## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Zoran Rakić

### 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 ${\cal K}(x^{''},t^{''};x',t')$ 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 ${\mathbb R} \leftrightarrow {\mathbb Q}_p$ and ${\mathbb Q}_p \leftrightarrow {\mathbb Q}_{p'} \, ,\, p\neq p'$. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena. This talk is based on joint work with Branko Dragovich, see [1].

[1] B. Dragovich and Z. Rakic, Path Integrals for Quadratic Lagrangians on $p$-Adic and Adelic Spaces'', $p$-Adic Numbers, Ultrametric Analysis and Applications \textbf{2} (4), 322--340 (2010), [arXiv:1011.6589 [math-ph]].