We now turn to the problem of renormalizing our field theory to all orders in perturbation theory. To this end, we shall define a flow of effective actions, by defining an effective action that depends on a scale parameter ∧. The differential equation obtained by applying ∂/∂∧ to the functional integral for the effective action is the renormalization group (RG) equation. The graphical expansion derived in Chap. 2 provides an explicit solution of the RG equation. However, the differential equation itself is very useful for deriving bounds for the solutions without using the details of the graphical representation. We shall learn more about the connection between the two approaches as we develop the theory.