Minimum-length deformed QM/QFT, issues and problems

TL;DR

This paper explores the implications of minimum-length deformed quantum mechanics and quantum field theory, highlighting issues with singularities, the nature of wave-functions, and the classical limit, revealing fundamental challenges and insights.

Contribution

It provides a detailed analysis of how minimum-length deformed quantum mechanics affects singularity resolution, wave-function interpretation, and the classical limit, identifying key issues and limitations.

Findings

Singularity resolution is linked to nonlocalizable states beneath the Planck length.

Corrections to the Hamiltonian do not universally prevent singularities.

The classical limit of the theory has significant and problematic consequences.

Abstract

Using a particular Hilbert space representation of minimum-length deformed quantum mechanics, we show that the resolution of the wave-function singularities for strongly attractive potentials, as well as cosmological singularity in the framework of a minisuperspace approximation, is uniquely tied to the fact that this sort of quantum mechanics implies the reduced Hilbert space of state-vectors consisting of the functions nonlocalizable beneath the Planck length. (Corrections to the Hamiltonian do not provide such an universal mechanism for avoiding singularities.) Following this discussion, as a next step we take a critical view of the meaning of wave-function in such a quantum theory. For this reason we focus on the construction of current vector and the subsequent continuity equation. Some issues gained in the framework of this discussion are then considered in the context of field theory. Finally, we discuss the classical limit of the minimum-length deformed quantum mechanics and its dramatic consequences.