Synopses & Reviews
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.
Review
`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)
Synopsis
defined as elements of Grassmann algebra (an algebra with anticom- muting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used. Later, during the next twenty years, the algebraic apparatus de- veloped by Martin was used in all mathematical works. We must point out here the considerable contribution made by F. A. Berezin, G 1. Kac, D. A. Leites, B. Kostant. In their works, they constructed a new division of mathematics which can naturally be called an algebraic superanalysis. Following the example of physicists, researchers called the investigations carried out with the use of commuting and anticom- muting coordinates supermathematics; all mathematical objects that appeared in supermathematics were called superobjects, although, of course, there is nothing "super" in supermathematics. However, despite the great achievements in algebraic superanaly- sis, this formalism could not be regarded as a generalization to the case of commuting and anticommuting variables from the ordinary Newton analysis. What is more, Schwinger's formalism was still used in practically all physical works, on an intuitive level, and physicists regarded functions of anticommuting variables as "real functions" == maps of sets and not as elements of Grassmann algebras. In 1974, Salam and Strathdee proposed a very apt name for a set of super- points. They called this set a superspace.
Description
Includes bibliographical references (p. [329]-343) and index.
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.