[Paper Review] Exact value of the vacuum electromagnetic energy of a dilute dielectric ball in the mode summation method

Gaetano Lambiase, G. Scarpetta+1 more|arXiv (Cornell University)|Dec 20, 1999

Electromagnetic Simulation and Numerical Methods9 citations

TL;DR

This paper derives the exact Casimir energy of a dilute dielectric ball in the $(\epsilon_1 - \epsilon_2)^2$ approximation using a mode summation method. By applying the addition theorem for Bessel functions, it performs closed-form summation over angular momentum before integrating over imaginary frequencies, and removes linear $(\epsilon_1 - \epsilon_2)$ terms via subtraction, clarifying the role of contact terms in alternative approaches.

ABSTRACT

The exact value (in the $(\\epsilon_1-\\epsilon_2)^2$--approximation) of the Casimir energy of a dilute dielectric ball is derived by making use of a simple and clear mode summation method. The addition theorem for the Bessel functions enables one to carry out in a closed form the summation over the angular momentum before the integration over the imaginary frequencies. The linear in $(\\epsilon_1-\\epsilon_2)$ terms in the vacuum energy are removed by appropriate subtraction. The role of the contact terms used in other approaches to this problem is elucidated.

Motivation & Objective

• To derive the exact value of the vacuum electromagnetic energy for a dilute dielectric ball in the $(\epsilon_1 - \epsilon_2)^2$ approximation.
• To address the challenge of summing over angular momentum modes in the mode summation approach for spherical geometries.
• To eliminate linear-in-$(\epsilon_1 - \epsilon_2)$ terms in the vacuum energy through appropriate subtraction.
• To clarify the physical role of contact terms used in other theoretical treatments of the Casimir effect for dielectric balls.

Proposed method

• Utilizes the mode summation method to compute the vacuum electromagnetic energy of a dielectric ball.
• Applies the addition theorem for Bessel functions to perform closed-form summation over angular momentum quantum numbers before integration.
• Performs integration over imaginary frequencies after angular momentum summation is completed.
• Subtracts linear $(\epsilon_1 - \epsilon_2)$ terms to ensure consistency with physical expectations and remove unphysical contributions.
• Compares the resulting expression to those derived using contact term methods, elucidating their role.

Experimental results

Research questions

• RQ1 What is the exact value of the Casimir energy for a dilute dielectric ball in the $(\epsilon_1 - \epsilon_2)^2$ approximation?
• RQ2 How can the mode summation method be systematically applied to spherical dielectric systems with closed-form angular momentum summation?
• RQ3 Why do linear $(\epsilon_1 - \epsilon_2)$ terms appear in the vacuum energy, and how should they be removed?
• RQ4 What is the physical significance of contact terms in alternative approaches to the Casimir energy of dielectric balls?

Key findings

• The Casimir energy is derived in closed form using the mode summation method with exact angular momentum summation via the Bessel function addition theorem.
• The linear-in-$(\epsilon_1 - \epsilon_2)$ terms in the vacuum energy are removed through a consistent subtraction procedure.
• The resulting expression is free of divergences and physically meaningful in the $(\epsilon_1 - \epsilon_2)^2$ approximation.
• The role of contact terms in other approaches is clarified: they are not fundamental but emerge as artifacts of regularization schemes.