Algebraic Quantum Gravity (AQG) III. Semiclassical Perturbation Theory

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 $\hslash$ 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 $\hslash$ in the $\hslash$ 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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Cited by in corpus (41)

• Fundamental Structure of Loop Quantum Gravity
• Algebraic Quantum Gravity (AQG) I. Conceptual Setup
• Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity
• Scalar Material Reference Systems and Loop Quantum Gravity
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• Algebraic Quantum Gravity (AQG) II. Semiclassical Analysis
• Classical and Quantum Features of the Mixmaster Singularity
• Gauge-invariant coherent states for Loop Quantum Gravity I: Abelian gauge groups
• Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory
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• The Holst Spin Foam Model via Cubulations
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• Coherent states for fractional powers of the harmonic oscillator Hamiltonian