Properties of the Rovelli-Smolin-DePietri volume operator in the spaces of monochromatic intertwiners

Abstract

We study some properties of the Rovelli-Smolin-DePietri volume operator in loop quantum gravity, which significantly simplify the diagonalization problem and shed some light on the pattern of degeneracy of the eigenstates. The operator is defined by its action in the spaces of tensor products $\mathcal{H}_{j_1}\otimes \ldots \otimes \mathcal{H}_{j_N}$ of the irreducible SU(2) representation spaces $\mathcal{H}_{j_i}, i=1,\ldots,N$, labelled with spins $j_i\in \frac{1}{2}\mathbb{N}$. We restrict to the spaces of SU(2) invariant tensors (intertwiners) with all spins equal $j_1=\ldots=j_N=j$. We call them spin $j$ monochromatic intertwiners. Such spaces are important in the study of SU(2) gauge invariant states that are isotropic and can be applied to extract the cosmological sector of the theory. In the case of spin $1/2$ we solve the eigenvalue problem completely: we show that the volume operator is proportional to identity and calculate the proportionality factor.