5. LOCALITY 13

4. A Wilsonian definition of a quantum field theory

Any detector one could imagine has some finite resolution, and so only

probes some low-energy effective theory, described by some

Seff

[Λ]. How-

ever, one could imagine building detectors of arbitrarily high (but finite)

resolution, and so one could imagine probing

Seff

[Λ] for arbitrarily high

(but finite) Λ.

As is usual in physics, one should only consider those objects which can

in principle be observed. Thus, one should say that all aspects of a quantum

field theory are encoded in its various low-energy effective theories.

Let us make this into a (rough) definition. A more precise version of this

definition is given later in this introduction; a completely precise version is

given in the body of the book.

Definition 4.0.1. A (continuum) quantum field theory is:

(1) An effective action

Seff

[Λ] :

C∞(M)[0,Λ]

→ R[[ ]]

for all Λ ∈ (0, ∞). More precisely, Seff [Λ] should be a formal

power series both in the field φ ∈

C∞(M)[0,Λ]

and in the variable

.

(2) Modulo , each

Seff

[Λ] must be of the form

Seff

[Λ](φ) = −

1

2

M

φ D φ + cubic and higher terms.

where D is the positive-definite Laplacian. (If we want to consider

a massive scalar field theory, we can replace D by D

+m2).

(3) If Λ Λ,

Seff

[Λ ] is determined from

Seff

[Λ] by the renormaliza-

tion group equation (which makes sense in the formal power series

setting).

(4) The effective actions Seff [Λ] satisfy a locality axiom, which we will

sketch below.

Earlier I described several different versions of the renormalization group

equation; one based on the world-line formulation of quantum field theory,

and one defined by considering arbitrary parametrices for the Laplacian.

One gets an equivalent definition of quantum field theory using either of

these versions of the renormalization group flow.

5. Locality

Locality is one of the fundamental principles of quantum field theory.

Roughly, locality says that any interaction between fundamental particles

occurs at a point. Two particles at different points of space-time cannot

spontaneously affect each other. They can only interact through the medium

of other particles. The locality requirement thus excludes any “spooky ac-

tion at a distance”.