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 of the irreducible SU(2) representation spaces , labelled with spins . We restrict to spaces of SU(2) invariant tensors (intertwiners) with all spins equal j ₁ =⋯= 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.

我们研究了循环量子引力中 Rovelli-Smolin-DePietri 体积算符的一些性质，这显着简化了对角化问题并揭示了本征态的简并模式。操作符由其在不可约 SU(2) 表示空间的张量积空间中的动作定义，标记为自旋。我们限制为 SU(2) 不变张量（交织器）的空间，所有自旋都等于j ₁ =⋯= j _N = j。我们称它们为自旋j 单色交织器。这些空间在研究各向同性的 SU(2) 规范不变态中很重要，可用于提取理论的宇宙学部分。在自旋 1/2 的情况下，我们完全解决了特征值问题：我们证明了体积算子与恒等式成正比并计算了比例因子。