76 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
Let me explain more precisely what I mean by saying there is a small L
asymptotic expansion
Ii,k[L]
j∈Z≥0
gr(L)Φr.
Without loss of generality, we can require that the local action functionals
Φr appearing here are homogeneous of degree k in the field e E .
Recall that A is the global sections of some bundle of algebras A on a
manifold with corners X. Let Ax denote the fibre of A at x X. For every
element α A , let αx Ax denote the value of α at x.
The statement that there is such an asymptotic expansion means that
there is a non-decreasing sequence dR Z, tending to infinity, such that for
all R, for all fields e E , for all x X,
lim
L→0
L−dR
αx Ii,k[L](e)
R
r=0
gr(L)Φr(e) = 0
in the finite dimensional vector space Ax.
Then, as before, the theorem is:
Theorem 13.4.3. The space T
(n+1)(E
) has the structure of a principal
Oloc(E , A ) bundle over T
(n)(E
), in a canonical way. Further, T
(0)(E
)
is canonically isomorphic to the space Oloc(E
+
, A ) of A -valued local action
functionals on E which are at least cubic modulo the ideal I A .
Further, the choice of renormalization scheme gives rise to a section
T
(n)(E
) T
(n+1)(E
) of each torsor, and so a bijection between T
(∞)(E
)
and the space
Oloc(E
+
, A )[[ ]]
of local action functionals with values in A , which are at least cubic modulo
and modulo the ideal I A .
Proof. The proof is essentially the same as before. The extra difficul-
ties are of two kinds: working with an auxiliary parameter space X intro-
duces extra analytical difficulties, and working with quadratic terms in our
interaction forces us to use Artinian induction with respect to the powers of
the ideal I A .
For simplicity, I will only give the proof when the effective interactions
I[L] are all at least cubic modulo . The argument in the general case is the
same, except that we also must perform Artinian induction with respect to
the powers of the ideal I A .
As before, we will prove the renormalization scheme dependent ver-
sion of the theorem, saying that there is a bijection between T
(∞)(E
) and
Oloc(E
+
, A )[[ ]]. The renormalization scheme independent formulation is an
easy corollary.
Let us start by showing how to construct a theory associated to a local
interaction
I =
iI(i,k)
Oloc(E , A )[[ ]].
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