The Casimir Effect in Minimal Length Theories based on a Generalized Uncertainity Principle

TL;DR

This paper investigates how a minimal length, derived from a Generalized Uncertainty Principle, affects the Casimir effect between metallic plates, revealing a correction that scales as the fifth power of the separation distance and remains attractive.

Contribution

It provides the first calculation of minimal length corrections to the Casimir effect within specific GUP models, showing a distinct $a^{-5}$ scaling and consistent attractive nature.

Findings

Correction scales as $a^{-5}$, different from the standard $a^{-3}$ QED result.

The correction is always attractive across models.

Models can be distinguished by the magnitude of the correction.

Abstract

We study the corrections to the Casimir effect in the classical geometry of two parallel metallic plates, separated by a distance $a$, due to the presence of a minimal length ($\hbar\sqrt{(\beta)}$) arising from quantum mechanical models based on a Generalized Uncertainty Principle (GUP). The approach for the quantization of the electromagnetic field is based on projecting onto the maximally localized states of a few specific GUP models and was previously developed to study the Casimir-Polder effect. For each model we compute the lowest order correction in the minimal length to the Casimir energy and find that it scales with the fifth power of the distance between the plates $a^{-5}$ as opposed to the well known QED result which scales as $a^{-3}$ and, contrary to previous claims, we find that it is always attractive. The various GUP models can be in principle differentiated by the strength of the correction to the Casimir energy as every model is characterized by a specific multiplicative numerical constant.