From Geometry to Quantum Mechanics (eBook)

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

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.