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Operator calculus for \(p\)-adic valued symbols and quantization. (English) Zbl 1244.47065
The paper gives a brief review of the quantum formalism with \(p\)-adic variables and \(p\)-adic valued wave functions. In particular, the authors discuss \(p\)-adic Banach spaces, including those equipped with an inner product (note that in these Hilbert-like spaces the inner product does not agree with the norm), groups of operators preserving the inner product, spaces of square integrable functions with respect to the \(p\)-adic Gauss distribution; \(p\)-adic position and momentum operators and their properties. See also another survey by the same authors, S. Albeverio, R. Cianci and A. Yu. Khrennikov [\(p\)-Adic Numbers Ultrametric Anal. Appl. 1, No. 2, 91–104 (2009; Zbl 1187.81137)].
In the authors’ words, “\(p\)-adic valued quantum theory suffers from the absence of a ‘good spectral theorem’ for symmetric operators.” Subsequently, a spectral theorem for a class of operators on \(p\)-adic Banach spaces was proved by the reviewer; see A. N. Kochubei [“Non-Archimedean normal operators”, J. Math. Phys. 51, No. 2, 023526 (2010; doi:10.1063/1.3293980)].

47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11E95 \(p\)-adic theory
47N50 Applications of operator theory in the physical sciences
81S99 General quantum mechanics and problems of quantization
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