Path integrals for quadratic lagrangians on p-adic and adelic spaces

Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized to any finite-dimensional spaces. These general formulas are presented in the form which is invariant under interchange of the number fields ℝ ↔ ℚ _p and ℚ ↔ ℚ _p , p ≠ p′. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena.