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Complex Analysis in the Future Tube

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Book cover Several Complex Variables II

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 8))

Abstract

The future tube in ℂn +1 is the unbounded domain τ+ = z = (z o,...,z n ) ∈ ℂn +1 : (Im z o)2 > (Imz 1)2 +...+ (Im z n )2 Im z o > 0. In other words, τ+ is a tube domain over the future cone V + = y ∈ ℝn +1 : y 2 0 > y 2 1 +...+ y 2 n , y 0 > 0. The domain τ+ is biholomorphically equivalent to a classical Cartan domain of the IVth type, hence to a bounded symmetric domain in ℂn +1. The future tube τ+ in ℂ4 (n = 3) is important in mathematical physics, especially in axiomatic quantum field theory, being the natural domain of definition of holomorphic relativistic fields. These specific features of the future tube motivated its investigation by mathematicians and physicists. Beginning with Elie Cartan’s classification of bounded symmetric domains, these domains were examined in many papers where the complex structure of their boundaries, integral representations, boundary values of holomorphic functions and so on were considered. The proof of the “edge-of-the-wedge” theorem by N.N. Bogolubov generated the rapid development of applications of the theory of several complex variables to axiomatic quantum field theory. During this period the “C-convex hull” and “finite covariance” theorems were proved, the Jost-Lehmann-Dyson representation was found et cetera. Recently R. Penrose has proposed a transformation connecting holomorphic solutions of the basic equations of field theory with analytic sheaf cohomologies of domains in ℝℙ3. These two directions developed, to a large extent, independently from each other, and some important results obtained in axiomatic quantum field theory still remain unknown to specialists in several complex variables and differential geometry. One of the goals of this paper is to give a unified presentation of advances in complex analysis in the future tube and related domains achieved in both of these directions.

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  1. A. G. Sergeev
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  2. V. S. Vladimirov
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  1. Central Economic and Mathematical Institute of the Russian Academy of Sciences, ul. Krasikova 32, 117418, Moscow, Russia

    G. M. Khenkin

  2. Steklov Mathematical Institute, ul. Vavilova 42, 117966, Moscow, Russia

    A. G. Vitushkin

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Sergeev, A.G., Vladimirov, V.S. (1994). Complex Analysis in the Future Tube. In: Khenkin, G.M., Vitushkin, A.G. (eds) Several Complex Variables II. Encyclopaedia of Mathematical Sciences, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57882-3_4

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