New length operator for loop quantum gravity

An alternative expression for the length operator in loop quantum gravity is presented. The operator is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space. The expression for the regularized length operator can moreover be understood both from a simple geometrical perspective as the average of a formula relating the length to area, volume and flux operators, and also consistently as the result of direct substitution of the densitized triad operator with the functional derivative operator into the regularized expression of the length. Both these derivations are discussed, and the origin of an undetermined overall factor in each case is also elucidated.

Figure 1

(a) A partition of the neighborhood $R_c$ of the curve $c$ in $\Sigma$ into cubes. (b) The figure illustrates the way two 2-surfaces of a partition are adapted to the curve $c$ according to conditions (i) and (ii) with the vertex $v$ at the center.