Maeda / Michor / Ochiai From Geometry to Quantum Mechanics

In Honor of Hideki Omori
1. Auflage 2007
ISBN: 978-0-8176-4530-4
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 252, 324 Seiten

Reihe: Progress in Mathematics

* Invited articles in differential geometry and mathematical physics in honor of Hideki Omori * Focus on recent trends and future directions in symplectic and Poisson geometry, global analysis, Lie group theory, quantizations and noncommutative geometry, as well as applications of PDEs and variational methods to geometry * Will appeal to graduate students in mathematics and quantum mechanics; also a reference

Autoren/Hrsg.

Maeda, Yoshiaki

Michor, Peter

Ochiai, Takushiro

Yoshioka, Akira

Inhaltsverzeichnis

1;Contents;7
2;Preface;9
3;Curriculum Vitae;13
4;Part I Global Analysis and In.nite-Dimensional Lie Groups;19
4.1;Aspects of Stochastic Global Analysis;20
4.1.1;1 Introduction;20
4.1.2;2 Convolution semi-groups and Brownian motions;21
4.1.3;3 Reproducing kernel Hilbert spaces, connections, and stochastic .ows;29
4.1.4;4 Heat semi-groups on differential forms;32
4.1.5;5 Analysis on path spaces;34
4.2;A Lie Group Structure for Automorphisms of a Contact Weyl Manifold;42
4.2.1;1 Introduction;42
4.2.2;2 Infinite-dimensional Lie groups;44
4.2.3;3 Deformation quantization;46
4.2.4;4 ContactWeyl manifold over a symplectic manifold;47
4.2.5;5 A Lie group structure of Aut((M, );49
4.2.6;6 Concluding remarks;57
5;Part II Riemannian Geometry;63
5.1;Projective Structures of a Curve in a Conformal Space;64
5.1.1;1 Introduction;64
5.1.2;2 Projective structures of a curve;65
5.2;Deformations of Surfaces Preserving Conformal or Similarity Invariants;70
5.2.1;1 Deformation of surfaces preserving conformal invariants;71
5.2.2;2 Deformation of surfaces preserving similarity invariants;78
5.3;Global Structures of Compact Conformally Flat Semi-Symmetric Spaces of Dimension 3 and of Non-Constant Curvature;86
5.3.1;1 Introduction;86
5.3.2;2 Preliminaries;88
5.3.3;3 Geometric structures;91
5.3.4;4 Limit sets;93
5.3.5;5 Discrete holonomy groups;94
5.3.6;6 Indiscrete holonomy groups;95
5.4;Differential Geometry of Analytic Surfaces with Singularities;102
5.4.1;1 Curves at singularities;102
5.4.2;2 Surfaces around singularities;104
6;Part III Symplectic Geometry and Poisson Geometry;108
6.1;The Integration Problem for Complex Lie Algebroids;110
6.1.1;1 Introduction;110
6.1.2;2 Complexifications of real Lie algebroids;112
6.1.3;3 Involutive structures;115
6.1.4;4 Boundary Lie algebroids;119
6.1.5;5 Generalized complex structures;121
6.1.6;6 Further topics and questions;122
6.2;Reduction, Induction and Ricci Flat Symplectic Connections;128
6.2.1;1 Induction and contact quadruples;130
6.2.2;2 Lift of hamiltonian vector fields and of conformal vector fields;134
6.2.3;3 Conformally homogeneous symplectic manifolds;137
6.2.4;4 Induced connections;141
6.2.5;5 A reduction construction;143
6.3;Local Lie Algebra Determines Base Manifold;148
6.3.1;1 Introduction;148
6.3.2;2 Jacobi modules;149
6.3.3;3 Useful facts about associative algebras;153
6.3.4;4 Spectra of Jacobi modules;155
6.3.5;5 Isomorphisms;158
6.4;Lie Algebroids Associated with Deformed Schouten Bracket of 2-Vector Fields;164
6.4.1;1 Introduction;164
6.4.2;2 Lie algebroids and Jacobi–Lie algebroids;165
6.4.3;3 Deformed bracket on 1-forms;169
6.5;Parabolic Geometries Associated with Differential Equations of Finite Type;178
6.5.1;1 Introduction;178
6.5.2;2 Pseudo-product GLA;181
6.5.3;3 Symbol of the classical cases;189
6.5.4;4 Symbol of the exceptional cases;203
6.5.5;5 Equivalence of Parabolic Geometries;221
7;Part IV Quantizations and Noncommutative Geometry;228
7.1;Toward Geometric Quantum Theory;230
7.1.1;1 µ-regulated algebras;231
7.1.2;2 Deformation quantizations and localizations;235
7.1.3;3 Limit, extremal localizations, infinitesimal intertwiners;245
7.1.4;4 Deformation by one variable;250
7.1.5;5 The case where D is the ordinary differential;253
7.1.6;6 Special localizations;260
7.2;Resonance Gyrons and Quantum Geometry;270
7.2.1;1 Introduction;270
7.2.2;2 Commutation relations and Poisson brackets for l : m resonance;273
7.2.3;3 Irreducible representations of l : m resonance algebra;276
7.2.4;4 Quantum geometry of the l : m resonance;279
7.2.5;5 Coherent states and gyron spectrum;285
7.3;A Secondary Invariant of Foliated Spaces and Type III.von Neumann Algebras;294
7.3.1;1 Foliations that yield type III. factors;295
7.3.2;2 Lifted Anosov foliations and foliated T 2-bundles;296
7.3.3;3 A secondary invariant associated to (Mµ,Fµ);298
7.4;The Geometry of Space-Time and Its Deformations from a Physical Perspective;304
7.4.1;1 Epistemological introduction;304
7.4.2;2 From Atlas to Galileo and Newton to Einstein and Planck;306
7.4.3;3 Possible quantized anti de Sitter structures in the microworld;310
7.5;Geometric Objects in an Approach to Quantum Geometry;320
7.5.1;1 Introduction;320
7.5.2;2 Deformation of a commutative product;322
7.5.3;3 Bundle gerbes as a non-cohomological notion;327
7.5.4;4 Broken associative products and extensions;334
7.5.5;5 The notion of q-number functions;336

Leseproben

Aspects of Stochastic Global Analysis (P. 3)

K. D. Elworthy

Mathematics Institute, Warwick University, Coventry CV4 7AL, England
Dedicated to Hideki Omori
Summary.
This is a survey of some topics where global and stochastic analysis play a role. An introduction to analysis on Banach spaces with Gaussian measure leads to an analysis of the geometry of stochastic differential equations, stochastic flows, and their associated connections, with reference to some related topological vanishing theorems. Following that, there is a description of the construction of Sobolev calculi over path and loop spaces with diffusion measures, and also of a possible L2 de Rham and Hodge-Kodaira theory on path spaces. Diffeomorphism groups and diffusion measures on their path spaces are central to much of the discussion. Knowledge of stochastic analysis is not assumed.

Keywords:
Path space, diffeomorphism group, Hodge-Kodaira theory, in.nite dimensions, universal connection, stochastic differential equations, Malliavin calculus, Gaussian measures, differential forms, Weitzenbock formula, sub-Riemannian.

1 Introduction

Stochastic and global analysis come together in several distinct ways. One is from the fact that the basic objects of finite dimensional stochastic analysis naturally live on manifolds and often induce Riemannian or sub-Riemannian structures on those manifolds, so they have their own intrinsic geometry.

Another is that stochastic analysis is expected to be a major tool in infinite dimensional analysis because of the singularity of the operators which arise there, a fairly prevalent assumption has been that in this situation stochastic methods are more likely to be successful than direct attempts to extend PDE techniques to in.nite dimensional situations.

(Ironically that situation has been reversed in recent work on the stochastic 3D Navier–Stokes equation, [DPD03].) Stimulated particularly by the approach of Bismut to index theorems, [Bis84], and by other ideas from topology, representation theory, and theoretical physics, this has been extended to attempts to use stochastic analysis in the construction of ininite dimensional geometric structures, for example on loop spaces of Riemannian manifolds.

As examples see [AMT04], and [L´ea05]. In any case global analysis was firmly embedded in stochastic analysis with the advent of Malliavin calculus, a theory of Sobolev spaces and calculus on the space of continuous paths on Rn, as described briefly below, and especially its relationships with diffusion operators and processes on finite dimensional manifolds.

In this introductory selection of topics, both of these aspects of the intersection are touched on. After a brief introduction to analysis on spaces with Gaussian measure there is a discussion of the geometry of stochastic differential equations, stochastic flows, and their associated connections, with reference to some related topological vanishing theorems. Following that, there is a discussion of the construction of Sobolev calculi over path and loop groups with diffusion measures, and also of de Rham and Hodge-Kodaira theory on path spaces.