an:05681120
Zbl 1244.47065
Albeverio, S.; Cianci, R.; Khrennikov, A. Yu.
Operator calculus for \(p\)-adic valued symbols and quantization
EN
Rend. Semin. Mat., Univ. Politec. Torino 67, No. 2, 137-150 (2009).
00259249
2009
j
47S10 11S80 11E95 47N50 81S99
\(p\)-adic valued wave functions; \(p\)-adic Gauss distribution; \(p\)-adic position operator; \(p\)-adic momentum operator
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, \textit{S. Albeverio, R. Cianci} and \textit{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 \textit{A. N. Kochubei} [``Non-Archimedean normal operators'', J.~Math. Phys. 51, No.~2, 023526 (2010; \url{doi:10.1063/1.3293980})].
Anatoly N. Kochubei (Ky??v)
Zbl 1187.81137