Superanalysis - Andrei Y. Khrennikov

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Résumé :
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. br/ 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 emp/em-adic coordinates, are interpreted as creation or annihilation operators of quantum field theory. br/ 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. br/ emAudience:/em 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.

Biographie:
Prof. Andrei Khrennikov is the director of International center for mathematical modeling in physics, engineering and cognitive science, University of V?xj?, Sweden, which was created 8 years ago to perform interdisciplinary research. Two series of conferences on quantum foundations (especially probabilistic aspects) were established on the basis of this center: Foundations of Probability and Physics and Quantum Theory: Reconsideration of Foundations. These series became well known in the quantum community (including quantum information groups). Hundreds of theoreticians (physicists and mathematicians), experimenters and even philosophers participated in these conferences presenting a huge diversity of views to quantum foundations. Contacts with these people played the crucial role in creation of the present book. Prof. Andrei Khrennikov published about 300 papers in internationally recognized journals in mathematics, physics and biology and 9 monographs - in p-adic and non-Archimedean analysis with applications to mathematical physics and cognitive sciences as well as foundations of probabilityu theory....

Sommaire:
I Analysis on a Superspace over Banach Superalge-bras.- 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. Differentiale 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 Super-space.- 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.