Abstract
The effective action in quantum general relativity is strongly dependent on the gauge-fixing and parametrization of the quantum metric. As a consequence, in the effective approach to quantum gravity, there is no possibility to introduce the renormalization-group framework in a consistent way. On the other hand, the version of effective action proposed by Vilkovisky and DeWitt does not depend on the gauge-fixing and parametrization off- shell, opening the way to explore the running of the cosmological and Newton constants as well as the coefficients of the higher-derivative terms of the total action. We argue that in the effective framework the one-loop beta functions for the zero-, two- and four-derivative terms can be regarded as exact, that means, free from corrections coming from the higher loops. In this perspective, the running describes the renormalization group flow between the present-day Hubble scale in the IR and the Planck scale in the UV.
Highlights
The gauge-fixing independence of the Vilkovisky-DeWitt effective action has been proved in a general setting and was confirmed by one-loop direct calculations [6,7,8,9,10,11,12,13]
We argue that in the effective framework the one-loop beta functions for the zero, two- and four-derivative terms can be regarded as exact, that means, free from corrections coming from the higher loops
We shall follow the previous publication on the subject [16] and consider the renormalization group equations in the effective quantum gravity based on general relativity in the unique effective action formalism
Summary
Let us start by formulating general definitions, valid for any gauge field theory, which we subsequently particularise for quantum gravity. This issue is especially relevant for quantum gravity, where there is a one-parameter family of such metrics, characterised by the parametera = −1/4, given by [23] It was shown by explicit calculations [13] (see [24, 25]) that the Vilkovisky-DeWitt effective action depends on the choice ofa. In the parallel paper [14] we have shown that this prescription works perfectly well even under changes of the parametrization of the quantum field, which modifies the parametera It defines a unique off-shell effective action. We shall come back to this reasoning and use it intensively in the two sections when discussing the renormalization group
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