We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables — which should lead to fractal patterns on the manifold of squeezed states.

中文翻译：

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量子斑点上的随机游走

我们描述了辛群对压缩态齐次空间（量子斑点）的作用，并将该作用扩展到半群。然后，我们将 metaplectic 表示扩展到 metaplectic（或振荡器）半群，并使用 Bargmann-Fock 空间研究这种扩展的性质。分析了挤压的形状几何，并提出将辛半群中的非对易元素用于同时监测非对易量子变量——这应该会导致挤压态流形上的分形模式。