Path Integral Quantization of Volume

Abstract

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.