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
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