On path integrals for wave functions taking $p$-adic values

TL;DR

This paper develops a $p$-adic path integral framework for wave functions valued in $p$-adic fields, computing the Feynman propagator for free particles and showing parallels with classical quantum mechanics.

Contribution

It introduces a novel $p$-adic path integral approach for wave functions and derives the Feynman propagator in this context, extending quantum formalism into $p$-adic analysis.

Findings

$p$-adic path integral constructed for wave functions

Feynman propagator computed for free particles

Results resemble classical quantum mechanics counterparts

Abstract

In this paper, we construct a $p$-adic path integral via $p$-adic multiple integrals. This integral describes the evolution of a wave function $\Psi(x)$, which is defined as a map from a domain in $\mathbb{C}_{p}$ to $\mathbb{C}_{p}$. We also compute the Feynman propagator for free particles, demonstrating that the result obtained is similar to the classical counterparts.