Superanalysis

1. Front Matter

   Pages i-ix

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2. Introduction

   • Andrei Khrennikov

   Pages 1-5

3. Analysis on a Superspace over Banach Superalgebras

   • Andrei Khrennikov

   Pages 7-56

4. Generalized Functions on a Superspace

   • Andrei Khrennikov

   Pages 57-107

5. Distribution Theory on an Infinite-Dimensional Superspace

   • Andrei Khrennikov

   Pages 109-182

6. Pseudodifferential Operators in Superanalysis

   • Andrei Khrennikov

   Pages 183-225

7. Fundamentals of the Probability Theory on a Superspace

   • Andrei Khrennikov

   Pages 227-256

8. Non-Archimedean Superanalysis

   • Andrei Khrennikov

   Pages 257-291

9. Noncommutative Analysis

   • Andrei Khrennikov

   Pages 293-311

10. Applications in Physics

    • Andrei Khrennikov

    Pages 313-328

11. Back Matter

    Pages 329-357

    PDF

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.

• Banach algebra
• calculus
• derivative
• differential equation
• functional analysis
• mechanics

`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)

• Department of Mathematics, Statistics and Computer Sciences, University of Växjö, Växjö, Sweden

  Andrei Khrennikov

• Department of Mathematics, Moscow State University of Electronic Engineering, Zelenograd, Russia

  Andrei Khrennikov

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