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Superanalysis, I. Differential calculus. (English. Russian original) Zbl 0552.46023
Theor. Math. Phys. 59, 317-335 (1984); translation from Teor. Mat. Fiz. 59, No. 1, 3-27 (1984).
The author studies differential calculus in superspaces over commutative Banach superalgebras. A Banach superalgebra is an associative Banach algebra \(\Lambda\) over the reals, with an identity, which is decomposed in a direct sum as a linear space, \(\Lambda =\Lambda_ 0\oplus \Lambda_ 1,\) admits a function \(p: \Lambda \to \{0,1\}\) such that \(p(a)=0\) for \(a\in \Lambda_ 0\), \(p(a)=1\) for \(a\in \Lambda_ 1\) and \(p(ab)=p(a)+p(b)(mod. 2).\) If \(ab-(-1)^{p(a)p(b)}=0\) then \(\Lambda\) is called a commutative superalgebra. The notion includes commutative Banach algebras and Grassmann algebras. A superspace of dimension (m,n) over \(\Lambda\) is a Banach space \[ R_{\Lambda}^{m,n}=\Lambda_ 0\times...\times \Lambda_ 0(m-times)\times \Lambda_ 1\times...\times \Lambda_ 1(n-times) \] with the norm \(\| (x_ 1,...,x_{m+n}\| =\sum^{m+n}_{i=1}\| x_ i\|\) and plays a role in superanalysis, analogous to that of \({\mathbb{R}}^ m\) in classical analysis. A superderivative is defined and a generalization of Cauchy-Riemann equations of complex analysis are derived. The basic rules of calculus are proved, including the chain rule, derivative of product, implicit function theorem and Taylor’s expansion. The rules hold also over the field of complex numbers, and some of them remain unchanged over arbitrary non-discrete complete field. The paper contains many examples and it is to be the first one in a sequence of the author’s papers on superanalysis.
Reviewer: A.Sterna-Karwat

46G05 Derivatives of functions in infinite-dimensional spaces
46H05 General theory of topological algebras
32H99 Holomorphic mappings and correspondences
Full Text: DOI
[1] N. N. Dogolyubov, A. A. Logunov, and I. T. Todorov, Introduction to Axiomatic Quantum Field Theory, Benjamin, New York (1975).
[2] R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin, New York (1964). · Zbl 0135.44305
[3] R. Jost, The General Theory of Quantized Fields, AMS, Providence, R.I. (1965). · Zbl 0127.19105
[4] V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables, M.I.T. Press, Cambr., Mass. (1966).
[5] Twistors and Gauge Fields (Collection of Papers by R. Penrose and others; Russian translations), Mir, Moscow (1983).
[6] M. F. Atiyah, ?Geometry of Yang-Mills fields,? Preprint, Pisa (1979).
[7] E. Witten, Phys. Lett. B,77, 394 (1978).
[8] I. V. Volovich, Teor. Mat. Fiz.,54, 89 (1983);55, 39 (1983).
[9] V. I. Ogievetskii and L. Mezincescu, Usp. Fiz. Nauk,117, 637 (1975).
[10] A. A. Slavnov, Usp. Fiz. Nauk,124, 487 (1978).
[11] P. Van Nieuwenhuizen, Phys. Rep.,68, 189 (1981).
[12] Superspace and Supergravity (eds. S. W. Hawking and M. Rocek), Cambridge University Press, Cambridge (1981); B. Zumino, ?Supersymmetry and supergravity,? UC Berkeley Report No. UCB-PTH-83/2 (1983).
[13] A. Salam and J. Strathdee, Nucl. Phys. B,71, 51 (1974).
[14] Yu. A. Gol’fand and E. P. Likhtman, Pis’ma Zh. Eksp. Teor. Fiz.,13, 452 (1971).
[15] D. V. Volkov and V. P. Akulov, Pis’ma Zh. Eksp. Teor. Fiz.,16, 621 (1972).
[16] J. Wess and B. Zumino, Nucl. Phys. B,70, 39 (1974).
[17] J. Schwinger, Theory of Quantized Fields [Russian translation], IL. (1956); N. N. Bogolyubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, Interscience (1959); F. A. Berezin, The Method of Second Quantization, New York (1966).
[18] Yu. V. Novozhilov and A. V. Tulub, Usp. Fiz. Nauk,61, 53 (1957).
[19] I. L. Martin, Proc. R. Soc. London, Ser. A,251, 536, 543 (1959). · Zbl 0086.22203
[20] I. V. Volovich, Dokl. Akad. Nauk SSSR,269, 524 (1983).
[21] V. S. Vladimirov and I. V. Volovich, Dokl. Akad. Nauk SSSR,273, 26 (1983).
[22] A. Rogers, J. Math. Phys.,21, 1352 (1980). · Zbl 0447.58003
[23] A. Jadczyk and K. Pilch, Commun. Math. Phys.,78, 373 (1981). · Zbl 0464.58006
[24] J. Hoyos, M. Quirós, J. Ramírez Mittelbrunn, and F. J. Urríes, J. Math. Phys.,23, 1504 (1982). · Zbl 0503.53049
[25] G. Scheffers, Der. Verhandl. Sächsisch. Acad. Wiss. Leipzig, Math. Phys. Kl.,45, 828 (1893);46, 120 (1894).
[26] P. W. Ketchum, Trans. Am. Math. Soc.,30, 641 (1928). · JFM 55.0787.02
[27] E. R. Lorch, Trans. Am. Math. Soc.,54, 414 (1943).
[28] E. K. Blum, Trans. Am. Math. Soc.,78, 343 (1955).
[29] N. M. Krylov, Dokl. Akad. Nauk SSSR,60, 687 (1947).
[30] J. A. Ward, J. Duke Math.,7, 233 (1940). · Zbl 0024.24602
[31] V. S. Fedorov, Mat. Sb.,18, 352 (1946).
[32] I. M. Gel’fand, D. A. Raikov, and G. E. Shilov, Commutative Normed Rings [in Russian], Fizmatgiz, Moscow (1960).
[33] V. M. Maksimov, Dokl. Akad. Nauk SSSR,267, 48 (1982).
[34] F. A. Berezin, Introduction to Algebra and Analysis with Noncommuting Variables [in Russian], Moscow State University, Moscow (1983). · Zbl 0527.15020
[35] D. A. Leites, Usp. Mat. Nauk,35, 3 (1980).
[36] B. Kostant, Lect. Notes Math., No. 570, 177 (1977).
[37] F. A. Berezin and G. I. Kats, Mat. Sb.,3, 343 (1970).
[38] A. S. Schwarz, Commun. Math. Phys.,87, 37 (1982). · Zbl 0503.53048
[39] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1972). · Zbl 0235.46001
[40] G. E. Shilov, Mathematical Analysis. Functions of Several Real Variables [in Russian], Nauka, Moscow (1972). · Zbl 0245.46078
[41] J. Dieudonné, Foundations of Modern Analysis, New York (1960).
[42] É. Cartan, Differential Calculus. Differential Forms [Russian translation], Mir, Moscow (1971). · Zbl 0223.35004
[43] L. Schwartz, Analysis, Vols. 1 and 2 [Russian translation], Mir, Moscow (1972).
[44] S. Lang, Differentiable Manifolds [Russian translation], IL., Moscow (1967). · Zbl 0146.17503
[45] N. Bourbaki, Differentiable and Analytic Manifolds. Summary of Results [Russian translation], Mir, Moscow (1975).
[46] A. I. Kostrikin and Yu. I. Manin, Linear Algebra and Geometry [in Russian], Moscow State University, Moscow (1980). · Zbl 0532.00002
[47] N. M. Krylov, Dokl. Akad. Nauk SSSR,60, 799 (1947).
[48] F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Pitnam Publishing Inc., Boston (1982).
[49] K. Yosida, Functional Analysis, Springer, Berlin (1965). · Zbl 0126.11504
[50] J. Mikusinski, The Bochner Integral, Birkhäuser Verlag, Basel (1978). · Zbl 0369.28010
[51] N. Bourbaki, Integration [Russian translation], Nauka, Moscow (1977).
[52] V. S. Vladimirov, Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1979). · Zbl 0515.46033
[53] G. I. Kats and A. I. Koronkevich, Funktsional. Analiz i Ego Prilozhen.,5, 78 (1971).
[54] Z. I. Borevich and I. R. Shafarevich, Theory of Numbers [in Russian], Nauka, Moscow (1972). · Zbl 0121.04202
[55] J. P. Serre, Lie Algebras and Lie Groups, Benjamin (1965). · Zbl 0132.27803
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