Towards Adelic Noncommutative Quantum Mechanics

TL;DR

This paper explores the integration of adelic, noncommutative, and nonarchimedean geometries into quantum mechanics, proposing a framework for adelic noncommutative quantum theory and highlighting its mathematical foundations and potential research directions.

Contribution

It introduces a novel approach combining adelic and noncommutative geometries in quantum mechanics, extending the Moyal product and suggesting future research avenues.

Findings

Relations between noncommutativity and nonarchimedean spaces identified

Extended Moyal product proposed for adelic noncommutative quantum mechanics

Foundations and potential research questions outlined

Abstract

A motivation of using noncommutative and nonarchimedean geometry on very short distances is given. Besides some mathematical preliminaries, we give a short introduction in adelic quantum mechanics. We also recall to basic ideas and tools embedded in q-deformed and noncommutative quantum mechanics. A rather fundamental approach, called deformation quantization, is noted. A few relations between noncommutativity and nonarchimedean spaces as well as similarities between corresponding quantum theories on them are pointed out. An extended Moyal product in a proposed form of adelic noncommutative quantum mechanics is considered. We suggest some question for future investigations.