GEOMETRY EIGENVALUES AND THE SCALAR PRODUCT FROM RECOUPLING THEORY INLOOP QUANTUM-GRAVITY

We summarize the basics of the loop representation of quantum gravity and describe the main aspects of the formalism, including its latest d evelopments, in a reorganized and consistent form. Recoupling theory, in its graphical tangle-theoretic Temperley-Lieb formulation, provides a powerful calculation tool in this context. We describe its applicat ion to the loop representation in detail. Using recoupling theory, we derive general expressions for the spectrum of the quantum area and th e quantum volume operators. We compute several volume eigenvalues expl icitly. We introduce a scalar product with respect to which area and v olume are symmetric operators, and (the trivalent expansions of) the s pin network states are orthonormal.