标题： 关于($2+1$)维Duffin-Kemmer-Petiau振荡子在外部磁场中的注记 显示英文标题

标题： Remarks on the ($2+1$)-dimensional Duffin-Kemmer-Petiau oscillator in an external magnetic field

摘要： 这项工作重新审视了在存在外部磁场的情况下，($2+1$) 维度的 Duffin-Kemmer-Petiau 振荡器 (DKPO) 中自旋为$1$粒子的问题。通过遵循 Duffin-Kemmer-Petiau (DKP) 理论中自旋$1$部分的适当方法，文献中先前使用的$6\times 6$表示被证明可以约化为$3\times 3$不可约表示。这种方法使我们能够发现最近在各种研究中传播的结果的新方面，以及之前被忽略且需要修订的其他考虑因素。最后，我们介绍了二维 DKP 理论在凝聚态系统中的某些应用，特别是在 Lieb 点阵中的应用。 显示英文摘要

摘要： This work re-examines the issue of spin-$1$ particles in a ($2+1$)-dimensional Duffin-Kemmer-Petiau oscillator (DKPO) in the presence of an external magnetic field. By following the appropriate procedure for the spin-$1$ sector of the Duffin-Kemmer-Petiau (DKP) theory, the previously used $6\times 6$ representation in the literature is shown to be reducible to a $3\times 3$ irreducible representation. This approach enabled us to find new aspects of the results recently disseminated in various studies, as well as other considerations overlooked and requiring revision. Finally, we present some applications of two-dimensional DKP theory in condensed matter systems, particularly in Lieb lattices.