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

TL;DR

This work engineers a p-adic quantum-dynamics framework by constructing a Cp-valued path integral and defining a propagator kernel using p-adic measures and additive characters. It leverages Zelenov's line-segment subdivision to realize p-adic integration over discretized trajectories and handles Cp-valued wavefunctions in the absence of a general Laplacian theory. The main result for free particles is a concrete propagator expressed as a Cp-exponential depending on the squared endpoint separation with a p-adic scale, aligning with classical intuition for Green-function-like evolution. Overall, the paper demonstrates a viable non-Archimedean approach to quantum evolution and highlights how p-adic techniques can yield tractable expressions analogous to their real-number 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 $Ψ(x)$, 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.

Figures (2)

• Figure 1: The process of $p$-adic path integral
• Figure :