteitsplein 1, 2610 Antwerpen, Belgium.

Entropy (Basel, Switzerland)
|February 26, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a novel framework for analyzing Riemannian manifolds by uniquely decomposing tangent vectors. This approach enables complexification of tangent spaces, allowing application of modular operator theory and Kubo-Mori concepts.

Keywords:
Kubo–Mori theoryadmittance functioncomplex connection coefficientscomplexified tangent spacesfluctuation–dissipation theoremmodular operatorparallel transport

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Area of Science:

  • Differential Geometry
  • Mathematical Physics

Background:

  • Riemannian manifolds possess tangent spaces with unique vector decomposition properties.
  • Complexification of tangent spaces is a key technique in advanced geometric analysis.

Purpose of the Study:

  • To extend Kubo-Mori theory concepts to Riemannian manifolds with specific tangent vector decompositions.
  • To investigate the application of modular operator theory to complexified tangent spaces.

Main Methods:

  • Unique decomposition of tangent vectors into two subspaces.
  • Complexification of tangent spaces and subspaces.
  • Generalization of Kubo-Mori admittance function and inner product.
  • Complexification of parallel transport operators.

Main Results:

  • Demonstrated that complexified tangent spaces are invariant under the modular automorphism group.
  • Introduced generalized admittance functions and inner products.
  • Identified real and imaginary contributions to connection coefficients.
  • Established a fluctuation-dissipation theorem linking admittance to spectral path dependence.

Conclusions:

  • The study provides a generalized framework for analyzing geometric structures using concepts from quantum statistical mechanics.
  • The developed methods offer new insights into the relationship between geometric properties and physical theories.
  • The fluctuation-dissipation theorem highlights a deep connection between spectral properties and transport phenomena.