68 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
functionals implies that this quantity is local. Thus, let
Ii,k = Ii,k[L] Wi,k
R

⎝P
(0,L),
(r,s) (i,k)
rIr,s⎠

.
Then,
Wa,b
R

⎝P
(0,L),
(r,s)≤(i,k)
rIr,s⎠

= Ia,b[L]
for all (a, b) (i, k), and we can continue the induction.
11.1. Recall that we defined T
(n)
to be the space of theories defined
modulo
n+1,
and T
(∞)
to be the space of theories. Thus, we have shown
that the choice of renormalization scheme sets up a bijection between T
(∞)
and functionals I
Oloc(C∞(M
+
))[[ ]], and between T
(n)
and functionals
I
Oloc(C∞(M
+
))[ ]/
n+1.
The more fundamental statement of the theorem on the bijection be-
tween theories and local action functionals is that the map T
(n+1)
T
(n)
makes T
(n+1)
into a principal bundle for the group
Oloc(C∞(M
)), in a
canonical way (independent of any arbitrary choices, such as that of a renor-
malization scheme). In this subsection we will use the bijection constructed
above to prove this statement.
The bijection between theories and Lagrangians shows that the map
T
(n+1)
T
(n)
is surjective; this is the only place the bijection is used.
To show that T
(n+1)
T
(n)
is a principal bundle, suppose that {I[L]},
{J[L]} are two theories which are defined modulo
n+2
and which agree
modulo
n+1.
Let I0[L] T
(0)
be the classical theory corresponding to both I[L] and
J[L]. Let us consider the tangent space to T
(0)
at I[L], which includes
infinitesimal deformations of classical theories which do not have to be at
least cubic. More precisely, let TI0[L]T
(0)
be the set of H[L]
O(C∞(M
))
such that
I0[L] + δH[L]
satisfies the classical renormalization group equation modulo δ2,
I0,i[L] + δHi[L] = W0,i P (ε, L), I0,j[ε] + δHj[ε] modulo
δ2,
and which satisfy the usual locality axiom, that H[L] has a small L asymp-
totic expansion in terms of local action functionals.
The bijection between classical theories and local action functionals is
canonical, independent of the choice of renormalization scheme. This is true
even if we include non-cubic terms in our effective interaction, as long as
these non-cubic terms are accompanied by nilpotent parameters.
Thus, we have a canonical isomorphism of vector spaces
TI0[L]T
(0)

=
Oloc(C∞(M
)).
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