**Proceedings of Symposia in Pure Mathematics**

Volume: 59;
1996;
224 pp;
Hardcover

MSC: Primary 81; 35; 47; 58; 46;

Print ISBN: 978-0-8218-0381-3

Product Code: PSPUM/59

List Price: $70.00

Individual Member Price: $56.00

**Electronic ISBN: 978-0-8218-9362-3
Product Code: PSPUM/59.E**

List Price: $70.00

Individual Member Price: $56.00

# Quantization, Nonlinear Partial Differential Equations, and Operator Algebra

Share this page *Edited by *
*William B Arveson; Thomas P Branson; Irving E Segal*

Recent inroads in higher-dimensional nonlinear quantum
field theory and in the global theory of relevant nonlinear wave
equations have been accompanied by very interesting cognate
developments. These developments include symplectic quantization theory
on manifolds and in group representations, the operator algebraic
implementation of quantum dynamics, and differential geometric, general
relativistic, and purely algebraic aspects.

This book describes the outstanding recent progress in this important and challenging field and presents general background for the scientific context and specifics regarding key difficulties. Quantization is developed in the context of rigorous nonlinear quantum field theory in four dimensions and in connection with symplectic manifold theory and random Schrödinger operators. Nonlinear wave equations are exposed in relation to recent important progress in general relativity, in purely mathematical terms of microlocal analysis, and as represented by progress on the relativistic Boltzmann equation.

Most of the developments in this volume appear in book form for the first time. The resulting work is a concise and informative way to explore the field and the spectrum of methods available for its investigation.

#### Table of Contents

# Table of Contents

## Quantization, Nonlinear Partial Differential Equations, and Operator Algebra

- Contents vii8 free
- Preface ix10 free
- E[sub(0)]-semigroups in quantum field theory 112 free
- Nonlinear phenomena in the spectral theory of geometric linear differential operators 2738
- Existence theorem for solutions of Einstein's equations with 1 parameter spacelike isometry groups 6778
- Quantum stochastic calculus, evolutions and flows 8192
- Endomorphisms of B(H) 93104
- Absolutely continuous spectrum in random Schrödinger operators 139150
- Quantization by deformation and statistical mechanics 149160
- Possible classification of continuous spatial semigroups of *-endomorphisms of B(h) 161172
- Rigorous covariant form of the correspondence principle 175186
- The relativistic Boltzmann equation 203214
- Microlocal analysis and nonlinear PDE 211222

#### Readership

Graduate students, research mathematicians, and physicists interested in quantum theory, functional analysis, nonlinear wave equations, and statistical mechanics.