Geometric Structures in Nonlinear Physics (Hermann, Robert//Interdisciplinary Mathematics): Robert Hermann

Description

As a result of a Kuhnian revolution, geometry has transformed our ways of thinking about physics. In this book, I return to the foundational work I began in the 1960's in the books published by Benjamin, Academic Press, and Dekker. In these my emphasis was on the linear problems involving vector bundles, linear differential operators, etc. In contrast, this volume emphasizes the type of nonlinear physics exemplified by fluid mechanics, vortex dynamics, interacting quantum field theory and Einsteinian gravitation. As in the previous book in this series, "Geometric Computing Science: First Steps", Interdisciplinary Mathematics Vol. 2, much of this material is preliminary in nature.

My own mathematical roots are grounded in the theory of geometric stroctures, as developed in Paris and Princeton in the 1950's. It is also striking how elementary particle physics tied in with these ideas in the period 1965 -1980, emphasizing Lie groups and fiber bundles. As a result of this revolution in the ways of thinking about fundamental physics, it is now conventional wisdom that a Lie group is at the foundation. For example, when elementary particle physicists talk about the quark model, they are positing such a choice as a finite dimensional compact Lie group. String theory is based on another type of structure, one of tbe infinite Lie pseudogroups studied by Lie, Cartan and Spencer. This volume suggests that a more complicated geometric structure, a deformation structure, is involved. Using deformation theory, I connect such topics as quantum field renormalization theory, Heisenberg-picture quantum field theory and the Colombeau-Rosinger theory of multiplication of Distributions. (In fact, the latter subject is related

to nonstandard analysis, hence ultimately logic is at the bottom of it all). From approximately 1964 to 1974, I worked on particle physics. My 1966 book "Lie Groups for Physicisis" couulbuted to the particle physicisis' evolution from the viewpoint of 19th century mathematics to that of Lie groups and fiber bundles. In this volume, my aim is to further catalyze the development of new mathematical tools and ideas which can aid in the futher development of physics along nonlinear lines. Physicists now use the term 'nonlinear physics' as a shorthand for mathematically going beyond the theory of linear differential equations. It seems to me that the heart of nonlinear physics consists of the problems of multiplication of distributions and tbe singularity structure of generalized solutions of nonlinear partial differemial equations.

Contents:
PREFACE
1
quantum field theory, Riemannian metric, Pseudogroups
The variational problem associated with LieSpencer
11
pseudogroupes, Lie Algebras, Quantum Field Theory
A DEFORMATIONTHEORETIC STRUCTURE BASED ON
25
quantum field theory, Poisson Bracket, Dirac Delta Functions
Computation of the simplest 1D field equations
36
Poisson Bracket, Canonical Commutation Relations, dual space
The Euler field equations
46
vector bundle, quotient set, equivalence relation
The Euler equations and variational principles for
53
pseudogroups, vector fields, Lie algebra
SOME AIMS IN RESEARCH IN THE GEOMETRY OF FLUIDS
57
LIE ALGEBRA DEFORMATIONS AND QUATUM FIELD THEORY
59
Deformation Theory, Lie Algebras, Category Theory
Some reserach areas
66
Lagrangian field theories, associative algebra, QUANTUM FIELD THEORY
Bibliography
74
Pseudogroup, Cartan connections, Current algebra
FRACTALS AND HEGEL
81
Pfaffian system, fiber bundle, direct sum
RIEMANNIAN METRICS AFFINE CONNECTIONS
93
Moving Frame, tensor field, symmetric matrix
Affine connections
100
affine connection, covariant derivative, moving frame
THE VORTICITY GRADIENT CONVECTION AND DIVERGENCE
109
Lie Derivative, PSEUDOGROUP, linear differential operator
The Spencer operators for the Lie algebra of
115
Lie derivative, local diffeomorphisms, Lie subalgebra
The dual Spencer operators for the Lie algebra
121
differential operator, Vorticity, convection operator
The Conservation of Vorticity in the Euler Perfect
127
Conservation Laws, RIEMANNIAN GEOMETRY, VARIATIONAL PRINCIPLE
Integrability conditions for infinitesimal variations
133
Affine Connection, Variational Principle, Diffeomorphisms
Variation of the Action generated by a differential
142
Affine Connection, adjoint representation, algebra structure
DIFFERENTIAL FORM METHODS IN THE THEORY OF
163
calculus of variations, string theories, Grassmann algebra
Extremals of constrained variational problems
41
variational problem, solder forms, orthonormal frame bundle
Minimal submanifolds of Riemannian manifolds
56
Hodge dual, Jacobi identity, inner product
The Cartan form in YangMills theory
72
Lagrangian Field Theories, Differential Forms, Vector Bundles
DIFFERENTIAL GEOMETRY OF THE HILBERT VARIATIONAL
207
scalar curvature, curvature forms, metric tensor
The vanishing of the Ricci curvature and the first
220
Ricci curvature, metric variation, Palatini Action
field in General Relativity
225
diffeomorphisms, inverse matrix, tensor fields
The Cartan Form for variational problems determined
238
metric tensor, Einstein equations, variational principle
THE TENSOR ALGEBRA ON A MANIFOLD AND ITS
247
tensor algebra, AFFINE CONNECTION, covariant differentiation
Total covariant derivatives
255
Tensor Fields, Moving Frame, Tensor Analysis
The total covariant derivative operator acting on
265
Riemannian Metrics, Levi-Civita Connection, Transition Relations
Tensorvalued differential forms
272
associative algebra, Tensor Algebra, Covariant Derivative
CLASSICAL AND QUANTUM MECHANICS AND THEIR
283
Poisson bracket, quantum mechanics, Hilbert space

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