Introduction to the Functional Renormalization Group

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Springer Science & Business Media, 03.05.2010 - 380 Seiten
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This book, based on a graduate course given by the authors, is a pedagogic and self-contained introduction to the renormalization group with special emphasis on the functional renormalization group. The functional renormalization group is a modern formulation of the Wilsonian renormalization group in terms of formally exact functional differential equations for generating functionals.

In Part I the reader is introduced to the basic concepts of the renormalization group idea, requiring only basic knowledge of equilibrium statistical mechanics. More advanced methods, such as diagrammatic perturbation theory, are introduced step by step.

Part II then gives a self-contained introduction to the functional renormalization group. After a careful definition of various types of generating functionals, the renormalization group flow equations for these functionals are derived. This procedure is shown to encompass the traditional method of the mode elimination steps of the Wilsonian renormalization group procedure. Then, approximate solutions of these flow equations using expansions in powers of irreducible vertices or in powers of derivatives are given.

Finally, in Part III the exact hierarchy of functional renormalization group flow equations for the irreducible vertices is used to study various aspects of non-relativistic fermions, including the so-called BCS-BEC crossover, thereby making the link to contemporary research topics.

 

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Inhalt

Foundations of the Renormalization Group
3
MeanField Theory and the Gaussian Approximation
23
Wilsonian Renormalization Group
53
Critical Behavior of the Ising Model Close to Four Dimensions 91
90
FieldTheoretical Renormalization Group
123
Introduction to the Functional Renormalization Group
140
Exact FRG Flow Equations
181
Vacuum Expectation Values
200
Vertex Expansion
209
Derivative Expansion
233
Functional Renormalization Group Approach to Fermions
248
Fermionic Functional Renormalization Group
255
Partial Bosonization in the Forward Scattering
305
Partial Bosonization in the ParticleParticle
327
Index
369
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