Casimir Forces between Beads on Strings and Membranes

TL;DR

This paper develops a formalism to calculate quantum fluctuation-induced forces between beads on membranes, revealing how these forces depend on membrane properties, dimension, and temperature, with interactions turning off beyond a critical dimension.

Contribution

The paper introduces a general method to compute Casimir-like forces on beads attached to membranes, accounting for various physical parameters and identifying a critical dimension where interactions vanish.

Findings

Interactions are attractive when present.

For finite temperature, forces decay exponentially at large distances.

Interactions cease beyond a certain critical membrane dimension.

Abstract

We develop a general formalism to calculate the force between beads attached to a flat $d$-dimensional membrane due to the quantum fluctuations of the membrane. The interaction potential is derived as a function of $d$ and the membrane energy density, tension, stiffness and temperature. We find that the induced interactions turn off when $d$ exceeds a certain critical dimension. The potential is attractive in all cases where it is non-zero and at finite temperature falls off exponentially for large distances.