Average Effective Potential for the Conformal Factor

In a four dimensional theory of gravity with lagrangian quadratic in curvature and torsion, we compute the effective action for metrics of the form gμν = ρ δμν , with ρ constant. Using standard field-theoretic methods we find that one loop quantum effects produce a nontrivial effective potential for ρ. We explain this unexpected result by showing how our regularization procedure differs from the one that is usually adopted in Quantum Gravity. Using the method of the average effective potential, we compute the scale dependence of the v.e.v. of the conformal factor. ∗ [email protected] ∗∗ [email protected] 1 In quantum field theory the vacuum expectation value (v.e.v.) of the fields is usually determined by the effective potential (the nonderivative part of the effective action). In classical theories of gravity the possible form of a potential for the metric is severely constrained by general covariance: the only allowed local term in the Lagrangian depending on the metric but not on its derivatives is the cosmological term. We have suggested elsewhere that in Quantum Gravity the v.e.v. of the metric could be fixed by an effective potential [1]. The particular dynamics that we employed there was based on a bimetric Lagrangian, which one could think of as a mean field approximation to an ordinary gravitational Lagrangian quadratic in curvature and torsion. We observed that in the presence of two metrics one could obtain a genuine potential term whose minimum fixes the v.e.v. of the metric. One could think that this result was due to the unconventional dynamics that we started with. The main point we want to make in this note is that the same result can be obtained starting from an ordinary Lagrangian quadratic in curvature and torsion and using the familiar background field method. We will restrict our attention to the conformal sector and write gμν = ρ γμν (1) where γμν is a fixed fiducial metric. In order to simplify the discussion as much as possible we will present calculations only for the case γμν = δμν , but our results hold more generally. The effective dynamics of the conformal factor ρ induced by the conformal anomaly of matter fields has been the subject of recent investigations [2,3]. In this work we will discuss the effective potential for ρ in the framework of a gauge theory of gravity. From standard Quantum Gravity arguments, one would expect to find only a cosmological term, i.e. a potential proportional to ρ. Instead, we find an effective potential of the Coleman–Weinberg form, with the minimum occurring for nonzero ρ. We will explain the origin of this result: it lies in the way in which the regularization is defined. We then discuss the renormalization group flow of the minimum of the potential. We do this by computing the average effective potential for ρ. The average effective action is a continuum version of the block-spin action of lattice theories, which has been recently applied to scalar and gauge theories [4,5]. We find that the v.e.v. of ρ (and therefore of the metric) is essentially constant up to Planck’s energy, and scales according to its canonical dimension (mass squared) above Planck’s energy, up to logarithmic corrections. In the conclusion we offer some speculations on the physical meaning of this behavior. In the model we shall consider, the independent dynamical variables are the vierbein θμ and an O(4) gauge field Aμ a b (we shall concentrate on the Euclidean theory, where a, b = 1, 2, 3, 4 are internal indices and μ, ν = 1, 2, 3, 4 are spacetime indices). With θ and A we can construct metric, curvature and torsion fields: gμν = θ a μ θ b ν δab , (2a) Fμν a b = ∂μAν a b − ∂νAμb + eAμcAνb − eAνcAμb , (2b) Θμ a ν = ∂μθ a ν − ∂νθμ + eAμbθν − eAνbθμ , (2c) where e is the gauge coupling constant. As an action we take S(θ, A) = 1 4 ∫ dx √ | det g| gg [ δacδ Fμν a b Fρσ c d + δabΘμ a νΘρ b σ ]