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.
In this paper, we construct a
p-adic path integral via
p-adic multiple integrals. This integral describes the evolution of a wave function
Ψ(x), defined as a map from a domain in
Cp to
Cp. Unlike the standard
p-adic quantum mechanics formulated for functions
Qp→C, our construction works directly on
Cp-valued wave functions. Using
p-adic Dirac delta measures and
p-adic additive characters, we define the propagator as a finite-partition multiple integral, which allows explicit iterative integration. As an application, we compute the Feynman propagator for free particles and obtain a closed-form expression analogous to the classical counterpart. Our method provides a discrete, constructive alternative that complements the existing distribution-theoretic frameworks for
p-adic functional integration.