Diieomorphism-invariant Spin Network States

We extend the theory of diieomorphism-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 nitely many piecewise smoothly immersed curves in M. We construct diieomorphism-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 thègroup averaging' technique of Ashtekar, Marolf, Mour~ ao and Thiemann, we equip the space spanned by thesèdiieomorphism-invariant spin network states' with a natural inner product.