Quantum states of elementary three-geometry

We introduce a quantum volume operator K—which could play a significant role in discretized quantum gravity models—by taking into account a symmetrical coupling scheme of three SU(2) angular momenta. The spectrum of K is discrete and defines a complete set of eigenvectors which is alternative with respect to the complete sets employed when the usual binary coupling schemes of angular momenta are considered. Each of these states, which we call quantum bubbles, represents an interference of three-dimensional geometrical configurations. We study the generalized recoupling coefficients connecting the symmetrical and the binary basis vectors, and provide an explicit recursive solution for such coefficients by analysing also their asymptotic limit.