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Journal of Mathematical Physics

On path integrals for wave functions taking p -adic values

Su Hu, Min-Soo Kim

Published Oct 1, 2026• 0 citations
Publisher

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), defined as a map from a domain in Cp to Cp. Unlike the standard p-adic quantum mechanics formulated for functions Qp→C, our construction works directly on Cp-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.

DOI
10.1063/5.0316612

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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), defined as a map from a domain in Cp to Cp. Unlike the standard p-adic quantum mechanics formulated for functions Qp→C, our construction works directly on Cp-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.
In this paper, we construct a ppp-adic path integral via ppp-adic multiple integrals. This integral describes the evolution of a wave function Ψ(x)Ψ(x)Ψ(x), defined as a map from a domain in Cp\mathbb{C}_pCp​ to Cp\mathbb{C}_pCp​. Unlike the standard ppp-adic quantum mechanics formulated for functions Qp→C\mathbb{Q}_p \to \mathbb{C}Qp​→C, our construction works directly on Cp\mathbb{C}_pCp​-valued wave functions. Using ppp-adic Dirac delta measures and ppp-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 ppp-adic functional integration.
The definition of a path integral is proposed. The method suggested is analogous to Lagrange’s formulation of a path integral used in ordinary quantum mechanics. The notation of linear order on the set of p-adic number, p-adic segment, p-adic Lagrangian, integral of p-adic function of one variable and classical action are introduced. It is proven that if the action is stationary at some trajectory then the Euler–Lagrange equations are satisfied on this trajectory. A finite approximation of a path integral is constructed and the kernel of the operator of evolution is calculated for the case of p-adic harmonic oscillator.
A formulation of quantum mechanics on p-adic number fields is presented. Quantum amplitudes are taken as complex functions of p-adic variables and it is shown how the Weyl approach to quantum mechanics can be generalized to the p-adic case. The p-adic analogs of simple one-dimensional systems (free particle, compact and noncompact oscillators) are defined by a ‘‘group of motion,’’ which is an Abelian subgroup of SL (2,Qp). In each case the evolution operator is a unitary representation of the appropriate group. Its spectrum is given by characters and its eigenstates are calculated.
We propose a natural, parameter-free, discrete-variable formulation of Feynman path integrals. We show that for discrete-variable quantum systems, Feynman path integrals take the form of walks on the graph whose weighted adjacency matrix is the Hamiltonian. By working out expressions for the partition function and transition amplitudes of discretized versions of continuous-variable quantum systems, and then taking the continuum limit, we explicitly recover Feynman's continuous-variable path integrals. We also discuss the implications of our result.