Diffeomorphism-Invariant Spin Network States

Details Diffeomorphism-Invariant Spin Network States Diffeomorphism-Invariant Spin Network States

1997-08-04

arxiv.org/abs/q-alg/9708005v2

Jour. Funct. Analysis, 158 (1998) 253-266

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

13 pages, LaTeX, one encapsulated Postscript figure, some corrections in the definition of the inner product

Scientific paper

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.

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