This paper provides a review of the complicated problems in Lifshitz theory describing the Casimir force between real material plates composed of metals and dielectrics, including different approaches to their resolution. For both metallic plates with perfect crystal lattices and any dielectric plates, we show that the Casimir entropy calculated in the framework of Lifshitz theory violates the Nernst heat theorem when the well-approved dielectric functions are used in computations. The respective theoretical Casimir forces are excluded by the measurement data of numerous precision experiments. In the literature, this situation has been called the Casimir puzzle and the Casimir conundrum for the cases of metallic and dielectric plates, respectively. This review presents a summary of both the main theoretical and experimental findings on this subject. Next, a discussion is provided of the main approaches proposed in the literature to bring the Lifshitz theory into agreement with the measurement data and with the laws of thermodynamics. Special attention is paid to the recently suggested spatially nonlocal Drude-like response functions, which consider the relaxation properties of conduction electrons, as does the standard Drude model, but lead to the theoretical results being in agreement with both thermodynamics and the measurement data through the alternative response to quantum fluctuations of the mass shell. Further advances and trends in this field of research are discussed.

中文翻译：

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卡西米尔难题和卡西米尔难题：发现和寻找解决方案

本文回顾了Lifshitz理论中的复杂问题，这些问题描述了由金属和电介质组成的真实材料板之间的卡西米尔力，包括不同的解析方法。对于具有完美晶格的金属板和任何介电板，我们都表明，当在计算中使用公认的介电函数时，在Lifshitz理论框架内计算出的Casimir熵违反了能斯特定理。大量精确实验的测量数据排除了相应的理论卡西米尔力。在文献中，对于金属板和电介质板，这种情况分别称为卡西米尔难题和卡西米尔难题。这篇综述总结了关于该主题的主要理论和实验发现。接下来，讨论了文献中提出的使Lifshitz理论与测量数据和热力学定律一致的主要方法。特别注意最近建议的空间非局部类Drude响应函数，该函数考虑了传导电子的弛豫特性，这与标准Drude模型一样，但通过热力学和测量数据得出的理论结果与热力学和测量数据一致对质量壳的量子涨落的另一种反应。讨论了该研究领域的进一步进展和趋势。讨论了文献中提出的使Lifshitz理论与测量数据和热力学定律一致的主要方法。特别注意最近建议的空间非局部类Drude响应函数，该函数考虑了传导电子的弛豫特性，这与标准Drude模型一样，但通过热力学和测量数据得出的理论结果与热力学和测量数据一致对质量壳的量子涨落的另一种反应。讨论了该研究领域的进一步进展和趋势。讨论了文献中提出的使Lifshitz理论与测量数据和热力学定律一致的主要方法。特别注意最近建议的空间非局部类Drude响应函数，该函数考虑了传导电子的弛豫特性，这与标准Drude模型一样，但通过热力学和测量数据得出的理论结果与热力学和测量数据一致对质量壳的量子涨落的另一种反应。讨论了该研究领域的进一步进展和趋势。与标准Drude模型一样，它们考虑了传导电子的弛豫特性，但通过对质壳的量子涨落的交替响应，得出的理论结果与热力学和测量数据一致。讨论了该研究领域的进一步进展和趋势。它考虑了传导电子的弛豫特性，就像标准的Drude模型一样，但是通过对质壳的量子涨落的交替响应，使得理论结果与热力学和测量数据相吻合。讨论了该研究领域的进一步进展和趋势。