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
] 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”.
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