2. THE EFFECTIVE ACTION 33

Therefore the effective interaction form of the renormalization group equa-

tion (RGE) is

I[Λ ](a)

= log

φ∈C∞(M)

[Λ ,Λ)

exp −

1

2

φ, (D

+m2)φ

+

1

I[Λ](φ + a) .

Note that in this expression the field a no longer has to be low-energy.

We obtain a variant of the renormalization group equation by considering

effective interactions I[Λ] which are functionals of all fields, not just low-

energy fields, and using the equation above. This equation is invertible; it

is valid even if Λ Λ.

We will always deal with this invertible effective interaction form of the

RGE. Henceforth, it will simply be referred to as the RGE.

2.2. We will often deal with integrals of the form

x∈U

exp (Φ(x)/ + I(x + a)/ )

over a vector space U, where Φ is a quadratic form (normally negative

definite) on U. We will use the convention that the “measure” on U will be

the Lebesgue measure normalised so that

x∈U

exp (Φ(x)/ ) = 1.

Thus, the measure depends on .

2.3. Normally, in quantum field theory textbooks, one starts with an

action functional

S(φ) = −

1

2

φ, (D

+m2)φ

+ I(φ),

where the interacting term I(φ) is a local action functional. This means that

it can be written as a sum

I(φ) =

iIi,k(φ)

where

Ii,k(a) =

s

j=1

M

D1,j(a) · · · Dk,j(a)

and the Di,j are differential operators on M. We also require that I(φ) is

at least cubic modulo .

As I mentioned before, the local interaction I is supposed to be thought

of as the scale ∞ effective interaction. Then the effective interaction at scale

Λ is obtained by applying the renormalization group flow from energy ∞

down to energy Λ. This is expressed in the functional integral

I[Λ](a) = log

φ∈C∞(M)[Λ,∞)

exp −

1

2

φ, (D

+m2)φ

+

1

I(φ + a) .