A riveting study of Ward-Takahashi identities is presented for a broken dilatation or scale symmetry in a generalized quantum Hall system. On using the “Peierls substitution” scheme, it is shown that noncommutativity between spatial coordinates emerges naturally at a large magnetic field limit. Thereafter, we derive a path-integral action for the corresponding noncommutative quantum system and discuss the equivalence between the considered noncommutative system and the generalized Landau problem thus rendering an effective commmutative description. We then derive an expression for the unintegrated scale or dilatation anomaly for the generalized Landau system using Fujikawa's method and is subsequently renormalised. In fact, we identify the Ward-Takahashi identities associated with broken dilatation symmetry to be anomalous which is a purely quantum effect induced from the noncommutative structure between spatial coordinates and therefore non-intuitive in nature.

对 Ward-Takahashi 恒等式进行了引人入胜的研究，以发现广义量子霍尔系统中膨胀或尺度对称性的破坏。在使用“Peierls 替代”方案时，表明空间坐标之间的非对易性在大磁场限制下自然出现。此后，我们推导出相应非对易量子系统的路径积分作用，并讨论所考虑的非对易系统与广义朗道问题之间的等价性，从而提供有效的对易描述。然后，我们使用 Fujikawa 的方法推导出广义朗道系统的未积分尺度或膨胀异常的表达式，并随后进行重整化。实际上，