Koslowski-Sahlmann表示：量子位形空间
The Koslowski-Sahlmann representation: Quantum Configuration Space
摘要：
Koslowski-Sahlmann（KS）表象是圈量子引力（LQG）离散空间几何表象的推广，以适应光滑空间几何标记的状态。如最近所示，KS表示支持，除了完整性和通量算子的行动，行动的运营商，这是量子对应的某些连接依赖功能被称为“背景指数”。在这里，我们证明KS表示显示了以下性质，这些性质与LQG性质完全对应：（i）由$SU(2)$完整性和“$U(1)$”背景指数构成的交换$*$代数可完备为$C^*$代数（ii）半解析$SU(2)$联络空间在该代数的谱中是拓扑稠密的（iii）在这个谱上存在一个测度，对于这个测度，KS Hilbert空间被实现为谱上的平方可积函数的空间（iv）这个谱允许作为$SU(2)$和$U(1)$的有限个拷贝的投影极限的表征（v）以与LQG（holonomy-flux）代数完全相同的方式从柱函数和它们的导子构造作为KS表示基础的代数，除了KS柱函数依赖于holonomy和背景指数之外，这种额外的依赖性是KS和LQG代数之间的差异的原因。虽然这些结果是在紧空间中得到的，但它们有望在渐近平坦的情况下用于KS表示的构造。
英文摘要：
The Koslowski-Sahlmann (KS) representation is a generalization of the representation underlying the discrete spatial geometry of Loop Quantum Gravity (LQG), to accommodate states labelled by smooth spatial geometries. As shown recently, the KS representation supports, in addition to the action of the holonomy and flux operators, the action of operators which are the quantum counterparts of certain connection dependent functions known as "background exponentials". Here we show that the KS representation displays the following properties which are the exact counterparts of LQG ones: (i) the abelian $*$ algebra of $SU(2)$ holonomies and `$U(1)$' background exponentials can be completed to a $C^*$ algebra (ii) the space of semianalytic $SU(2)$ connections is topologically dense in the spectrum of this algebra (iii) there exists a measure on this spectrum for which the KS Hilbert space is realised as the space of square integrable functions on the spectrum (iv) the spectrum admits a characterization as a projective limit of finite numbers of copies of $SU(2)$ and $U(1)$ (v) the algebra underlying the KS representation is constructed from cylindrical functions and their derivations in exactly the same way as the LQG (holonomy-flux) algebra except that the KS cylindrical functions depend on the holonomies and the background exponentials, this extra dependence being responsible for the differences between the KS and LQG algebras. While these results are obtained for compact spaces, they are expected to be of use for the construction of the KS representation in the asymptotically flat case.
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