11. PROOF OF THE MAIN THEOREM 67

Since

Wi,k

⎛

⎝P

(L, L ),W

(i,k)

⎛

⎝P

(ε, L),I −

(r,s) (i,k)

rIr,s CT

(ε)⎠⎠

⎞⎞

is non-singular, the last equation reduces to

Ii,k

CT

(L , ε) = Singε Wi,k

⎛

⎝P

(ε, L),I −

(r,s) (i,k)

rIr,s CT

(ε)⎠

⎞

= Ii,k

CT

(L, ε)

as desired.

Thus, Ii,k

CT

(L, ε) is independent of L, and can be written as Ii,k

CT

(ε); and

we can continue the induction.

11. Proof of the main theorem

Let I ∈

Oloc(C∞(M +

))[[ ]] be a local action functional. What we have

shown so far allows us to construct the corresponding theory. Let

W

R

(P (0,L),I) = lim

ε→0

W

(

P (ε, L),I −

ICT

(ε)

)

be the renormalized version of the renormalization group flow from scale

0 to scale L applied to I. The theory associated to I has scale L effective

interaction W

R

(P (0,L),I). It is easy to see that this satisfies all the axioms

of a perturbative quantum field theory, as given in Definition 7. The locality

axiom of the definition of perturbative quantum field theory follows from

Theorem 9.5.1.

We need to show the converse: that to each theory there is a correspond-

ing local action functional. This is a simple induction. Let {I[L]} denote

a collection of effective interactions defining a theory. Let us assume, by

induction on the lexicographic ordering as before, that we have constructed

local action functionals Ir,s

for (r, s) (i, k) such that

Wa,b

R

⎛

⎝P

(0,L),

(r,s) (i,k)

rIr,s⎠

⎞

= Ia,b[L]

for all (a, b) (i, k).

Then, the infinitesimal renormalization group equation implies that

Wi,k

R

⎛

⎝P

(0,L),

(r,s) (i,k)

rIr,s⎠

⎞

− Ii,k[L]

is independent of L. The locality axiom for the theory {I[L]} – which says

that each Ii,k[L] has a small L asymptotic expansion in terms of local action