The Geroch group is an infinite dimensional transitive group of symmetries of classical cylindrically symmetric gravitational waves which acts by non-canonical transformations on the phase space of these waves. Here this symmetry is rederived and the unique Poisson bracket on the Geroch group which makes its action on the gravitational phase space Lie-Poisson is obtained. Two possible notions of asymptotic flatness are proposed that are compatible with the Poisson bracket on the phase space, and corresponding asymptotic flatness preserving subgroups of the Geroch group are defined which turn out to be compatible with the Poisson bracket on the group. A quantization of the Geroch group is proposed that is similar to, but distinct from, the $\mathfrak{sl}_2$ Yangian, and a certain action of this quantum Geroch group on gravitational observables is shown to preserve the commutation relations of Korotkin and Samtleben's quantization of asymptotically flat cylindrically symmetric gravitational waves. The action also preserves three of the additional conditions that define their quantization. It is conjectured that the action preserves the remaining two conditions (asymptotic flatness and a unit determinant condition on a certain basic field) as well and is, in fact, a symmetry of their model. Our results on the quantum theory are formal, but a possible rigorous formulation based on algebraic quantum theory is outlined.

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关于经典和量子 Geroch 群

Geroch 群是经典圆柱对称引力波对称性的无限维传递群，它通过对这些波的相空间的非正则变换起作用。这里重新推导了这种对称性，并获得了 Geroch 群上唯一的 Poisson 括号，它对引力相空间 Lie-Poisson 起作用。提出了两种可能的渐近平坦度概念，它们与相空间上的泊松括号兼容，并定义了 Geroch 群的相应渐近平坦度保持子群，结果证明它们与群上的泊松括号兼容。提出了 Geroch 群的量化，它类似于但不同于 $\mathfrak{sl}_2$ Yangian，并且这个量子 Geroch 群对引力可观测量的某种作用被证明保持了 Korotkin 和 Samtleben 对渐近平坦圆柱对称引力波的量子化的交换关系。该操作还保留了定义其量化的三个附加条件。据推测，该动作也保留了剩余的两个条件（渐近平坦度和某个基本场上的单位行列式条件），并且实际上是他们模型的对称性。我们在量子理论上的结果是形式化的，但概述了基于代数量子理论的可能的严格公式。该操作还保留了定义其量化的三个附加条件。据推测，该动作也保留了剩余的两个条件（渐近平坦度和某个基本场上的单位行列式条件），并且实际上是他们模型的对称性。我们在量子理论上的结果是形式化的，但概述了基于代数量子理论的可能的严格公式。该操作还保留了定义其量化的三个附加条件。据推测，该动作也保留了剩余的两个条件（渐近平坦度和某个基本场上的单位行列式条件），并且实际上是他们模型的对称性。我们在量子理论上的结果是形式化的，但概述了基于代数量子理论的可能的严格公式。