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Algebraic Quantum Gravity (AQG) III. Semiclassical

Giesel, K., & Thiemann, T. (2007). Algebraic Quantum Gravity (AQG) III. Semiclassical. Classical and Quantum Gravity, 24(10), 2565-2588. doi:10.1088/0264-9381/24/10/005.

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Creators:
Giesel, Kristina1, Author
Thiemann, Thomas1, Author
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1Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24014

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Abstract: In the two previous papers of this series we defined a new combinatorical approach to quantum gravity, Algebraic Quantum Gravity (AQG). We showed that AQG reproduces the correct infinitesimal dynamics in the semiclassical limit, provided one incorrectly substitutes the non -- Abelean group SU(2) by the Abelean group \$U(1)^3\$ in the calculations. The mere reason why that substitution was performed at all is that in the non -- Abelean case the volume operator, pivotal for the definition of the dynamics, is not diagonisable by analytical methods. This, in contrast to the Abelean case, so far prohibited semiclassical computations. In this paper we show why this unjustified substitution nevertheless reproduces the correct physical result: Namely, we introduce for the first time semiclassical perturbation theory within AQG (and LQG) which allows to compute expectation values of interesting operators such as the master constraint as a power series in \$\hbar\$ with error control. That is, in particular matrix elements of fractional powers of the volume operator can be computed with extremely high precision for sufficiently large power of \$\hbar\$ in the \$\hbar\$ expansion. With this new tool, the non -- Abelean calculation, although technically more involved, is then exactly analogous to the Abelean calculation, thus justifying the Abelean analysis in retrospect. The results of this paper turn AQG into a calculational discipline.

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Dates: 2007
Publication Status: Published in print
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Identifiers: eDoc: 286218
Other: arXiv:gr-qc/0607101
DOI: 10.1088/0264-9381/24/10/005
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Title: Classical and Quantum Gravity
Source Genre: Journal
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Pages: - Volume / Issue: 24 (10) Sequence Number: - Start / End Page: 2565 - 2588 Identifier: -