Hopfield模型再访：协方差和量化
The Hopfield model revisited: Covariance and Quantization
摘要：
量子电动力学在电介质中有几种可能的应用，它们需要对电磁场与物质场的相互作用进行量子描述。相关的量子模型可以指宏观电磁场，或者，在替代方案中，介观场（极化场）描述电磁场和物质场之间的有效相互作用。我们采用后一种方法，并专注于Hopfield模型的电磁场在一个框架中的电介质色散介质中的时空相关的介观参数发生，如磁化率，物质共振频率，以及电磁场和极化场之间的耦合。我们最直接的目标是用唯象的方法描述非线性介质中由克尔效应引起的时空变化的介电扰动。这种扩展的模型是通过一个洛伦兹不变的拉格朗日，恒定的微观参数，并在休息帧，符合标准的装置。此外，我们推导出一个协变标量积，并提供了一个协变量化方案，它保持考虑到隐含在模型中的约束。指出了可行的应用实例。
英文摘要：
There are several possible applications of quantum electrodynamics in dielectric media which require a quantum description for the electromagnetic field interacting with matter fields. The associated quantum models can refer to macroscopic electromagnetic fields or, in alternative, to mesoscopic fields (polarization fields) describing an effective interaction between electromagnetic field and matter fields. We adopt the latter approach, and focus on the Hopfield model for the electromagnetic field in a dielectric dispersive medium in a framework in which space-time dependent mesoscopic parameters occur, like susceptibility, matter resonance frequency, and also coupling between electromagnetic field and polarization field. Our most direct goal is to describe in a phenomenological way a space-time varying dielectric perturbation induced by means of the Kerr effect in nonlinear dielectric media. This extension of the model is implemented by means of a Lorentz-invariant Lagrangian which, for constant microscopic parameters, and in the rest frame, coincides with the standard one. Moreover, we deduce a covariant scalar product and provide a covariant quantization scheme which keeps into account the constraints implicit in the model. Examples of viable applications are indicated.
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