In this paper we formulate the quantum theory for longitudinal (rotational free) collective electron modes, the so-called plasmons. Starting from the jellium dispersion relation, a first-quantized wave-mechanical theory is established. We show that plasmon quantum (quasi)particles are charged bosons described by complex Klein-Gordon fields satisfying a “relativistic” scalar Klein-Gordon equation in which the speed of light $c$ is replaced by a velocity $a$ on the order of the Fermi velocity. Based on a formally ($c\to a$) covariant description, the first-quantized theory is extended to the second-quantized level via a Lagrangian formalism. We show how the second-quantized theory enables the study of the plasmon quantum particles interaction with an electromagnetic gauge field via a modification of the free plasmon Lagrangian density by the minimal coupling principle. Utilizing the Weyl expansion, first- and second-quantized theories for surface-plasmon quantum particles are established in close analogy to those of the bulk plasmon quantum particle. This work opens up for the study of, e.g., squeezed, entangled, and coherent plasmon states in line with what has been studied theoretically and experimentally for photons for many years.

中文翻译：

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体和表面等离子体：波力学和二次量子化理论

在本文中，我们为纵向（自由旋转）集体电子模式（即所谓的等离子体激元）制定了量子理论。从果冻色散关系出发，建立了第一量子化波力学理论。我们表明，等离子体量子（准）粒子是由复杂的 Klein-Gordon 场描述的带电玻色子，满足“相对论”标量 Klein-Gordon 方程，其中光速$C$ 被速度取代 $\mathrm{一种}$费米速度的数量级。基于正式的 ($C\to \mathrm{一种}$) 协变描述，第一量化理论通过拉格朗日形式主义扩展到第二量化水平。我们展示了二次量子化理论如何通过最小耦合原理修改自由等离子体拉格朗日密度来研究等离子体量子粒子与电磁规范场的相互作用。利用 Weyl 展开式，建立了表面等离子体量子粒子的第一和第二量子化理论，与体等离子体量子粒子的理论非常相似。这项工作为研究例如挤压、纠缠和相干等离子体状态开辟了道路，这与多年来对光子的理论和实验研究一致。