54 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
Thus, local action functionals are the same as Lagrangians modulo those
Lagrangians which are a total derivative.
Definition 7.1.2. A perturbative quantum field theory, with space of
fields
C∞(M)
and kinetic action
1
2
φ, (D
+m2)φ
, is given by a set of
effective interactions I[L]
O+(C∞(M
))[[ ]] for all L (0, ∞], such that
(1) The renormalization group equation
I[L] = W (P (ε, L),I[ε])
is satisfied, for all ε, L (0, ∞].
(2) For each i, k, there is a small L asymptotic expansion
Ii,k[L]
r∈Z≥0
gr(L)Φr
where gr(L)
C∞((0,
∞)L) and Φr
Oloc(C∞(M
)).
Let T
(∞)
denote the set of perturbative quantum field theories, and let T
(n)
denote the set of theories defined modulo
n+1.
Thus, T
(∞)
= lim
←−
T
(n).
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 a.
Then, 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, and for all fields a
C∞(M),
lim
L→0
L−dR
Ii,k[L](a)
R
r=0
gr(L)Φr(a) = 0.
In other words, we are asking that the asymptotic expansion exists in the
weak topology on
Hom(C∞(M)⊗k,
R)Sk .
The main theorem of this chapter (in the case of scalar field theories) is
the following.
Theorem A. Let T
(n)
denote the set of perturbative quantum field the-
ories defined modulo
n+1.
Then T
(n+1)
is, in a canonical way, a principal
bundle over T
(n)
for the abelian group
Oloc(C∞(M
)) of local action func-
tionals on M.
Further, T (0) is canonically isomorphic to the space Oloc(C∞(M
+
)) of
local action functionals which are at least cubic.
There is a variant of this theorem, which states that there is a bijection
between theories and Lagrangians once we choose a renormalization scheme,
which is a way to extract the singular part of certain functions of one vari-
able. The concept of renormalization scheme will be discussed in Section 9;
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