ISSN:
1572-9036
Keywords:
non-Archimedean Hilbert space
;
p-adic quantization
;
precision of a measurement
;
symmetric and orthogonal operators
;
isometric orthogonal operators
;
Cauchy–Buniakovski–Schwarz inequality
;
majorant and self-polar norms
;
p-adic Gaussian distribution
;
p-adic analiticity
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1006219101760
