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The Schrödinger and Bargmann-Fock representations in non-Archimedean quantum mechanics. (English. Russian original) Zbl 0749.46046

Sov. Phys., Dokl. 35, No. 7, 638-640 (1990); translation from Dokl. Akad. Nauk SSSR 313, No. 2, 325-329 (1990).
Summary: We develop the mathematical apparatus of quantization with wave functions taking values in a non-Archimedean field \(K\). We construct a theory of generalized functions ( distributions) and integration in a non- Archimedean space \(K\). We introduce the function spaces \(L_ 2(K^ n,dx)\) and \(L_ 2(Z^ n,e^{-z\bar z} dz d\bar z)\), where \(Z\) is a quadratic extension of the field \(K\), and in those spaces we realize (with the aid of a calculus of pseudo-differential operators) the Schrödinger and Bargmann-Fock representations.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46N50 Applications of functional analysis in quantum physics
81S99 General quantum mechanics and problems of quantization
46F10 Operations with distributions and generalized functions
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