dorsal/arxiv
View SchemaDiffeomorphism-Invariant Spin Network States
| Authors | John C. Baez, Stephen Sawin |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9708005 |
| URL | https://arxiv.org/abs/q-alg/9708005 |
| Journal | Jour. Funct. Analysis, 158 (1998) 253-266 |
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, Mourao and Thiemann, we equip the space spanned by these `diffeomorphism-invariant spin network states' with a natural inner product.
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"abstract": "We extend the theory of diffeomorphism-invariant spin network states from the\nreal-analytic category to the smooth category. Suppose that G is a compact\nconnected semisimple Lie group and P -\u003e M is a smooth principal G-bundle. A\n`cylinder function\u0027 on the space of smooth connections on P is a continuous\ncomplex function of the holonomies along finitely many piecewise smoothly\nimmersed curves in M. We construct diffeomorphism-invariant functionals on the\nspace of cylinder functions from `spin networks\u0027: graphs in M with edges\nlabeled by representations of G and vertices labeled by intertwining operators.\nUsing the `group averaging\u0027 technique of Ashtekar, Marolf, Mourao and Thiemann,\nwe equip the space spanned by these `diffeomorphism-invariant spin network\nstates\u0027 with a natural inner product.",
"arxiv_id": "q-alg/9708005",
"authors": [
"John C. Baez",
"Stephen Sawin"
],
"categories": [
"q-alg",
"gr-qc",
"math.QA"
],
"journal_ref": "Jour. Funct. Analysis, 158 (1998) 253-266",
"title": "Diffeomorphism-Invariant Spin Network States",
"url": "https://arxiv.org/abs/q-alg/9708005"
},
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