3. WILSONIAN LOW ENERGY THEORIES 7
Detectors that can only see phenomena of energy at most Λ can be
represented by functions
→ R[[ ]],
which are extended to
via the projection
Let us denote by Obs≤Λ the space of all functions on
in this way. Elements of Obs≤Λ will be referred to as observables of energy
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
Thus, we will assume that
there is a measure on
of the form
where D φ is the Lebesgue measure, and
[Λ] is a function on Obs≤Λ,
O1(φ) · · · On(φ)Dφ
for all low-energy observables Oi ∈ Obs≤Λ.
[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
[Λ] appears in
the functional integral via
, instead of
as is more usual.
We will assume the quadratic part of
[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.
O1(φ) · · · On(φ)dμ
O1(φ) · · · On(φ)dμ.
It follows from this that
[Λ ](φL) = log
[Λ](φL + φH )