Abstract
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.
Similar content being viewed by others
References
V. S. Vladimirov and I. V. Volovich, “Superanalysis I. Differential calculus,” Teor. Mat. Fiz. 59, 3–27 (1984).
V. S. Vladimirov and I.V. Volovich, “Superanalysis II. Integral calculus,” Teor. Mat. Fiz. 60, 169–198 (1984).
S. Albeverio, R. Cianci and A. Yu. Khrennikov, “p-Adic valued quantization,” p-Adic Numbers Ultrametric Anal. Appl. 1N(2), 91–104 (2009).
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adicmathematical physics,” p-Adic Numbers Ultrametric Anal. Appl. 1(1), 1–17 (2009).
B. Dragovich, “Some p-adic aspects of superanalysis,” Proc. Int. Workshop Supersymmetry and Quantum Symmetries, SQS’03, pp. 181–186 (JINR, Dubna, 2004), [arXiv:hep-th/0401044].
B. Dragovich and A. Khrennikov, “p-Adic and adelic superanalysis,” Bulgarian J. Phys. 33(s2), 159–173 (2006), [arXiv: hep-th/0512318].
B. Dragovich and A. Khrennikov, “Adelic superanalysis,” Proc. Int. Workshop Supersymmetry and Quantum Symmetries, SQS’05, pp. 384–392 (JINR, Dubna, 2006).
J. Schwinger, “A note to the quantum dynamical principle,” Phil. Mag. 44, 1171–1193 (1953).
J. L. Martin, “Generalized classical dynamics and “classical analogue” of a Fermi oscillator,” Proc. Royal Soc. A. 251, 536–542 (1959).
J. L. Martin, “The Feynman principle for a Fermi system,” Proc. Royal Soc. A 251, 543–549 (1959).
A. Yu. Khrennikov, “Functional superanalysis,” Uspekhi Matem. Nauk 43, 87–114 (1988); English translation: Russian Math. Surveys 43, 103 (1988).
A. Yu. Khrennikov, Supernalysis (Nauka, Fizmatlit, Moscow, 1997) [in Russian]; English translation: (Kluwer, Dordreht, 1999).
A. Salam and J. Strathdee, “Feynman rules for superfields,” Nucl. Phys. B 86, 142–152 (1975).
J. Wess and B. Zumino, “Supergauge transformations in four dimensions,” Nucl. Phys. B 70, 39–50 (1974).
J. Dell and I. Smolin, “Graded manifolds theory as the geometry of supersymmetry,” Commun. Math. Phys. 66, 197–222 (1979).
B. S. De Witt, Supermanifolds (Cambridge, U. P., 1984).
A. Rogers, “Super Lie groups: global topology and local structure,” J. Math. Phys. 21, 724–731 (1980).
A. Rogers, “A global theory of supermanifolds,” J. Math. Phys. 22, 939–945 (1981).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English.
Rights and permissions
About this article
Cite this article
Khrennikov, A.Y. Foundations of analysis on superspace — 1: Differential calculus. P-Adic Num Ultrametr Anal Appl 7, 96–110 (2015). https://doi.org/10.1134/S2070046615020028
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046615020028