Revisiting Quantum Volume Operator

Abstract: In this paper we introduce the n-dimensional hypersurface quantum volume operator by using the n-dimensional holonomy variation formula. Instead of trying to construct the n-dimensional hypersurface volume operator by using the n-1 dimensional hypersufrace volume operators, as it is usually done in 3d case, we introduce the n-dimensional volume operator directly. We use two facts - first, that the area of the n-dimensional hypersurface of the n+1 dimensional manifold is the volume of the n dimensional induced metric and secondly that the holonomy variation formula is valid for the n-dimensional hypersufrace in the n+1 manifold with connection values in any Lie algebra.