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Quantization as a dimensional reduction phenomenon. (English) Zbl 1130.81043

Bassi, Angelo (ed.) et al., Quantum mechanics. Are there quantum jumps? Trieste, Italy, 5 September 2005 and on the present status of quantum mechanics, Lošinj, Croatia, 7–9 September 2005. Proceedings. In honor of Giancarlo Ghirardi 70th birthday. Melville, NY: American Institute of Physics (AIP) (ISBN 0-7354-0337-6/hbk). AIP Conference Proceedings 844, 158-176 (2006).
Classical mechanics, in the operatorial formulation of Koopman and von Neumann, can be written also in a functional form. In this form two Grassmann partners of time make their natural appearance extending in this manner time to a three dimensional supermanifold. Quantization is then achieved by a process of dimensional reduction of this supermanifold. We prove that this procedure is equivalent to the well-known method of geometric quantization.
For the entire collection see [Zbl 1103.81006].

MSC:

81S10 Geometry and quantization, symplectic methods
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