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

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 $\Psi (x)$, defined as a map from a domain in ${\mathbb{C}}_p$ to ${\mathbb{C}}_p$. Unlike the standard $p$-adic quantum mechanics formulated for functions ${\mathbb{Q}}_p\to \mathbb{C}$, our construction works directly on ${\mathbb{C}}_p$-valued wave functions. Using $p$-adic Dirac delta measures and $p$-adic additive characters, we define the propagator as a finite-partition multiple integral, which allows explicit iterative integration. As an application, we compute the Feynman propagator for free particles and obtain a closed-form expression analogous to the classical counterpart. Our method provides a discrete, constructive alternative that complements the existing distribution-theoretic frameworks for $p$-adic functional integration.

12 pages