Revisiting Quantum Volume Operator

TL;DR

This paper introduces a new n-dimensional hypersurface quantum volume operator using the holonomy variation formula, directly defining it without relying on lower-dimensional operators, advancing quantum geometry methods.

Contribution

It presents a novel n-dimensional volume operator constructed directly via holonomy variation, bypassing traditional recursive approaches used in lower dimensions.

Findings

Defines the n-dimensional hypersurface quantum volume operator.

Utilizes the holonomy variation formula in n+1 dimensions.

Provides a framework for higher-dimensional quantum geometry.

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.