Macroscopic Variables and Hilbert C^*-Modules in Algebraic Quantum Field Theory

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In 1999 together with Prof. Maumary (Univ. de Lausanne, Switzerland) I clarified deep connections between supersymmetry and homotopy formulated in an algebraic and categorical form (S.Maumary and I.Ojima, "Supersymmetry and Homotopy" which is now under revision). Along the general formulation of macroscopic variables arising from spontaneous symmetry breaking (SSB), I investigated the Josephson effect as the most important ingredient in superconductivity : In contrast to the traditional derivation whose general validity is unclear, I succeeded in obtaining a concise derivation of the Josephson formulae which clarifies the essence of Josephson current as the physical effect due to the difference in order parameters accross two adjacent phases. In close connection with this theme, I have continued the research on the general formulation of local temperature states in algebraic QFT together with Profs.D.Buchholz, H.Roos in Germany (to be completed by the coming summer) and on the generali … More zation of superselction theory to SSB cases the latter of which constitutes the main contents of my research in 2000.Regarding the later, I have worked on mathematical preparations for it : In contrast to Tannaka-Krein duality for compact groups playing the essential roles in the cases of unbroken symmetry, the extension of duality to homogeneous spaces becomes important in our case of SSB.What I obtained with Prof. Maumary is a result for this extension in use of the homotopy-fiber category, on the basis of which the research has been going on. To completie the project, we need to combine this result with a method which allows the extension of the algebra with unbroken subgroup symmetry to a larger one with full symmetry. For this purpose, the investigation on Goldstone multiplets, order parameters and the centres of represented algebras done in preparation for my series lectures "Thermal Instability of Supersymmetry I, II") at Borel Seminar 2000 is expected to play important roles. Along these lines, the project is expected to be completed this year. Less