arXiv:hep-th/9210046 v2
8 Jun 1999
NSF-ITP-92-132
UTTG-20-92
hep-th/9210046
Effective Field Theory and the Fermi Surface
Joseph Polchinski
∗
Institute for Theoretical Physics
University of California
Santa Barbara, California 93106-4030
and
Theory Group
Department of Physics
University of Texas
Austin, Texas 78712
ABSTRACT
This is an introduction to the method of effective field theory. As an
application, I derive the effective field theory of low energy excita-
tions in a conductor, the Landau theory of Fermi liquids, and explain
why the high-
T
c
superconductors must be described by a different
effective field theory.
Lectures presented at TASI 1992.
*
[email protected]

Effective field theory is a very powerful tool in quantum field theory, and
in particular gives a new point of view about the meaning of renormalization.
It is approximately two decades old and appears throughout the literature,
but it is ignored in most textbooks. It is therefore appropriate to devote two
lectures to effective field theory here at the TASI school.
1
In the first lecture I will describe the general method and ideology. In the
second I will develop in detail one application—the effective field theory of
the low-energy excitations in a metal, which is known as the Landau theory
of Fermi liquids. This is an unusual subject for a school on particle physics,
but you will see that it is a beautiful example of the main themes in effective
field theory. As a bonus, we will be able to understand something about the
high temperature superconducting materials, and why it appears that they
require a new idea in quantum field theory.
Lecture 1: Effective Field Theory
Consider a quantum field theory with a characteristic energy scale
E
0
, and
suppose we are interested in the physics at some lower scale
E <<E
0
. Of
course, most systems have several characteristic scales, but we can consider
them one at a time. The next lecture will illustrate the treatment of a system
with two scales.
Effective field theory is a general method for analyzing this situation.
Choose a cutoff Λ at or slightly below
E
0
, and divide the fields in the path
integral into high- and low- frequency parts,
φ
=
φ
H
+
φ
L
φ
H
:
ω>
Λ
φ
L
:
ω<
Λ
.
(1)
1
See also the lecture by Peter Lepage in the 1989 TASI Proceedings.
1

For the rather general purposes of these lectures, we do not need to specify
how this is done—whether the separation is sharp or smooth, how Lorentz
and other symmetries are preserved, and so on. Now do the path integral
over the high-frequency part,
integraldisplay
D
φ
L
D
φ
H
e
iS
(
φ
L
,φ
H
)
=
integraldisplay
D
φ
L
e
iS
Λ
(
φ
L
)
,
(2)
where
e
iS
Λ
(
φ
L
)
=
integraldisplay
D
φ
H
e
iS
(
φ
L
,φ
H
)
.
(3)
We are left with an integral with an upper frequency cutoff Λ and an effective
action
S
Λ
(
φ
L
). This is known as a
low energy
or
Wilsonian
effective action,
to distinguish it from the 1PI effective action generated by integrating over
all frequencies but keeping only 1PI graphs. This point of view is identified
with Wilson, though it has many roots in the literature; see the Bibliography.

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