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

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.

💡 Research Summary

The paper “On path integrals for wave functions taking p‑adic values” develops a rigorous formulation of path integrals for quantum states whose wave functions take values in the p‑adic field Cₚ. Traditional p‑adic quantum mechanics, as introduced by Vladimirov and Volovich, treats wave functions as maps from the rational p‑adic field Qₚ to the complex numbers ℂ, and the dynamics are expressed through a formal path integral that lacks a mathematically precise definition. In contrast, the authors consider wave functions Ψ(x) defined on a domain of Cₚ and taking values in Cₚ. Because a Laplacian operator for Cₚ‑valued functions is not available, the usual Schrödinger equation cannot be used. Consequently, the authors turn to a path‑integral approach.

The construction relies on the p‑adic line integral theory of Zelenov. A linear order on Qₚ is introduced via a continuous injective map ϕ: Qₚ → ℝ, which allows one to speak of intervals