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On the extended tube conjecture. (English) Zbl 0855.32006

Etant donné les entiers \(n \geq 2\), \(N \geq 1\), on note: d’abord \(L (\mathbb{C}^n)\) le groupe des endomorphismes de \(\mathbb{C}^n\) qui conservent la forme quadratique \(z = (z_1, \dots, z_n) \mapsto z^2_1 - \sum^n_{j = 2} z^2_j\) et \(T(n)\) le cône supérieur de lumière \(\{z = x + iy \in \mathbb{C}^n : y_1 > (\sum^n_{j = 2} y^2_j)^{1/2}\}\); puis \(T(n;N)\) le domaine dans \((\mathbb{C}^n)^N\) formé des points \((Az^1, \dots, Az^N)\), où \(A \in L (\mathbb{C}^n)\), \(\text{det} A = 1\) et \(z^1, \dots, z^N \in T(n)\). Le tube \(T(n;1)\) est domaine d’holomorphie, car \(T(n,1) = \{z \in \mathbb{C}^n\): \(z^2_1 - \sum^n_{j = 2} z^2_j \in \mathbb{C} \backslash [0, \infty[\}\); mais le même question pour le tube \(T(n;N)\) est ouverte si \(N \geq 2\). Les Auteurs considèrent donc le tube \(T_0 (n;N)\) obtenu en ajoutant \(A\) unitaire aux 2 conditions imposées ci-dessus à \(A\), et montrent: d’une part l’équivalence des 3 propriétés \(T_0 (n;N) = T(n;N)\), \(n = 2\) ou \(N = 1\), \(T_0 (n; N)\) domaine d’holomorphie; d’autre part que \(T(n;2)\) est domaine d’holomorphie \(\forall n \geq 2\).
Reviewer: M.Hervé (Paris)

MSC:

32D05 Domains of holomorphy
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References:

[1] Bogoljubov, N. N., Vladimirov, V. S.: Representation ofn-point functions, Proc. Steklov Inst. Math.112, 1–18 (1971)
[2] Bros, J., Epstein, H., Glaser, V.: On the connection between analyticity and Lorentz covariance of Wightman functions, Commun. math. Phys.6, 77–100 (1967) · Zbl 0155.32304
[3] Gheorghe, A., Mihul, El.: On the geometry of Lorentz orbit spaces, Commun. math. Phys.43, 89–108 (1975) · Zbl 0303.57022
[4] Hall, D., Wightman, A. S.: A theorem on the invariant analytic functions with applications to relativistic quantum field theory, Mat. Fys. Medd. Dan. Vid. Selsk.31, 1–41 (1957) · Zbl 0078.44302
[5] Hepp, K.: Lorentz-kovariante analytische Funktionen, Helv. Phys. Acta36, 355–376 (1963) · Zbl 0138.45503
[6] Jost, R.: The general theory of quantized fields, Amer. Math. Soc. 1965 · Zbl 0127.19105
[7] Loeffel, J. J.: Quelques propriétés de l’espace de Minkowski sur les complexes, Helv. Phys. Acta36, 216–236 (1963) · Zbl 0127.19303
[8] Rockafeller, R. T.: Convex analysis, Princeton University Press 1970
[9] Sergeev, A. G., Heinzner, P.: The extended matrix disk is a domain of holomorphy, Math. USSR Izvestiya38, 637–645 (1992) · Zbl 0788.32006
[10] Sergeev, A. G., Vladimirov, V. S.: Complex analysis in the future tube, Encyclopaedia of Math. Sciences8, 179–253 (1994)
[11] Vladimirov, V. S.: Methods of the theory of functions of many complex variables, The M.I.T. Press 1966
[12] Vladimirov, V. S.: Several complex variables in mathematical physics, Lecture Notes in Math.919, 358–386 (1982) · Zbl 0493.32014
[13] Wightman, A. S.: Quantum field theory and analytic functions of several complex variables, J. Indian Math. Soc.24, 625–677 (1961) · Zbl 0105.22101
[14] Zavyalov, B. I., Trushin, V. B.: On the extendedn-point tube, Teor. i Matem. Fizika27, 3–15 (1976)
[15] Zharinov, V. V.: Integral representation of the Wightman functions in two-dimensional space-time, Teor. i Matem. Fizika9, 232–239 (1971)
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