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Mathematical methods of non-Archimedean physics. (English. Russian original) Zbl 0722.46040
Russ. Math. Surv. 45, No. 4, 87-125 (1990); translation from Usp. Mat. Nauk 45, No. 4(274), 79-110 (1990).
A review is presented of mathematical methods and results concerning non- archimedean quantum theory. It is assumed that the wave functions are defined over the non-archimedean space and take their values in non- archimedean fields. The models of nonarchimedean quantum mechanics, field theory and string theory are considered. The counterparts of Bargmann as well as Schrödinger representations are constructed. The existence and uniqueness theorems for the Schrödinger, Heisenberg and Liouville equation are proven. Non-archimedean scalar boson field is quantized with the help of Feynman path integrals. The non-archimedean counterpart of Klein-Gordon equation is investigated. It is shown that within this framework a new characteristic of electron arises, the so called non- archimedean color. Finally, a non-archimedean bosonic string theory is quantized in the light cone.

##### 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 81T05 Axiomatic quantum field theory; operator algebras 81S40 Path integrals in quantum mechanics 47S10 Operator theory over fields other than $$\mathbb{R}$$, $$\mathbb{C}$$ or the quaternions; non-Archimedean operator theory 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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