Giuseppe Benfatto, Giovanni Gallavotti

Renormalization Group (eBook, PDF)

• Format: PDF

Produktbeschreibung

Scaling and self-similarity ideas and methods in theoretical physics have, in the last twenty-five years, coalesced into renormalization-group methods. This book analyzes, from a single perspective, some of the most important applications: the critical-point theory in classical statistical mechanics, the scalar quantum field theories in two and three space-time dimensions, and Tomonaga's theory of the ground state of one-dimensional Fermi systems.

The dimension dependence is discussed together with the related existence of anomalies (in Tomonaga's theory and in 4 -e dimensions for the critical point). The theory of Bose condensation at zero temperature in three space dimensions is also considered. Attention is focused on results that can in principle be formally established from a mathematical point of view. The 4 -e dimensions theory, Bose condensation, as well as a few other statements are exceptions to this rule, because no complete treatment is yet available. However, the truly mathematical details are intentionally omitted and only referred to. This is done with the purpose of stressing the unifying conceptual structure rather than the technical differences or subtleties.

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Produktdetails

• Verlag: University of Pennsylvania Press, Inc.
• Erscheinungstermin: 10. November 2020
• Englisch
• ISBN-13: 9780691221694
• Artikelnr.: 60201066

Autorenporträt

Giuseppe Benfatto is Professor of Mathematical Physics at the University of Rome-Tor Vergata, and Giovanni Gallavotti is Professor of Rational Mechanics at the University of Rome-La Sapienza.

Inhaltsangabe

Preface
Ch. 1
Introduction
3
Ch. 2
Problems Equivalent to the Analysis of Suitable Functional
Integrals: Critical Point and Field Theory
5
Ch. 3
Other Functional Integrals: Fermi Sphere and Bose Condensation
12
Ch. 4
Effective Potentials and Schwinger Functions
19
Ch. 5
Multiscale Decomposition of Propagators and Fields: Running
Effective Potentials
22
Ch. 6
Renormalization Group: Relevant and Irrelevant Components of the
Effective Potentials
27
Ch. 7
Asymptotic Freedom: Upper Critical Dimension
34
Ch. 8
Beyond the Linear Approximations: The Beta Function and Perturbation
Theory
38
Ch. 9
The Beta Function as a Dynamical System: Asymptotic Freedom of
Marginal Theories
52
Ch. 10
Anomalous Dimension
56
Ch. 11
The Fermi Liquid and the Luttinger Model
66
Ch. 12
The Generic Critical Point for d = 3, [Gamma] = 0: The
[Epsilon]--Expansion
70
Ch. 13
Bose Condensation: Reformulation
75
Ch. 14
Bose Condensation: Effective Potentials
81
Ch. 15
The Beta Function for the Bose Condensation
87
A Brief Historical Note
96
Bibliographical Notes
100
Appendix 1. The Free Fermion Propagator
104
Appendix 2. Grassmannian Integration
106
Appendix 3. Trees and Feynman Graphs
111
Appendix 4. Schwinger Functions and Anomalous Dimension
120
Appendix 5. Propagators for the Bose Gas
124
Appendix 6. The Beta Function for the Bose Gas
126
References
135
Subject Index
141
Citation Index
143