dorsal/arxiv
View SchemaFunctional Integration on Spaces of Connections
| Authors | John C. Baez, Stephen Sawin |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9507023 |
| URL | https://arxiv.org/abs/q-alg/9507023 |
| Journal | J. Funct. Anal., 150 (1997), 1-26 |
Abstract
Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the holonomies of $A$ along finitely many piecewise smoothly immersed curves in $M$, and let a generalized measure on $\A$ be a bounded linear functional on cylinder functions. We construct a generalized measure on the space of connections that extends the uniform measure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is invariant under all automorphisms of $P$, not necessarily the identity on the base space $M$. Using `spin networks' we construct explicit functions spanning the corresponding Hilbert space $L^2(\A/\G)$, where $\G$ is the group of gauge transformations.
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"abstract": "Let $G$ be a compact connected Lie group and $P \\to M$ a smooth principal\n$G$-bundle. Let a `cylinder function\u0027 on the space $\\A$ of smooth connections\non $P$ be a continuous function of the holonomies of $A$ along finitely many\npiecewise smoothly immersed curves in $M$, and let a generalized measure on\n$\\A$ be a bounded linear functional on cylinder functions. We construct a\ngeneralized measure on the space of connections that extends the uniform\nmeasure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is\ninvariant under all automorphisms of $P$, not necessarily the identity on the\nbase space $M$. Using `spin networks\u0027 we construct explicit functions spanning\nthe corresponding Hilbert space $L^2(\\A/\\G)$, where $\\G$ is the group of gauge\ntransformations.",
"arxiv_id": "q-alg/9507023",
"authors": [
"John C. Baez",
"Stephen Sawin"
],
"categories": [
"q-alg",
"gr-qc",
"math.QA"
],
"journal_ref": "J. Funct. Anal., 150 (1997), 1-26",
"title": "Functional Integration on Spaces of Connections",
"url": "https://arxiv.org/abs/q-alg/9507023"
},
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