32 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Feynman graphs. Section 10 uses this to construct the local counterterms as-

sociated to a Lagrangian, which are needed to render the functional integral

finite. Section 11 gives the proof of theorems A and B above.

Finally, we turn to generalizations of the main results. Section 13 shows

how everything generalizes, mutatis mutandis, to the case when our fields

are no longer just functions, but sections of some vector bundle. Section 14

shows how we can further generalize to deal with theories on non-compact

manifolds, as long as an appropriate infrared cut-off is introduced.

2. The effective interaction and background field functional

integrals

As in the introduction, a quantum field theory in our Wilsonian defini-

tion will be given by a collection of effective actions, related by the renor-

malization group flow. In this section we will write down a version of the

renormalization group flow, based on the effective interaction, which we will

use throughout the book.

2.1. Let us assume that our energy Λ effective action can be written as

S[Λ](φ) = −

1

2

φ, (D

+m2)φ

+ I[Λ](φ)

where:

(1) The function I[Λ] is a formal series in , I[Λ] = I0[Λ]+ I1[Λ]+··· ,

where the leading term I0 is at least cubic. Each Ii is a formal power

series on the vector space

C∞(M)

of fields (later, I will explain what

this means more precisely).

The function I[Λ] will be called the effective interaction.

(2) , denotes the

L2

inner product on

C∞(M,

R) defined by φ, ψ =

M

φψ.

(3) D

denotes1

the Laplacian on M, with signs chosen so that the

eigenvalues of D are non-negative; and m ∈ R 0.

Recall that the renormalization group equation relating S[Λ] and S[Λ ]

can be written

S[Λ ](φL) = log

φH ∈C∞(M)[Λ

,Λ)

eS[Λ](φL+φH )/

.

We can rewrite this in terms of the effective interactions, as follows. The

spaces

C∞(M)

Λ

and

C∞(M)[Λ

,Λ)

are orthogonal. It follows that

S[Λ](φL + φH )

= −

1

2

φL, (D

+m2)φL

−

1

2

φH , (D

+m2)φH

+ I[Λ](φL + φH ).

1The

symbol Δ will be reserved for the BV Laplacian