Abstract
We show that with suitable choices of parametrization, gauge fixing and cutoff, the anomalous variation of the effective action under global rescalings of the background metric is identical to the derivative with respect to the cutoff, i.e. to the beta functional, as defined by the exact RG equation. The Ward identity and the RG equation can be combined, resulting in a modified flow equation that is manifestly invariant under global background rescalings.
Highlights
One of the most vexing challenges facing the asymptotic safety approach to quantum gravity has been the double dependence of the effective action on two fields, the background metric and the fluctuation field
We show that with suitable choices of parametrization, gauge fixing and cutoff, the anomalous variation of the effective action under global rescalings of the background metric is identical to the derivative with respect to the cutoff, i.e. to the beta functional, as defined by the exact RG equation
We have arrived at a remarkably simple result: the anomalous variation in the background scale Ward identity is exactly the “beta functional” of the theory induced by the coarsegraining procedure, as expressed by the r.h.s. of the RG equation: δ k = ∂t k, (4.1)
Summary
One of the most vexing challenges facing the asymptotic safety approach to quantum gravity has been the double dependence of the effective action on two fields, the background metric and the fluctuation field It is only when both dependences are taken into account that one can write an exact flow equation [1]. A step forward has recently been made by Morris for the special case when μν = 2 gμν, i.e. when the background is rescaled by a constant factor [19] He derived the modified Ward identity for this transformation and showed that in six dimensions the anomalous terms coming from the cutoff have the same form as the RG equation.
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