A hyperlink is a finite set of non-intersecting simple closed curves in \(\mathbb {R} \times \mathbb {R}^3\). Let R be a compact set inside \(\mathbb {R}^3\). The dynamical variables in General Relativity are the vierbein e and a \(\mathfrak {su}(2)\times \mathfrak {su}(2)\)-valued connection \(\omega \). Together with Minkowski metric, e will define a metric g on the manifold. Denote \(V_R(e)\) as the volume of R, for a given choice of e. The Einstein–Hilbert action \(S(e,\omega )\) is defined on e and \(\omega \). We will quantize the volume of R by integrating \(V_R(e)\) against a holonomy operator of a hyperlink L, disjoint from R, and the exponential of the Einstein–Hilbert action, over the space of vierbein e and \(\mathfrak {su}(2)\times \mathfrak {su}(2)\)-valued connection \(\omega \). Using our earlier work done on Chern–Simons path integrals in \(\mathbb {R}^3\), we will write this infinite-dimensional path integral as the limit of a sequence of Chern–Simons integrals. Our main result shows that the volume operator can be computed by counting the number of nodes on the projected hyperlink in \(\mathbb {R}^3\), which lie inside the interior of R. By assigning an irreducible representation of \(\mathfrak {su}(2)\times \mathfrak {su}(2)\) to each component of L, the volume operator gives the total kinetic energy, which comes from translational and angular momentum.

中文翻译：

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体积的路径积分量化

超链接是\（\ mathbb {R} \ times \ mathbb {R} ^ 3 \）中有限的一组非相交的简单闭合曲线。令R为\（\ mathbb {R} ^ 3 \）内部的紧集。广义相对论中的动力学变量是vierbein e和\（\ mathfrak {su}（2）\ times \ mathfrak {su}（2）\）值连接\（\ omega \）。e与Minkowski度量一起在流形上定义度量g。表示\（V_R（E）\）作为的体积- [R ，对于一个给定的选择ë。爱因斯坦–希尔伯特动作\（S（e，\ omega）\）在e和\（\ omega \）。我们将通过对超级链接L的完整算术运算符\（V_R（e）\）积分R （与R不相交）以及Einstein–Hilbert作用的指数，在verbein e和\（\ mathfrak {su}（2）\ times \ mathfrak {su}（2）\）值连接\（\ omega \）。使用我们先前在\（\ mathbb {R} ^ 3 \）中对Chern–Simons路径积分所做的工作，我们将把这个无限维路径积分写为Chern–Simons积分序列的极限。我们的主要结果表明，可以通过计算投影超链接中的节点数来计算体积算符\（\ mathbb {R} ^ 3 \），它位于R的内部。通过为L的每个分量分配\（\ mathfrak {su}（2）\ times \ mathfrak {su}（2）\）的不可约表示，体积算子给出了总动能，该动能来自平动和角动量。