Functional integration on spaces of connections

Stephen F. Sawin; John C. Baez

doi:10.1006/jfan.1997.3108

Functional integration on spaces of connections

Stephen F. Sawin, John C. Baez

Fairfield University

University of California - Riverside

Research output: Contribution to journal › Article › peer-review

Abstract

Suppose that G is a compact connected Lie group and P → M is a smooth principal G -bundle. We define a “cylinder function” on the space of smooth connections on P to be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M . Completing the algebra of cylinder functions in the sup norm, we obtain a commutative C*-algebra Fun(). Let a “generalized measure” on be a bounded linear functional on Fun(). We construct a generalized measure μ 0 on that is invariant under all automorphisms of the bundle P (not necessarily fixing the base M ). This result extends previous work which assumed M was real-analytic and used only piecewise analytic curves in the definition of cylinder functions. As before, any graph with n edges embedded in M determines a C*-subalgebra of Fun() isomorphic to C ( G n ), and the generalized measure μ 0 : Fun()→ restricts to the linear functional on C ( G n ) given by integration against normalized Haar measure on G n . Our result implies that the group of gauge transformations acts as unitary operators on L 2 (), the Hilbert space completion of Fun() in the norm &Vert; F &Vert; 2 = μ 0 (| F | 2 ) 1/2 . Using “spin networks,” we construct explicit functions spanning the subspace L 2 (/)&sube; L 2 () consisting of vectors invariant under the action of .

Original language	American English
Journal	Journal of Functional Analysis
Volume	150
DOIs	https://doi.org/10.1006/jfan.1997.3108
State	Published - Jan 1 1997

Disciplines

Computer Sciences
Physical Sciences and Mathematics

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Functional integration on spaces of connectionsFinal published version, 292 KBLicense: CC BY

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title = "Functional integration on spaces of connections",

abstract = " Suppose that G is a compact connected Lie group and P \→ M is a smooth principal G -bundle. We define a \“cylinder function\” on the space of smooth connections on P to be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M . Completing the algebra of cylinder functions in the sup norm, we obtain a commutative C*-algebra Fun(). Let a \“generalized measure\” on be a bounded linear functional on Fun(). We construct a generalized measure \μ 0 on that is invariant under all automorphisms of the bundle P (not necessarily fixing the base M ). This result extends previous work which assumed M was real-analytic and used only piecewise analytic curves in the definition of cylinder functions. As before, any graph with n edges embedded in M determines a C*-subalgebra of Fun() isomorphic to C ( G n ), and the generalized measure \μ 0 : Fun()\→ restricts to the linear functional on C ( G n ) given by integration against normalized Haar measure on G n . Our result implies that the group of gauge transformations acts as unitary operators on L 2 (), the Hilbert space completion of Fun() in the norm \&Vert; F \&Vert; 2 = \μ 0 (| F | 2 ) 1/2 . Using \“spin networks,\” we construct explicit functions spanning the subspace L 2 (/)\&sube; L 2 () consisting of vectors invariant under the action of .",

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doi = "10.1006/jfan.1997.3108",

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AU - Baez, John C.

PY - 1997/1/1

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N2 - Suppose that G is a compact connected Lie group and P → M is a smooth principal G -bundle. We define a “cylinder function” on the space of smooth connections on P to be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M . Completing the algebra of cylinder functions in the sup norm, we obtain a commutative C*-algebra Fun(). Let a “generalized measure” on be a bounded linear functional on Fun(). We construct a generalized measure μ 0 on that is invariant under all automorphisms of the bundle P (not necessarily fixing the base M ). This result extends previous work which assumed M was real-analytic and used only piecewise analytic curves in the definition of cylinder functions. As before, any graph with n edges embedded in M determines a C*-subalgebra of Fun() isomorphic to C ( G n ), and the generalized measure μ 0 : Fun()→ restricts to the linear functional on C ( G n ) given by integration against normalized Haar measure on G n . Our result implies that the group of gauge transformations acts as unitary operators on L 2 (), the Hilbert space completion of Fun() in the norm &Vert; F &Vert; 2 = μ 0 (| F | 2 ) 1/2 . Using “spin networks,” we construct explicit functions spanning the subspace L 2 (/)&sube; L 2 () consisting of vectors invariant under the action of .

AB - Suppose that G is a compact connected Lie group and P → M is a smooth principal G -bundle. We define a “cylinder function” on the space of smooth connections on P to be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M . Completing the algebra of cylinder functions in the sup norm, we obtain a commutative C*-algebra Fun(). Let a “generalized measure” on be a bounded linear functional on Fun(). We construct a generalized measure μ 0 on that is invariant under all automorphisms of the bundle P (not necessarily fixing the base M ). This result extends previous work which assumed M was real-analytic and used only piecewise analytic curves in the definition of cylinder functions. As before, any graph with n edges embedded in M determines a C*-subalgebra of Fun() isomorphic to C ( G n ), and the generalized measure μ 0 : Fun()→ restricts to the linear functional on C ( G n ) given by integration against normalized Haar measure on G n . Our result implies that the group of gauge transformations acts as unitary operators on L 2 (), the Hilbert space completion of Fun() in the norm &Vert; F &Vert; 2 = μ 0 (| F | 2 ) 1/2 . Using “spin networks,” we construct explicit functions spanning the subspace L 2 (/)&sube; L 2 () consisting of vectors invariant under the action of .

UR - https://digitalcommons.fairfield.edu/mathandcomputerscience-facultypubs/16

U2 - 10.1006/jfan.1997.3108

DO - 10.1006/jfan.1997.3108

M3 - Article

VL - 150

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

ER -

Functional integration on spaces of connections

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