Diffeomorphism-Invariant Spin Network States

Abstract

We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose thatGis a compact connected semisimple Lie group andP→Mis a smooth principalG-bundle. A “cylinder function” on the space of smooth connections onPis a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves inM. We construct diffeomorphism-invariant functionals on the space of cylinder functions from “spin networks”: graphs inMwith edges labeled by representations ofGand vertices labeled by intertwining operators. Using the “group averaging” technique of Ashtekar, Marolf, Mourão, and Thiemann, we equip the space spanned by these diffeomorphism-invariant spin network states with a natural inner product.