AVERAGE ACTION AND THE RENORMALIZATION-GROUP EQUATIONS

We formulate an effective action GAMMA-k for averages of fields taken within a volume of size k(-d). In contrast to the block-spin approach on the lattice we work in continuous (euclidean) space, preserving all symmetries. We establish how expectation values of operators with momenta smaller than k can be computed from GAMMA-k. The average action at different scales is related by an exact renormalization group equation. We apply these ideas to the N-component-phi-4 theory in the spontaneously broken phase and derive the one-loop renormalization group equations for the average potential. The average potential becomes convex as k --> 0.