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
](φ)/
O1(φ) · · · On(φ)dμ
=
φ∈C∞(M)≤Λ
eSeff
[Λ](φ)/
O1(φ) · · · On(φ)dμ.
It follows from this that
Seff
](φL) = log
φH ∈C∞(M)(Λ
,Λ]
exp
1
Seff
[Λ](φL + φH )
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