Quantization, Nonlinear Partial Differential Equations, and Operator Algebra - 1994 John von Neumann Symposium on Quantization and Nonlinear Wave Equations - June 7-11, 1994 - Massachusetts Institute of Technology, Cambridge, Massachusetts - Proceeding...

by John von Neumann

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.

Hardcover

First published January 1, 1996

About the author

John von Neumann (Hungarian: margittai Neumann János Lajos) was a Hungarian American[1] mathematician who made major contributions to a vast range of fields,[2] including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields. He is generally regarded as one of the foremost mathematicians of the 20th century. The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians." Even in Budapest, in the time that produced Szilárd (1898), Wigner (1902), and Teller (1908) his brilliance stood out. Most notably, von Neumann was a pioneer of the application of operator theory to quantum mechanics, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory and the concepts of cellular automata and the universal constructor. Along with Edward Teller and Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.

Reviews

[Medhat ullah · Rating 5 out of 5]

The symposium served as a crucible for the interdisciplinary fusion of abstract operator-theoretic frameworks and the perturbative and non-perturbative methodologies governing nonlinear PDEs, particularly in the quantization of wave dynamics. Central themes revolved around the spectral decomposition of non-self-adjoint operators within infinite-dimensional Hilbert spaces, elucidating the correspondence principles that reconcile classical nonlinear field theories with their quantum analogs.Noncommutative Operator Algebra:Explored the intricate morphisms and automorphisms inherent in C*- and von Neumann algebras, particularly as they relate to the rigorous mathematical formalization of quantum mechanics and quantum field theory.
Nonlinear Wave Equations:Detailed the advanced asymptotic and variational techniques employed to tackle solitonic structures, blow-up phenomena, and the bifurcations emerging in solutions to nonlinear Klein-Gordon and Schrödinger equations.
Quantization and Geometric Analysis:Investigated the interplay between symplectic geometry and deformation quantization, emphasizing the phase-space representations of quantum states in nonlinear contexts.
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