Diffeomorphism-invariant spin network states

We extend the theory of diffeomorphism-invariant spin network states From the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P --> M is a smooth principal G-bundle. A "cylinder function" on the space of smooth connections on P is a continuous complex Function of the holonomies along finitely many piecewise smoothly immersed curves in M. We construct diffeomorphism-invariant functionals on the space of cylinder functions From "spin networks": graphs in M with edges labeled by representations of G and vertices labeled by intertwining operators. Using the "group averaging" technique of Ashtekar, Marolf, Mourao, and Thiemann, we equip the space spanned by these diffeomorphism-invariant spin network states with a natural inner product. (C) 1998 Academic Press.