Minimal Length and the Existence of Some Infinitesimal Quantities in Quantum Theory and Gravity

TL;DR

This paper argues that theories with a minimal length eliminate infinitesimal quantities, replacing them with finite minimal variations dependent on energy, thus requiring a fundamental reformulation of quantum theory and gravity.

Contribution

It demonstrates that minimal length theories inherently remove infinitesimal quantities and necessitate a new mathematical framework for quantum theory and gravity.

Findings

Infinitesimal variations are replaced by finite minimal variations.

Minimal length theories alter the mathematical structure of quantum and gravitational theories.

Results closely approximate low-energy theories but with fundamentally different mathematics.

Abstract

In this work it is demonstrated that, provided a theory involves a minimal length, this theory must be free from such infinitesimal quantities as infinitely small variations in surface of the holographic screen, its volume, and entropy. The corresponding infinitesimal quantities in this case must be replaced by the "minimal variations possible" -- finite quantities dependent on the existent energies. As a result, the initial low-energy theory (quantum theory or general relativity) inevitably must be replaced by a minimal-length theory that gives very close results but operates with absolutely other mathematical apparatus.