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](φ L + φH)

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