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Superanalysis

Overview

This work can be recommended as an extensive course in superanalysis, the theory of functions of commuting and anticommuting variables. It follows the so-called functional superanalysis which was developed by J. Schwinger, B. De Witt, A. Rogers, V.S. Vladimirov and I.V. Volovich, Yu. Kobayashi and S. Nagamashi, M. Batchelor, U. Buzzo and R. Cianci and the present author. In this approach, superspace is defined as a set of points on which commuting and anticommuting coordinates are given. Thus functional ...
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Paperback (Softcover reprint of the original 1st ed. 1999)
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Overview

This work can be recommended as an extensive course in superanalysis, the theory of functions of commuting and anticommuting variables. It follows the so-called functional superanalysis which was developed by J. Schwinger, B. De Witt, A. Rogers, V.S. Vladimirov and I.V. Volovich, Yu. Kobayashi and S. Nagamashi, M. Batchelor, U. Buzzo and R. Cianci and the present author. In this approach, superspace is defined as a set of points on which commuting and anticommuting coordinates are given. Thus functional superanalysis is a natural generalization of Newton's analysis (on real space) and strongly differs from the so-called algebraic analysis which has no functions of superpoints, and where 'functions' are just elements of Grassmann algebras.
This volume is important for quantum physics in that it offers the possibility of extending the notion of space, and of operating on spaces which are described by noncommuting coordinates. These supercoordinates, which are described by an infinite number of ordinary real, complex or p-adic coordinates, are interpreted as creation or annihilation operators of quantum field theory.
Subjects treated include differential calculus, including Cauchy-Riemann conditions, on superspaces over supercommutative Banach and topological superalgebras; integral calculus, including integration of differential forms; theory of distributions and linear partial differential equations with constant coefficients; calculus of pseudo-differential operators; analysis on infinite-dimensional superspaces over supercommutative Banach and topological supermodules; infinite-dimensional superdistributions and Feynman integrals with applications to superfield theory; noncommutative probabilities (central limit theorem); and non-Archimedean superanalysis.
Audience: This volume will be of interest to researchers and postgraduate students whose work involves functional analysis, Feynman integration and distribution theory on infinite-dimensional (super)spaces and its applications to quantum physics, supersymmetry, superfield theory and supergravity.
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Editorial Reviews

From the Publisher
'The book will probably make a useful addition to the office|library collection of every researcher or mathematical centre with interests in analysis with anticommuting variables.'
Mathematical Reviews Clippings (2001)
Booknews
Presents an extended course in superanalysis. This work is important for quantum physics in that it offers the possibility of extending the notion of space, and of operating on spaces which are described by noncommuting coordinates. Subjects treated include differential and integral calculus, theory of distributions and linear partial differential equations with constant coefficients, and calculus of pseudo-differential operators. For advanced students and researchers whose work involves functional analysis and Feynman integration and distribution theory. The author teaches mathematics at the Moscow State University of Electronic Engineering. This is a revised translation of a title first published in Russian in 1997. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9789401059480
  • Publisher: Springer Netherlands
  • Publication date: 12/31/2013
  • Series: Mathematics and Its Applications (closed) Series
  • Edition description: Softcover reprint of the original 1st ed. 1999
  • Pages: 357
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.77 (d)

Table of Contents

Introduction. I: Analysis on a Superspace over Banach Superalgebras. 1. Differential Calculus. 2. Cauchy-Riemann Conditions and the Condition of A-Linearity of Derivatives. 3. Integral Calculus. 4. Integration of Differential Forms of Commuting Variables. 5. Review of the Development of Superanalysis. 6. Unsolved Problems and Possible Generalizations. II: Generalized Functions on a Superspace. 1. Locally Convex Superalgebras and Supermodules. 2. Analytic Generalized Functions on the Vladimirov-Volovich Superspace. 3. Fourier Transformation of Superanalytic Generalized Functions. 4. Superanalog of the Theory of Schwartz Distributions. 5. Theorem of Existence of a Fundamental Solution. 6. Unsolved Problems and Possible Generalizations. III. Distribution Theory on an Infinite-Dimensional Superspace. 1. Polylinear Algebra over Commutative Supermodules. 2. Banach Supermodules. 3. Hilbert Supermodules. 4. Duality of Topological Supermodules. 5. Differential Calculus on a Superspace over Topological Supermodules. 6. Analytic Distributions on a Superspace over Topological Supermodules. 7. Gaussian and Feynman Distributions. 8. Unsolved Problems and Possible Generalizations. IV: Pseudodifferential Operators in Superanalysis. 1. Pseudodifferential Operators Calculus. 2. The Correspondence Principle. 3. The Feynman-Kac Formula for the Symbol of the Evolution Operator. 4. Unsolved Problems and Possible Generalizations. V: Fundamentals of the Probability Theory on a Superspace. 1. Limit Theorems on a Superspace. 2. Random Processes on a Superspace. 3. Axiomatics of the Probability Theory over Superalgebras. 4. Unsolved Problems and Possible Generalizations. VI: Non-Archimedean Superanalysis. 1. Differentiable and Analytic Functions. 2. Generalized Functions. 3. Laplace Transformation. 4. Gaussian Distributions. 5. Duhamel non-Archimedean Integral. Chronological Exponent. 6. Cauchy Problem for Partial Differential Equations with Variable Coefficients. 7. Non-Archimedean Supersymmetrical Quantum Mechanics. 8. Trotter Formula for non-Archimedean Banach Algebras. 9. Volkenborn Distribution on a non-Archimedean Superspace. 10. Infinite-Dimensional non-Archimedean Superanalysis. 11. Unsolved Problems and Possible Generalizations. VII: Noncommutative Analysis. 1. Differential Calculus on a Superspace over a Noncommutative Banach Algebra. 2. Differential Calculus on Noncommutative Banach Algebras and Modules. 3. Generalized Functions of Noncommuting Variables. VIII: Applications in Physics. 1. Quantization in Hilbert Supermodules. 2. Transition Amplitudes and Distributions on the Space of Schwinger Sources. References. Index.
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