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Abstract

In this short note we answer a long standing question about the asymptotic safety scenario for quantum gravity. The term asymptotic safety refers to the conjecture that (i) the quantum field theory of gravity admits a non-trivial ultraviolet fixed point, and that (ii) this has only a finite number of relevant perturbations, i.e. a finite number of UV-stable directions (or in other words, a finite number of free parameters to be fixed experimentally). Within the f(R) approximation of the functional renormalization group equation of gravity, we show that assuming the first half of the conjecture to be true, the remaining half follows from general arguments, that is, we show that assuming the existence of a non-trivial fixed point, the fact that the number of relevant directions is finite is a general consequence of the structure of the equations.