标题： 渐近平直时空的量子运动学 显示英文标题

标题： A quantum kinematics for asymptotically flat spacetimes

摘要： 我们基于Koslowski-Sahlmann(KS)表示法，构建了渐近平坦时空的量子运动学。 KS表示法是对环量子引力(LQG)所依赖表示的一种推广，它除了支持通常的LQG算子外，还支持由“背景”$su(2)$电场标记的“背景指数算子”，这些算子是与连接相关的算子。 KS态除了对应于三重矢量一维激发的LQG态标签外，还有一个对应于描述三重矢量三维激发的“背景”电场的标签。 在量子理论中，渐近行为通过标记{\em 状态}和标记{\em 算子}的背景电场的渐近条件来控制。 对三重矢量的渐近条件被施加为对背景电场态标签的条件，同时将LQG自旋网络图标签限制在紧集上。 我们证明KS态可以作为广义连接的量子配置空间上的波函数，并且每个这样的广义连接的渐近行为由标记背景指数算子的背景电场决定。 类似于空间紧致的情况，通过群平均技术施加高斯定律和微分同胚约束，以获得一个大的规范不变态的区域。 结果显示，这个区域支持渐近旋转和平移群的酉作用，并且如Friedman和Sorkin所预期的那样，对于适当的空间拓扑，该区域包含在$2\pi$旋转下表现出费米行为的态。 显示英文摘要

摘要： We construct a quantum kinematics for asymptotically flat spacetimes based on the Koslowski-Sahlmann (KS) representation. The KS representation is a generalization of the representation underlying Loop Quantum Gravity (LQG) which supports, in addition to the usual LQG operators, the action of `background exponential operators' which are connection dependent operators labelled by `background' $su(2)$ electric fields. KS states have, in addition to the LQG state label corresponding to 1 dimensional excitations of the triad, a label corresponding to a `background' electric field which describes 3 dimensional excitations of the triad. Asymptotic behaviour in quantum theory is controlled through asymptotic conditions on the background electric fields which label the {\em states} and the background electric fields which label the {\em operators}. Asymptotic conditions on the triad are imposed as conditions on the background electric field state label while confining the LQG spin net graph labels to compact sets. We show that KS states can be realised as wave functions on a quantum configuration space of generalized connections and that the asymptotic behaviour of each such generalized connection is determined by that of the background electric fields which label the background exponential operators. Similar to the spatially compact case, the Gauss Law and diffeomorphism constraints are then imposed through Group Averaging techniques to obtain a large sector of gauge invariant states. It is shown that this sector supports a unitary action of the group of asymptotic rotations and translations and that, as anticipated by Friedman and Sorkin, for appropriate spatial topology, this sector contains states which display fermionic behaviour under $2\pi$ rotations.