to be a constant for simplic-
ity. This is the Kondo problem. The nonvanishing beta-function means that
the coupling grows with decreasing energy (for
J
positive). This is vividly
seen in measurements of resistivity as a function of temperature, which in-
creases as
T
decreases rather than showing the simple constant behavior of
potential scattering. When the coupling gets strong, a number of behaviors
are possible, depending on the value of
s
, sign of
J
, and various general-
izations.
In particular, in some cases one finds fixed points with critical
behavior given by rather nontrivial conformal field theories: more examples
of the interesting things that can happen when a marginal coupling gets
strong!
Exercise:
Show that if the Fermi surface is right at a van Hove singular-
ity, then under scaling of the energy to zero and of the momenta
toward the
singular point,
the four-Fermi interaction is marginal in
two
space dimensions.
In other words, if all electron momenta in a graph lie near the singularity,
the graph is marginal: one does not have the usual simplifications of Landau
theory.
Acknowledgements
I would like to thank I. Affleck, B. Blok, N. Bulut, F. de Wette, V. Kaplunov-
sky, A. Ludwig, M. Marder, J. Markert, D. Minic, M. Natsuume, D. Scalapino,
E. Smith, and L. Susskind for helpful remarks and conversations.
I would
also like to thank R. Shankar for discussions of his work and S. Weinberg for
comments on the manuscript. This work was supported in part by National
31

Science Foundation grants PHY89-04035 and PHY90-09850, by the Robert
A. Welch Foundation, and by the Texas Advanced Research Foundation.
Bibliography
Lecture 1
Wilson’s approach to the effective action is developed in
K. G. Wilson, Phys. Rev.
B4
(1971) 3174, 3184.
For further developments see
K. G. Wilson and J. G. Kogut, Phys. Rep.
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(1974) 75;
F. J. Wegner, in
Phase Transitions and Critical Phenomena, Vol. 6,
ed. C. Domb and M. S. Green, Academic Press, London, 1976;
L. P. Kadanoff, Rev. Mod. Phys.
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K. G. Wilson, Rev. Mod. Phys.
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As I hope is clear from the discussion, these ideas do not depend on pertur-
bation theory, and have been used to prove the existence of the continuum
limit nonpertubatively in asymptotically free theories. This is done for the
Gross-Neveu model in
K. Gawedzki and A. Kupiainen, Comm. Math. Phys.
102
(1985) 1,
and for
D
= 4 non-Abelian gauge theories in a series of papers culminating
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(1989) 355.
The idea that pion physics can be encoded in a Lagrangian appeared in
32

S. Weinberg, Phys.
Rev.
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Rev.
166
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and was developed further in
S. Coleman, J. Wess, and B. Zumino, Phys. Rev.
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C. G. Callan, S. Coleman, J. Wess, and B. Zumino, Phys. Rev.
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S. Weinberg, Physica
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H. Georgi,
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jamin/Cummings, Menlo Park, 1984.

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