In this paper, Radiative Correction (RC) to the Casimir energy was computed for the self-interacting massive/massless Lifshitz-like scalar field, confined between a pair of plates with Dirichlet and Mixed boundary conditions in 3 + 1 dimensions. Moreover, using the results obtained for the Dirichlet Casimir energy, the RC to the Casimir energy for Periodic and Neumann boundary conditions were also draw outed. To renormalize the bare parameters of the Lagrangian, a systematic perturbation expansion was used in which the counterterms were automatically obtained in a position-dependent manner. In our view, the position dependency of the counterterm was allowed, since it reflected the effects of the boundary condition imposed or the background space in the problem. All the answers obtained for the Casimir energy were consistent with well-known physical expects. In a language of graphs, the Casimir energy for the massive Lifshitz-like scalar field confined with four boundary conditions (Dirichlet, Neumann, Mixed, and Periodic) was also compared to each other, and as a concluding remark, the sign and magnitude of their values were discussed.

中文翻译：

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类 Lifshitz 理论中 Casimir 能量的单环修正

在本文中，对卡西米尔能量的辐射校正 (RC) 计算了自相互作用的大质量/无质量类 Lifshitz 标量场，限制在具有狄利克雷和混合边界条件的一对板之间3 + 1方面。此外，利用 Dirichlet Casimir 能量获得的结果，还得出了周期性和 Neumann 边界条件下 Casimir 能量的 RC。为了重新归一化拉格朗日的裸参数，使用了系统的扰动扩展，其中反项以位置相关的方式自动获得。在我们看来，反项的位置依赖性是允许的，因为它反映了施加的边界条件或问题中背景空间的影响。卡西米尔能量获得的所有答案都与众所周知的物理预期一致。在图形语言中，还比较了受限于四个边界条件（Dirichlet、Neumann、Mixed 和 Periodic）的大型 Lifshitz 样标量场的 Casimir 能量，