20 1. INTRODUCTION
7.2. As our definition of renormalizability only makes sense on
Rn,
we
will now restrict to considering scalar field theories on
Rn.
We want to mea-
sure
Seff
[Λ] as Λ ∞, after we have changed units. Define
RGl(Seff
[Λ])
by
RGl(Seff
[Λ])(φ) =
Seff [l−2Λ](Rl(φ))
Thus,
RGl(Seff
[Λ]) is the effective action
Seff [l2Λ],
but measured in units
that have been rescaled by l.
We can use the map RGl to implement precisely the definition of renor-
malizability suggested above.
Definition 7.2.1. A theory
{Seff
[Λ]} is renormalizable if
RGl(Seff
[Λ])
grows at most logarithmically as l 0.
7.3. It turns out that the map RGl defines a flow on the space of theories.
Lemma 7.3.1. If
{Seff
[Λ]} satisfies the renormalization group equation,
then so does {RGl(Seff [Λ]}.
Thus, sending
{Seff
[Λ]}
{RGl(Seff
[Λ])}
defines a flow on the space of theories: this is the local renormalization group
flow.
Recall that the choice of a renormalization scheme leads to a bijection
between the space of theories and Lagrangians. Under this bijection, the
local renormalization group flow acts on the space of Lagrangians. The con-
stants appearing in a Lagrangian (the coupling constants) become functions
of l; the dependence of the coupling constants on the parameter l is called
the β function. Renormalizability means these coupling constants have at
most logarithmic growth in l.
The local renormalization group flow RGl, as l 0, can be interpreted
geometrically as focusing on smaller and smaller regions of space-time, while
always using units appropriate to the size of the region one is considering.
In energy terms, applying RGl as l 0 amounts to focusing on phenomena
of higher and higher energy.
The logarithmic growth condition thus says the theory doesn’t break
down completely when we probe high-energy phenomena. If the effective
actions displayed polynomial growth, for instance, then one would find that
the perturbative description of the theory wouldn’t make sense at high en-
ergy, because the terms in the perturbative expansion would increase with
the energy.
7.4. The definition of renormalizability given above can be viewed as a
perturbative approximation to an ideal non-perturbative definition.
Definition 7.4.1 (Ideal definition). A non-perturbative theory is renor-
malizable if, as we flow the theory under RGl and let l 0, we converge to
a fixed point.
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