| Diffeomorphism-Invariant Spin Network States (2007) | |||||||||||||||
Abstract | |||||||||||||||
| 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, Mour~ao and Thiemann, we equip the space spanned by these `diffeomorphism-invariant spin network states' with a natural inner product. Introduction In the `new variables' approach to quantizing gravity, the kinematical Hibert space of the theory should consist of functions on some completion of the space of connections on a pr... | |||||||||||||||
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