where the low-energy field φL is in
. This is a finite dimensional
integral, and so (under mild conditions) is well defined as formal power series
in .
This equation is called the renormalization group equation. It says that if
Λ Λ, then
] is obtained from
[Λ] by averaging over fluctuations
of the low-energy field φL
with energy between Λ and Λ.
3.3. Recall that in the naive functional-integral point of view, there is
supposed to be a measure on the space
of the form
where refers to the (non-existent) Lebesgue measure on the vector space
and S(φ) is a function of the field φ.
It is natural to ask what role the “original” action S plays in the Wilso-
nian low-energy picture. The answer is that S is supposed to be the “energy
infinity effective action”. The low energy effective action
[Λ] is supposed
to be obtained from S by integrating out all fields of energy greater than Λ,
that is
[Λ](φL) = log
φH ∈C∞(M)(Λ,∞)
S(φL + φH ) .
This is a functional integral over the infinite dimensional space of fields with
energy greater than Λ. This integral doesn’t make sense; the terms in its
Feynman graph expansion are divergent.
However, one would not expect this expression to be well-defined. The
infinite energy effective action should not be defined; one would not expect to
have a description of how particles behave at infinite energy. The infinities
in the naive functional integral picture arise because the classical action
functional S is treated as the infinite energy effective action.
3.4. So far, I have explained how to define a renormalization group
equation using the eigenvalues of the Laplacian. This picture is very easy to
explain, but it has many disadvantages. The principal disadvantage is that
this definition is not local on space-time. Thus, it is difficult to integrate
the locality requirements of quantum field theory into this version of the
renormalization group flow.
In the body of this book, I will use a version of the renormalization group
flow that is based on length rather than on energy. A complete account of
this will have to wait until Chapter 2, but I will give a brief description here.
The version of the renormalization group flow based on length is not
derived directly from Feynman’s functional integral formulation of quantum
field theory. Instead, it is derived from a different (though ultimately equiv-
alent) formulation of quantum field theory, again due to Feynman (Fey50).
Let us consider the propagator for a free scalar field φ, with action
Sfree(φ) = Sk(φ) =
This propagator P is defined to be
the integral kernel for the inverse of the operator D
appearing in the
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