Foundations of analysis on superspace -1: Differential calculus

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• 作者：A. Yu. Khrennikov
• 关键词：supersymmetry ; superspace ; Banach superalgebra ; superanalysis
• 刊名：P-Adic Numbers, Ultrametric Analysis, and Applications
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：7
• 期：2
• 页码：96-110
• 全文大小：579 KB
• 参考文献：1.V. S. Vladimirov and I. V. Volovich, “Superanalysis I. Differential calculus,-Teor. Mat. Fiz. 59, 3-7 (1984).View Article MathSciNet
  2.V. S. Vladimirov and I.V. Volovich, “Superanalysis II. Integral calculus,-Teor. Mat. Fiz. 60, 169-98 (1984).View Article MathSciNet
  3.S. Albeverio, R. Cianci and A. Yu. Khrennikov, -em class="EmphasisTypeItalic">p-Adic valued quantization,-p-Adic Numbers Ultrametric Anal. Appl. 1N(2), 91-04 (2009).View Article MATH MathSciNet
  4.B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adicmathematical physics,-p-Adic Numbers Ultrametric Anal. Appl. 1(1), 1-7 (2009).View Article MATH MathSciNet
  5.B. Dragovich, “Some p-adic aspects of superanalysis,-Proc. Int. Workshop Supersymmetry and Quantum Symmetries, SQS-3, pp. 181-86 (JINR, Dubna, 2004), [arXiv:hep-th/0401044].
  6.B. Dragovich and A. Khrennikov, -em class="EmphasisTypeItalic">p-Adic and adelic superanalysis,-Bulgarian J. Phys. 33(s2), 159-73 (2006), [arXiv: hep-th/0512318].MATH MathSciNet
  7.B. Dragovich and A. Khrennikov, “Adelic superanalysis,-Proc. Int. Workshop Supersymmetry and Quantum Symmetries, SQS-5, pp. 384-92 (JINR, Dubna, 2006).
  8.J. Schwinger, “A note to the quantum dynamical principle,-Phil. Mag. 44, 1171-193 (1953).View Article MATH MathSciNet
  9.J. L. Martin, “Generalized classical dynamics and “classical analogue-of a Fermi oscillator,-Proc. Royal Soc. A. 251, 536-42 (1959).View Article MATH
  10.J. L. Martin, “The Feynman principle for a Fermi system,-Proc. Royal Soc. A 251, 543-49 (1959).View Article MATH
  11.A. Yu. Khrennikov, “Functional superanalysis,-Uspekhi Matem. Nauk 43, 87-14 (1988); English translation: Russian Math. Surveys 43, 103 (1988).MathSciNet
  12.A. Yu. Khrennikov, Supernalysis (Nauka, Fizmatlit, Moscow, 1997) [in Russian]; English translation: (Kluwer, Dordreht, 1999).
  13.A. Salam and J. Strathdee, “Feynman rules for superfields,-Nucl. Phys. B 86, 142-52 (1975).View Article MathSciNet
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  15.J. Dell and I. Smolin, “Graded manifolds theory as the geometry of supersymmetry,-Commun. Math. Phys. 66, 197-22 (1979).View Article MATH MathSciNet
  16.B. S. De Witt, Supermanifolds (Cambridge, U. P., 1984).
  17.A. Rogers, “Super Lie groups: global topology and local structure,-J. Math. Phys. 21, 724-31 (1980).
  18.A. Rogers, “A global theory of supermanifolds,-J. Math. Phys. 22, 939-45 (1981).View Article MATH MathSciNet
  
• 作者单位：A. Yu. Khrennikov (1)
  
  1. International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science, Linnaeus University, V?xj?-Kalmar, Sweden
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algebra
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：2070-0474

文摘

The recent experimental confirmation of the existence of Higgs boson stimulates theoretical research on supersymmetric models; in particular, mathematics of such modeling. Therefore we plan to present essentials of one special approach to “super-mathematics- so called functional superanalysis (in the spirit of De Witt, Rogers, Vladimirov and Volovich, and the author of this review) in compact and clear form in series of review-type papers. This first paper is a review on super-differential calculus for the concrete model of the superspace (invented by Vladimirov and Volovich). In the next review we plan to present the integral supercalculus. The main distinguishing feature of functional superanalysis is that this is a real super-extension of analysis of Newton and Leibniz, opposite to algebraic models of Martin and Berezin. Here functions of commuting and anticommuting variables are no simply algebraic elements belonging to Grassmann algebras, but point-wise maps, from superspace into superspace. Finally, we remark that the first non-Archimedean physical model was based on invention by Vladimirov and Volovich of superspaces based on supercommutative Banach superalgebras over non-Archimedean (in particular, p-adic) fields. This model plays the basic role in theory of p-adic superstrings.