Unifying Theory for Casimir Forces: Bulk and Surface Formulations

TL;DR

This paper introduces a unified theoretical framework for Casimir forces that connects bulk and surface formulations using electromagnetic principles, enhancing computational efficiency and clarity.

Contribution

It presents a comprehensive approach combining Maxwell stress tensor and path integral methods, unifying existing techniques and deriving a new surface-based formula for Casimir free energy.

Findings

Derived a new surface operator formula for Casimir free energy

Unified bulk and surface approaches within a single framework

Validated methods by deriving the Lifshitz formula

Abstract

The principles of the electromagnetic fluctuation-induced phenomena such as Casimir forces are well understood. However, recent experimental advances require universal and efficient methods to compute these forces. While several approaches have been proposed in the literature, their connection is often not entirely clear, and some of them have been introduced as purely numerical techniques. Here we present a unifying approach for the Casimir force and free energy that builds on both the Maxwell stress tensor and path integral quantization. The result is presented in terms of either bulk or surface operators that describe corresponding current fluctuations. Our surface approach yields a novel formula for the Casimir free energy. The path integral is presented both within a Lagrange and Hamiltonian formulation yielding different surface operators and expressions for the free energy that are equivalent. We compare our approaches to previously developed numerical methods and the scattering approach. The practical application of our methods is exemplified by the derivation of the Lifshitz formula.