### Abstract

Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a nonperturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C*-algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollary of these general results is a precise formulation of the 'loop transform' proposed by Rovelli and Smolin (1990). Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C*-algebra. The structure of this space is investigated and it is shown how observables labelled by 'strips' arise naturally.

Original language | English (US) |
---|---|

Article number | 004 |

Pages (from-to) | 1433-1467 |

Number of pages | 35 |

Journal | Classical and Quantum Gravity |

Volume | 9 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 1992 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)

### Cite this

*Classical and Quantum Gravity*,

*9*(6), 1433-1467. [004]. https://doi.org/10.1088/0264-9381/9/6/004

}

*Classical and Quantum Gravity*, vol. 9, no. 6, 004, pp. 1433-1467. https://doi.org/10.1088/0264-9381/9/6/004

**Representations of the holonomy algebras of gravity and nonAbelian gauge theories.** / Ashtekar, Abhay; Isham, C. J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Representations of the holonomy algebras of gravity and nonAbelian gauge theories

AU - Ashtekar, Abhay

AU - Isham, C. J.

PY - 1992/12/1

Y1 - 1992/12/1

N2 - Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a nonperturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C*-algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollary of these general results is a precise formulation of the 'loop transform' proposed by Rovelli and Smolin (1990). Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C*-algebra. The structure of this space is investigated and it is shown how observables labelled by 'strips' arise naturally.

AB - Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a nonperturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C*-algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollary of these general results is a precise formulation of the 'loop transform' proposed by Rovelli and Smolin (1990). Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C*-algebra. The structure of this space is investigated and it is shown how observables labelled by 'strips' arise naturally.

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U2 - 10.1088/0264-9381/9/6/004

DO - 10.1088/0264-9381/9/6/004

M3 - Article

AN - SCOPUS:33845532740

VL - 9

SP - 1433

EP - 1467

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 6

M1 - 004

ER -