A hyperlink is a finite set of non-intersecting simple closed curves in R×R3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R} \times \mathbb {R}^3$$\end{document}. Let S be an orientable surface in R3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^3$$\end{document}. The dynamical variables in general relativity are the vierbein e and a su(2)×su(2)\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {su}(2)\times \mathfrak {su}(2)$$\end{document}-valued connection ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}. Together with Minkowski metric, e will define a metric g on the manifold. Denote AS(e)\documentclass[12pt]{minimal}
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\begin{document}$$A_S(e)$$\end{document} as the area of S, for a given choice of e. The Einstein–Hilbert action S(e,ω)\documentclass[12pt]{minimal}
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\begin{document}$$S(e,\omega )$$\end{document} is defined on e and ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}. We will quantize the area of the surface S by integrating AS(e)\documentclass[12pt]{minimal}
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\begin{document}$$A_S(e)$$\end{document} against a holonomy operator of a hyperlink L, disjoint from S, and the exponential of the Einstein–Hilbert action, over the space of vierbeins e and su(2)×su(2)\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {su}(2)\times \mathfrak {su}(2)$$\end{document}-valued connections ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}. Using our earlier work done on Chern–Simons path integrals in R3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^3$$\end{document}, we will write this infinite dimensional path integral as the limit of a sequence of Chern–Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between L and S. By assigning an irreducible representation of su(2)×su(2)\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {su}(2)\times \mathfrak {su}(2)$$\end{document} to each component of L, the area operator gives the total net momentum impact on the surface S.
