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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