Realizations of algebras in terms of canonical or bosonic variables can often be used to simplify calculations and to exhibit underlying properties. There is a long history of using such methods in order to study symmetry groups related to collective motion, for instance in nuclear shell models. Here, related questions are addressed for algebras obtained by turning the quantum commutator into a Poisson bracket on moments of a quantum state, truncated to a given order. In this application, canonical realizations allow one to express the quantum back-reaction of moments on basic expectation values by means of effective potentials. In order to match degrees of freedom, faithfulness of the realization is important, which requires that, at least locally, the space of moments as a Poisson manifold is realized by a complete set of Casimir-Darboux coordinates in local charts. A systematic method to derive such variables is presented and applied to certain sets of moments which are important for physical applications. If only second-order moments are considered, their Poisson-bracket relations are isomorphic to the Lie bracket of sp(2N,R), providing an interesting link with realizations of nuclear shell models.

中文翻译：

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半经典截断的忠实实现

在规范变量或玻色变量方面的代数实现通常可用于简化计算并展示潜在属性。使用此类方法研究与集体运动相关的对称群已有很长的历史，例如在核壳模型中。在这里，通过将量子换向器转换为量子态矩的泊松括号获得代数的相关问题得到解决，截断为给定阶数。在这个应用中，规范实现允许人们通过有效势来表达矩在基本期望值上的量子逆反应。为了匹配自由度，实现的忠实度很重要，这要求，至少在局部，作为泊松流形的矩空间由局部图表中的一组完整的 Casimir-Darboux 坐标实现。提出了一种导出此类变量的系统方法，并将其应用于某些对物理应用很重要的矩集。如果只考虑二阶矩，它们的泊松括号关系与 sp(2N,R) 的李括号同构，提供了与核壳模型实现的有趣联系。