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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.

##### MSC:
 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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