7. A DEFINITION OF A QUANTUM FIELD THEORY 53
Then, the correlation function for the effective interaction I[L] is defined
by
EI[L](f1, . . . , fn) =
γ∈Γn
1
Aut(γ)
n−g(γ)−1Cγ(I[L])(f1,...,fn).
Unlike the heuristic graphical expansion we gave for the correlation functions
of the
φ3
theory, this expression is well-defined.
We should interpret this expansion as saying that we can compute the
correlation functions from the effective interaction I[L] by allowing particles
to propagate in the usual way, and to interact by I[L]; except that in between
any two interactions, particles must travel for a proper time of at least L.
This accounts for the fact that edges which join to internal vertices are
labelled by P (L, ∞).
If we have a collection {I[L] | L (0, ∞)} of effective interactions, then
the renormalization group equation is equivalent to the statement that all
the correlation functions constructed from I[L] using the prescription given
above are independent of L.
7. A definition of a quantum field theory
7.1. Now we have some preliminary definitions and an understanding
of why the terms in the graphical expansion of a functional integral diverge.
This book will describe a method for renormalizing these functional integrals
to yield a finite answer.
This section will give a formal definition of a quantum field theory, based
on Wilson’s philosophy of the effective action; and a precise statement of the
main theorem, which says roughly that there’s a bijection between theories
and Lagrangians.
Definition 7.1.1. A local action functional I
O(C∞(M
)) is a func-
tional which arises as an integral of some Lagrangian. More precisely, if we
Taylor expand I as I =

k
Ik where
Ik(λa) =
λkIk(a)
(so that Ik is homogeneous of degree k of the variable a
C∞(M)),
then Ik
must be of the form
Ik(a) =
s
j=1
M
D1,j(a) · · · Dk,j(a)
where Di,j are arbitrary differential operators on M.
Let
Oloc(C∞(M
))
O(C∞(M
)) be the subspace of local action function-
als.
As before, let
Oloc(C∞(M +
))[[ ]]
Oloc(C∞(M
))[[ ]]
be the subspace of those local action functionals which are at least cubic
modulo .
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