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\(p\)-adic valued quantization. (English) Zbl 1187.81137
Summary: This review covers an important domain of \(p\)-adic mathematical physics – quantum mechanics with \(p\)-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of \(p\)-adic numbers \(\mathbb Q_p\) , operators – symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the \(p\)-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg and Weyl forms, spectral properties of the operator of \(p\)-adic coordinate.We also present postulates of \(p\)-adic valued quantization. Here observables as well as probabilities take values in \(\mathbb Q_p\). A physical interpretation of \(p\)-adic quantities is provided through approximation by rational numbers.

81Q65 Alternative quantum mechanics (including hidden variables, etc.)
81T10 Model quantum field theories
81S10 Geometry and quantization, symplectic methods
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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