On Uncertainty Relations at Planck's Scale

TL;DR

This paper explores the extension of uncertainty principles to p-adic and adelic quantum mechanics, proposing new algebraic frameworks that could enable probing space at scales smaller than the Planck length.

Contribution

It introduces p-adic and adelic analogs of the Heisenberg algebra and uncertainty relations, expanding the theoretical foundation of quantum mechanics at the Planck scale.

Findings

P-adic analogs of the Heisenberg algebra are formulated.

Adelic quantum mechanics offers a new framework for sub-Planckian space exploration.

The approach suggests potential for new insights into quantum gravity phenomena.

Abstract

A brief review of the previous research on the Heisenberg uncertainty relations at the Planck scale is given. In this work, investigation of the uncertainty principle extends to p-adic and adelic quantum mechanics. In particular, p-adic analogs of the Heisenberg algebra and uncertainty relation are introduced. Unlike ordinary quantum theory, adelic quantum approach provides a promising framework to probe space below the Planck length.