the most difficult step. Let us consider the bare connected n-point correlation function, Γ ( n ) B in φ 4 theory † . It is defined as where φ$_{0}$ ( x ) refers to the bare fields, before renormalization. They are related to the renormalized fields by a rescaling: φ$_{0}$ ( x ) = Z 1 / 2 φ φ ( x ). The bare quantity, Γ ( n ) B is a function of the bare coupling constant, g$_{0}$ , and some cutoff, Λ. The renormalized quantity, Γ ( n ) R is a function of the renormalized coupling constant g and the renormalization scale M . The bare theory is independent of the renormalization scale, M , so we can write It is easy to show that this equation can be rewritten as with the definitions $^{†}$φ 4 theory is one of the simplest examples of an interacting quantum field theory. Its Lagrangian is L = 1 $_{2}$( ∂$_{µ}$φ ) 2 − 1 $_{2}$m$^{2}$φ 2 − λ $_{4!}$φ $^{4}$. More details can be found in [14]. However, its exact nature is unimportant here. We use it because its renormalization group equation is particularly simple. 7 β = M ∂g ∂M (1.6) γ = M ∂ ln Z$_{φ}$ ∂M (1.7) [ M ∂ ∂M + β ∂ ∂g − n 2 γ ] Γ ( n $^{)}$( k$_{i}$ ; g, M ) = 0 (1.5) ( M d dM )( Z ( − n/ 2) φ Γ ( n $^{)}$( k$_{i}$ ; g, M ) ) = 0 (1.4) Γ ( n $^{)}$( k$_{i}$ ; g, M ) = Z ( n/ 2) φ Γ ( n ) B ( k$_{i}$ ; g$_{0}$, Λ) (1.3) Γ ( n ) B ( x$_{1}$, x$_{2}$, ..., x$_{n}$ ) ≡ 〈 0 | T φ$_{0}$ ( x$_{1}$ ) φ$_{0}$ ( x$_{2}$ ) ...φ$_{0}$ ( x$_{n}$ ) | 0 〉 (1.2)