$p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Zoran RakiÄ‡## Path Integrals for Quadratic Lagrangians on $p$-Adic and Adelic SpacesAbstract
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]]. |