On p-adic path integral

Feynman's path integral is generalized to quantum mechanics on p-adic space and time. Such p-adic path integral is analytically evaluated for quadratic Lagrangians. Obtained result has the same form as that one in ordinary quantum mechanics. 1. It is well known that dynamical evolution of any one-dimensional quantum-mechanical system, described by a wave function Ψ(x, t), is given by Ψ(x ′′ , t ′′) = K(x ′′ , t ′′ ; x ′ , t ′)Ψ(x ′ , t ′)dx ′ , (1) where K(x ′′ , t ′′ ; x ′ , t ′) is the kernel of the corresponding unitary operator acting as follows: Ψ(t ′′) = U(t ′′ , t ′)Ψ(t ′). (2) K(x ′′ , t ′′ ; x ′ , t ′) is also called Green's function, or the quantum-mechanical prop-agator, and the probability amplitude to go a particle from a point (x ′ , t ′) to a point (x ′′ , t ′′). One can easily deduce the following three general properties: