Path Integrals for Quadratic Lagrangians on p-Adic and Adelic Spaces

Abstract

Feynman's path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes $K(x^{"},t^{"};x^{\prime },t^{\prime })$ 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 $\mathbb{R}\leftrightarrow {\mathbb{Q}}_p$ and ${\mathbb{Q}}_p\leftrightarrow {\mathbb{Q}}_{p^{\prime }},p\ne p^{\prime }$. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena.

23 pages

References in corpus (1)

• p-Adic Mathematical Physics