A Generalization Of The BHW Theorem

BHW theorem arose from the study of axiomatic quantum field theory which ? was given by D. Hall and A. S. Wightman in 1957 based on a crucial lemma of V. Bargman.u. Although being of a background in physics, this theorem could be regarded as a pure result in several complex variables about holomorphic extentions. Axiomatic quantum field theory is a very important step in the development of quantum field theory. This subject was born after A. S. Weightman introduced an axiomatic approach to the quantum field theory during the end of 1950. This subject soon became very active. Many scholars joined the study of the subject, for example, N. N. Bogoliubov, V. S. Vladimirov et al. in the former Soviet Uniov, R. Jost, A. S. Wightman, F. Dyson, H. Araki, D. Rude et al in the westen countries. In a very recently published book uantum field theories for mathematicians?which was edited by the Fields Medalists E. Witten, P. Deligne et al, there is a special section fully devolted to the axiomatic quantum field theory written by a famous mathematician D. Kzada.n who is a professor of Harvard University. He said in the book that ightman axioms can be used to derive deep and unexpected results on behavior of quantum field theories? In their paper uantum Yang-Mills Theory? describing one of seven famous Millenium Problems, A. A. JaIiand E. Witten also mentioned the importance of axiomantic quantum field theory. Holomorphic extenson is a basic topic in complex analysis. One of significant difference between several complex variables and one complex variable is the Hartogs pheonomena. In several complex variables, there exist some domains e.g. punctured polydiscs on which all holomorphic functions can be extended to the holomorphic functions on a properly larger domain. However, there exist also some domains e.g. convex demains on which not all holomorpbic functions can be extended to the holomorphic functions on any properly larger domain, such domains are called domains of holornogphy. BHW theorem is foundational in describing Wightmnn axioms. For its applica- tions in relativistic quantum field theory, one can refer to Hall, Wightman classical paper in 1957. The theorem states that any holomorphic function invariant w.r.t. the connected restricted Loreatz group on the N-point future tube can be extended to a holomorphic function invariant w.r.t. the connected proper complex Lorentz group on the extended future tube. For the definitions of future tube, Loreutz group and extended future tube, one can refer to the above mentioned paper of D. Kaz- dan. It known that the future tube is a convex domain and therefore a domain of holomorphy, i.e., there exits a holomorphic function on the N-point future tube which can be extended to any larger domain. The significance of BHW tberoem is that the situation changes if one requires the hollmorphic functions invariant. he theorem is one Qf motivations of the so-called extented future tube conjecture, which means that the above extensions in BHW theroem are largest possible. This long-standing conjecture was solved be Prof. Xiang-Yu Zhou. By Pauli mapping, there is an equivalent version of BHW theroem in a matrix form. That is to say, if we denote by H = {Z E C[2 x 2]: ZZ?> O}, and consider the action of SL(2,C) on 2i...