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
and kinetic action −
, is given by a set of
effective interactions I[L] ∈
))[[ ]] 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
where gr(L) ∈
∞)L) and Φr ∈
denote the set of perturbative quantum field theories, and let T
denote the set of theories defined modulo
Let me explain more precisely what I mean by saying there is a small L
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 ∈
gr(L)Φr(a) = 0.
In other words, we are asking that the asymptotic expansion exists in the
weak topology on
The main theorem of this chapter (in the case of scalar field theories) is
Theorem A. Let T
denote the set of perturbative quantum field the-
ories defined modulo
is, in a canonical way, a principal
bundle over T
for the abelian group
)) of local action func-
tionals on M.
Further, T (0) is canonically isomorphic to the space Oloc(C∞(M
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;