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Is there a Jordan geometry underlying quantum physics? (English) Zbl 1160.81398

Summary: There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics, see, e.g., [A. Ashtekar and T. A. Schilling, in Alex Harvey (ed.) On Einstein’s Path, Springer, Berlin, essays in honor of Engelbert Schücking. A Symposium, New York Univ., New York, NY, USA, December 12–13 1996. New York, NY: Springer. 23–55 (1999; Zbl 0976.53088); D. C. Brody and L. P. Hughston, J. Geom. Phys. 38, No. 1, 19–53, (2001; Zbl 1067.81081( R. Cirelli, M. Gatti and A. Manià, J. Geom. Phys. 45, No. 3–4, 267–284, (2003; Zbl 1032.81021); T. W. B. Kibble, Commun. Math. Phys. 65, 189–201, 1979; Zbl 0412.58006)]. From a purely mathematical side, the point of view of Jordan algebra theory might give new strength to such approaches: there is a “Jordan geometry” belonging to the Jordan part of the algebra of observables, in the same way as Lie groups belong to the Lie part. Both the Lie geometry and the Jordan geometry are well-adapted to describe certain features of quantum theory. We concentrate here on the mathematical description of the Jordan geometry and raise some questions concerning possible relations with foundational issues of quantum theory.

MSC:

81R25 Spinor and twistor methods applied to problems in quantum theory
81R15 Operator algebra methods applied to problems in quantum theory
17A15 Noncommutative Jordan algebras
81P05 General and philosophical questions in quantum theory
22E70 Applications of Lie groups to the sciences; explicit representations
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