8 1. INTRODUCTION

where the low-energy field φL is in

C∞(M)≤Λ

. 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

Seff

[Λ ] is obtained from

Seff

[Λ] by averaging over fluctuations

of the low-energy field φL ∈

C∞(M)≤Λ

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

C∞(M)

of the form

eS(φ)/

dφ,

where dφ refers to the (non-existent) Lebesgue measure on the vector space

C∞(M),

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

Seff

[Λ] is supposed

to be obtained from S by integrating out all fields of energy greater than Λ,

that is

Seff

[Λ](φL) = log

φH ∈C∞(M)(Λ,∞)

exp

1

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) = −

1

2

φ(D

+m2)φ.

This propagator P is defined to be

the integral kernel for the inverse of the operator D

+m2

appearing in the