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

TL;DR

This paper analytically evaluates Feynman path integrals for quadratic Lagrangians in p-adic and adelic quantum mechanics, establishing invariant formulas across different number fields and highlighting the fundamental role of adelic path integrals.

Contribution

It provides explicit, invariant formulas for path integrals in p-adic and adelic spaces, extending quantum mechanics to these number fields.

Findings

Analytic expressions for quadratic Lagrangian path integrals in p-adic and adelic spaces.

Invariance of formulas under interchange of real and p-adic number fields.

Adelic path integral as a fundamental object in quantum physics.

Abstract

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 ${\mathbb R} \leftrightarrow {\mathbb Q}_p$ 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.