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.

中文翻译：

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卡西米尔力的统一理论：体积和表面公式

电磁波动引起的现象，如卡西米尔力的原理是众所周知的。然而，最近的实验进展需要通用且有效的方法来计算这些力。虽然文献中提出了几种方法，但它们之间的联系通常并不完全清楚，其中一些已作为纯数值技术引入。在这里，我们提出了一种基于麦克斯韦应力张量和路径积分量化的 Casimir 力和自由能的统一方法。结果以描述相应电流波动的体或表面算子形式呈现。我们的表面方法产生了 Casimir 自由能的新公式。路径积分在拉格朗日公式和哈密顿公式中都给出，产生不同的表面算子和等效的自由能表达式。我们将我们的方法与以前开发的数值方法和散射方法进行比较。我们方法的实际应用以 Lifshitz 公式的推导为例。