Adelic Harmonic Oscillator

TL;DR

This paper develops a unified adelic quantum mechanics framework using Weyl quantization, illustrating it with an exact harmonic oscillator model that connects to the Riemann zeta function and suggests high-energy adelic matter.

Contribution

It introduces a novel adelic quantum mechanics formulation that unifies ordinary and p-adic quantum theories, with an exact harmonic oscillator example and links to number theory.

Findings

Eigenstates are Schwartz-Bruhat functions.

Mellin transform relates to Riemann zeta function.

Suggests existence of adelic matter at high energies.

Abstract

Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and $p$-adic quantum mechanics on an equal footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple, exact and instructive adelic model. Eigenstates are Schwartz-Bruhat functions. The Mellin transform of a simplest vacuum state leads to the well known functional relation for the Riemann zeta function. Some expectation values are calculated. The existence of adelic matter at very high energies is suggested.