论文标题
$ \ operatotorname {su} _q（2）$ lattice量规理论中的本地可观察物
Local Observables in $\operatorname{SU}_q(2)$ Lattice Gauge Theory
论文作者
论文摘要
我们根据$ \ operatatorName {su}（2）$的Heisenberg double，在规范图片中考虑了3D晶格仪理论的变形，然后在量子级别上进行了变形。我们表明，经典的纺纱器可用于定义一组基本的局部观测值。它们是居住在晶格顶点上的不变数量，并用成对的事件边缘标记。经典相空间上的任何功能，例如威尔逊循环，可以根据这些可观察到的东西来重写。在量子水平上，我们表明纺纱器成为纺纱算子。然后，本地可观察物的量化需要使用量子$ \ Mathcal {r} $ - 矩阵，我们证明这与围绕顶点周围的特定平行传输相同。我们将本地观察者的代数作为泊松代数，然后以$ \ mathfrak {so}^*（2n）$在量子级别为$ q $ - 定义。这种形式主义可以与依赖晶格规程理论技术的任何理论有关，例如拓扑模型，循环量子重力或当然还有晶格仪理论本身。
We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\operatorname{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities which live on the vertices of the lattice and are labelled by pairs of incident edges. Any function on the classical phase space, e.g. Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$-matrix which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$-deformation of $\mathfrak{so}^*(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.
