Approaches to Quantum Gravity

Asymptotic safety 113field and the classical field; the effective action Ɣ k is then obtained by identifyingthese fields. See [30]or[36] for the case of gravity.The effective action Ɣ k [φ A ], used at tree level, gives an accurate description ofprocesses occurring at momentum scales of order k. In general it will have theform Ɣ k (φ A , g i ) = ∑ i g i(k)O i (φ A ), where g i are running coupling constants andO i are all possible operators constructed with the fields φ A and their derivatives,which are compatible with the symmetries of the theory. It can be thought of as afunctional on F ×Q× R + , where F is the configuration space of the fields, Q is aninfinite dimensional manifold parametrized by the coupling constants, and R + isthe space parametrized by k. The dependence of Ɣ k on k is given by ∂ t Ɣ k (φ A , g i ) =∑i β i(k)O i (φ A ) where t = log(k/k 0 ) and β i (g j , k) = ∂ t g i are the beta functions.Dimensional analysis implies the scaling propertyƔ k (φ A , g i ) = Ɣ bk (b d Aφ A , b d ig i ), (8.3)where d A is the canonical dimension of φ A , d i is the canonical dimension of g i ,andb ∈ R + is a positive real scaling parameter. 1 One can rewrite the theory in termsof dimensionless fields ˜φ A = φ A k −d Aand dimensionless couplings ˜g i = g i k −d i.A transformation (8.3) with parameter b = k −1 can be used to define a functional˜Ɣ on (F × Q × R + )/R + :˜Ɣ( ˜φ A , ˜g i ) := Ɣ 1 ( ˜φ A , ˜g i ) = Ɣ k (φ A , g i ). (8.4)Similarly, β i (g j , k) = k d ia i ( ˜g j ) where a i ( ˜g j ) = β i ( ˜g j , 1). There follows that thebeta functions of the dimensionless couplings,˜β i ( ˜g j ) ≡ ∂ t ˜g i = a i ( ˜g j ) − d i ˜g i (8.5)depend on k only implicitly via the ˜g j (t).The effective actions Ɣ k and Ɣ k−δk differ essentially by a functional integral overfield modes with momenta between k and k − δk. Such integration does not leadto divergences, so the beta functions are automatically finite. Once calculated at acertain scale k, they are automatically determined at any other scale by dimensionalanalysis. Thus, the scale k 0 and the “bare” action S act just as initial conditions:when the beta functions are known, one can start from an arbitrary initial point onQ and follow the RG trajectory in either direction. The effective action Ɣ k at anyscale k can be obtained integrating the flow. In particular, the UV behaviour can bestudied by taking the limit k → ∞.It often happens that the flow cannot be integrated beyond a certain limiting scale, defining the point at which some “new physics” has to make its appearance. In1 We assume that the coordinates are dimensionless, as is natural in curved space, resulting in unconventionalcanonical dimensions. The metric is an area.