3. WILSONIAN LOW ENERGY THEORIES 7

Detectors that can only see phenomena of energy at most Λ can be

represented by functions

O :

C∞(M)≤Λ

→ R[[ ]],

which are extended to

C∞(M)

via the projection

C∞(M)

→

C∞(M)≤Λ.

Let us denote by Obs≤Λ the space of all functions on

C∞(M)

that arise

in this way. Elements of Obs≤Λ will be referred to as observables of energy

≤ Λ.

The fundamental quantities of the low-energy effective theory are the

correlation functions O1,...,On between low-energy observables Oi ∈

Obs≤Λ. It is natural to expect that these correlation functions arise from

some kind of statistical system on

C∞(M)≤Λ.

Thus, we will assume that

there is a measure on

C∞(M)≤Λ,

of the form

eSeff

[Λ]/

D φ

where D φ is the Lebesgue measure, and

Seff

[Λ] is a function on Obs≤Λ,

such that

O1,...,On =

φ∈C∞(M)≤Λ

eSeff

[Λ](φ)/

O1(φ) · · · On(φ)Dφ

for all low-energy observables Oi ∈ Obs≤Λ.

The function

Seff

[Λ] is called the low-energy effective action. This ob-

ject completely describes all aspects of a quantum field theory that can be

seen using observables of energy ≤ Λ.

Note that our sign conventions are unusual, in that

Seff

[Λ] appears in

the functional integral via

eSeff

[Λ]/

, instead of

e−Seff

[Λ]/

as is more usual.

We will assume the quadratic part of

Seff

[Λ] is negative-definite.

3.2. If Λ ≤ Λ, any observable of energy at most Λ is in particular an

observable of energy at most Λ. Thus, there are inclusion maps

Obs≤Λ → Obs≤Λ

if Λ ≤ Λ.

Suppose we have a collection O1,...,On ∈ Obs≤Λ of observables of

energy at most Λ . The correlation functions between these observables

should be the same whether they are considered to lie in Obs≤Λ or Obs≤Λ.

That is,

φ∈C∞(M)≤Λ

eSeff

[Λ ](φ)/

O1(φ) · · · On(φ)dμ

=

φ∈C∞(M)≤Λ

eSeff

[Λ](φ)/

O1(φ) · · · On(φ)dμ.

It follows from this that

Seff

[Λ ](φL) = log

φH ∈C∞(M)(Λ

,Λ]

exp

1

Seff

[Λ](φL + φH )