Fundamental Aspects of Asymptotic Safety in Quantum Gravity by Zoë H. Slade

Author:Zoë H. Slade
Language: eng
Format: epub, pdf
ISBN: 9783030195076
Publisher: Springer International Publishing

3.2 Conformally Reduced Gravity at Order Derivative-Squared

In this section we give a quick review of the results we need and their context from reference [1]. Recall that we arrive at conformally reduced gravity (in Euclidean signature) by writing:

(3.2.1)

The metric is restricted to an overall conformal factor times a reference metric which in fact we set to flat: . Following the background field method, we split the total conformal factor field into a background conformal factor field and fluctuation conformal factor field . It is then the latter that is integrated over. Similarly, we parametrise the background metric in terms of the background conformal factor field .

Examples of parametrisations used previously in the literature include [13] and [14, 15]. However we leave the choice of parametrisation unspecified. It is important to note however that f cannot depend on k since it is introduced at the bare level and has no relation to the infrared cutoff (moreover if f depended on k, the flow equation (3.2.2) would no longer hold). Later we will change to dimensionless variables using k and in these variables it can be forced to depend on k (see Sects. 3.3.5 and 3.4.1).

By keeping only the conformal factor of the metric, diffeomorphism invariance is destroyed. Therefore gauge fixing and ghosts are not required in this setup. A remnant diffeomorphism is preserved however, a multiplicative rescaling, which constrains appearances of the background field.

Introducing the classical fluctuation field and total classical field , the effective action satisfies the flow equation

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