# zbMATH — the first resource for mathematics

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)].

##### MSC:
 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
Full Text: