In the asymptotic safety programme for quantum gravity, it is important to go beyond polynomial truncations. Three such approximations have been derived where the restriction is only to a general function f (R) of the curvature R > 0. We confront these with the requirement that a fixed point solution be smooth and exist for all R ≥ 0. Singularities induced by cutoff choices force the earlier versions to have no such solutions. However, we show that the most recent version has a number of lines of fixed points, each supporting a continuous spectrum of eigen-perturbations. We uncover and analyse the first five such lines. Sensible fixed point behaviour may be achieved if one consistently incorporates geometry/topology change. As an exploratory example, we analyse the equations analytically continued to R < 0, however we now find only partial solutions. We show how these results are always consistent with, and to some extent can be predicted from, a straightforward analysis of the constraints inherent in the equations.

Within the conformally reduced gravity model, where the metric is parametrised by a function f (φ) of the conformal factor φ, we keep dependence on both the background and fluctuation fields, to local potential approximation and O(∂ 2 ) respectively, making no other approximation. Explicit appearances of the background metric are then dictated by realising a remnant diffeomorphism invariance. The standard non-perturbative Renormalization Group (RG) scale k is inherently background dependent, which we show in general forbids the existence of RG fixed points with respect to k. By utilising transformations that follow from combining the flow equations with the modified split Ward identity, we uncover a unique background independent notion of RG scale,k. The corresponding RG flow equations are then not only explicitly background independent along the entire RG flow but also explicitly independent of the form of f . In general f (φ) is forced to be scale dependent and needs to be renormalised, but if this is avoided then k-fixed points are allowed and furthermore they coincide withk-fixed points.

The O(∂ 2 ) background independent flow equations for conformally reduced gravity are shown to be equivalent to flow equations naturally adapted to scalar field theory with a wrong sign kinetic term. This sign change is shown to have a profound effect on the renormalization group properties, broadly resulting in a continuum of fixed points supporting both a discrete and a continuous eigenoperator spectrum, the latter always including relevant directions. The properties at the Gaussian fixed point are understood in particular depth, but also detailed studies of the local potential approximation, and the full O(∂ 2 ) approximation are given. These results are related to evidence for asymptotic safety found by other authors.arXiv:1605.07636v2 [hep-th]

In this paper we review the definition and properties of redundant operators in the exact renormalisation group. We explain why it is important to require them to be eigenoperators and why generically they appear only as a consequence of symmetries of the particular choice of renormalisation group equations. This clarifies when Newton's constant and or the cosmological constant can be considered inessential. We then apply these ideas to the Local Potential Approximation and approximations of a similar spirit such as the f (R) approximation in the asymptotic safety programme in quantum gravity. We show that these approximations can break down if the fixed point does not support a 'vacuum' solution in the appropriate domain: all eigenoperators become redundant and the physical space of perturbations collapses to a point. We show that this is the case for the recently discovered lines of fixed points in the f (R) flow equations.

Working within the familiar local potential approximation, and concentrating on the example of a single scalar field in three dimensions, we show that the commonly used approximation method of identifying the total and background fields, leads to pathologies in the resulting fixed point structure and the associated spaces of eigenoperators. We then show how a consistent treatment of the background field through the corresponding modified shift Ward identity, can cure these pathologies, restoring universality of physical quantities with respect to the choice of dependence on the background field, even within the local potential approximation. Along the way we point out similarities to what has been previously found in the f (R) approximation in asymptotic safety for gravity.