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Quantization, Nonlinear Partial Differential Equations, and Operator Algebra
Edited by: William Arveson, University of California, Berkeley, CA, Thomas Branson, University of Iowa, Iowa City, IA, and Irving Segal, Massachusetts Institute of Technology, Cambridge, MA
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Proceedings of Symposia in Pure Mathematics
1996; 224 pp; hardcover
Volume: 59
ISBN-10: 0-8218-0381-6
ISBN-13: 978-0-8218-0381-3
List Price: US$70 Member Price: US$56
Order Code: PSPUM/59

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. Quantization and Nonlinear Wave Equations thus was highly appropriate as the theme for the first John von Neumann Symposium (June 1994) held at MIT. The symposium was intended to treat topics of emerging signifigance underlying future mathematical developments.

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.

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

• W. Arveson -- $$E_0$$-semigroups in quantum field theory
• T. Branson -- Nonlinear phenomena in the spectral theory of geometric linear differential operators
• Y. Choquet-Bruhat and V. Moncrief -- Existence theorem for solutions of Einstein's equations with 1 parametric spacelike isometry groups
• R. L. Hudson -- Quantum stochastic calculus, evolutions, and flows
• O. Bratteli, P. T. Jorgensen, and G. L. Price -- Endomorphisms of $$\mathcal B(\mathcal H)$$
• A. Klein -- Absolutely continuous spectrum in random Schrödinger operators
• A. Lichnerowicz -- Quantization by deformation and statistical mechanics
• R. T. Powers -- Possible classification of continuous spatial semigroups of *-endomorphisms of $$\mathfrak B(\mathfrak h)$$
• I. Segal -- Rigorous covariant form of the correspondence principle
• W. A. Strauss -- The relativistic Boltzmann equation
• M. E. Taylor -- Microlocal analysis and nonlinear PDE