Algebraic Quantum Gravity (AQG) III. Semiclassical Perturbation Theory

TL;DR

This paper introduces semiclassical perturbation theory within Algebraic Quantum Gravity (AQG), enabling precise calculations of operator expectation values and justifying previous Abelean approximations in non-Abelean quantum gravity.

Contribution

It develops the first semiclassical perturbation theory in AQG, allowing high-precision computations of operators and validating the Abelean approximation in non-Abelean cases.

Findings

Semiclassical perturbation theory enables power series expansion in ar.

High-precision computation of fractional volume operator powers.

Justification of Abelean approximation in non-Abelean quantum gravity.

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.