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Название: Geometric Structures in Nonlinear Physics
Автор: Hermann R.
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.
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