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p-adic path integrals. (English) Zbl 0729.58057

Summary: The definition of a path integral is proposed. The method suggested is analogous to Lagrange’s formulation of a path integral used in ordinary quantum mechanics. The notation of linear order on the set of p-adic number, p-adic segment, p-adic Lagrangian, integral of p-adic function of one variable and classical action are introduced. It is proven that if the action is stationary at some trajectory then the Euler-Lagrange equations are satisfied on this trajectory. A finite approximation of a path integral is constructed and the kernel of the operator of evolution is calculated for the case of p-adic harmonic oscillator.

MSC:

58Z05 Applications of global analysis to the sciences
81S40 Path integrals in quantum mechanics
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