Abstract

By systematically studying the infinite degeneracy and constants of motion in the Landau problem, we obtain a central extension of the Euclidean group in two dimension as a dynamical symmetry group, and \(Sp(2,\mathbb{R})\) as the spectrum generating group, irrespective of the choice of the gauge. The method of group contraction plays an important role. Dirac’s remarkable representation of the \(SO(3,2)\) group and the isomorphism of this group with \(Sp(4,\mathbb{R})\) are revisited. New insights are gained into the meaning of a two-oscillator system in the Dirac representation. It is argued that because even the two-dimensional isotropic oscillator with the \(SU(2)\) dynamical symmetry group does not arise in the Landau problem, the relevance or applicability of the \(SO(3,2)\) group is invalidated. A modified Landau–Zeeman model is discussed in which the \(SO(3,2)\) group isomorphic to \(Sp(4,\mathbb{R})\) can arise naturally.

中文翻译：

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$$SO(3,2)$$群的狄拉克表示和朗道问题

摘要

通过系统地研究朗道问题中的无限简并性和运动常数，我们得到了欧几里得群在二维上的中心扩张作为动力学对称群，并得到\(Sp(2,\mathbb{R})\ )作为频谱生成组，与仪表的选择无关。团体收缩的方法起着重要的作用。狄拉克对\(SO(3,2)\)群的显着表示以及该群与\(Sp(4,\mathbb{R})\)的同构被重新审视。对狄拉克表示中双振荡器系统的含义有了新的见解。有人认为，由于即使具有\(SU(2)\)动力学对称群的二维各向同性振子在朗道问题中也不会出现，因此\(SO(3,2)\)群的相关性或适用性无效。讨论了改进的朗道-塞曼模型，其中与\(Sp(4,\mathbb{R})\)同构的\(SO(3,2)\)群可以自然出现。