As a fundamental branch of theoretical physics, quantum field theory has led, in the last 20 years, to spectacular progress in our understanding of phase transitions and elementary particles. This textbook emphasizes the underlying unity of the concepts and methods used in both domains, and presents in clear language topics such as the perturbative expansion, Feynman diagrams, renormalization, and the renormalization group. It contains detailed applications of critical phenomena to condensed matter physics, such as the calculation of critical exponents and a discussion of the XY model. Applications to particle physics include quantum electrodynamics and chromodynamics, electroweak interactions, and lattice gauge theories. The book is based on courses given over several years on statistical mechanics and field theory, and is written at graduate level. It attempts to guide the reader through a somewhat difficult and sometimes intricate subject in as clear a manner as
possible, leading to a level of understanding where more advanced textbooks and research articles will be accessible. The only textbook covering the subject at this level, the work is thus an ideal guide for graduate and postgraduate students in physics, researchers in quantum and statistical field theory, and those from other fields of physics seeking an introduction to quantum field theory. A large number of problems are given to test the reader's grasp of the ideas.
Author(s): Michel Le Bellac, Gabriel Barton
Edition: 1
Publisher: Clarendon Press
Year: 1991
Language: English
Pages: 608
City: Oxford, New York
Cover
Title
Copyright
Preface
Preface to the English edition
Notation and conventions
Contents
Part I - Critical phenomena
1 Introduction to critical phenomena
1.1 The ferromagnetic transition
1.2 The Ising model
1.3 The mean field
1.4 Correlation functions
1.5 Qualitative description of critical phenomena
Problems
Further reading
2 Landau theory
2.1 The Ginzburg-Landau Hamiltonian and the Landau approximation
2.2 The Landau theory of phase transitions
2.3 Correlation functions
2.4 Critique of the Landau approximation, and the Ginzburg criterion
Problems
Further reading
3 The renormalization group
3.1 Basic concepts: blocks of spins, critical surface, and fixed points
3.2 Behaviour near a fixed point. Critical exponents
3.3 The Ising model on a triangular lattice, and the approximation by cumulants
3.4 The Gaussian model
3.5 Calculation of the critical exponents to order epsilon
3.6 Marginal fields, the function beta(g), and logarithmic corrections in 4D
Problems
Further reading
4 Two-dimensional models
4.1 The XY model: qualitative study
4.2 Renormalization-group analysis
4.3 Nonlinear sigma-models
Problems
Further reading
Part II - Perturbation theory and renormalization: the Euclidean scalar field
5 The perturbation expansion and Feynman diagrams
5.1 Wick's theorem and the generating functional
5.2 The perturbation expansion of G(2) and of G(4): Feynman diagrams
5.3 Connected correlation functions and proper vertices
5.4 The effective potential: the loop expansion
5.5 The evaluation of Feynman diagrams
5.6 Power-counting: ultraviolet and infrared divergences
Problems
Further reading
6 Renormalization
6.1 Introduction
6.2 The renormalization of mass and coupling constant
6.3 Field renormalization: counterterms
6.4 The general case
6.5 Composite operators and their renormalization
6.6 The minimal subtraction scheme (MS)
Problems
Further reading
7 The Callan-Symanzik equations
7.1 The Callan-Symanzik equations at the critical temperature
7.2 The Callan-Symanzik equations for T > Tc
7.3 Renormalization and the renormalization group
7.4 The renormalization group in D = 4 dimensions
7.5 The renormalization group in dimensions D < 4
Problems
Further reading
Part III - The quantum theory of scalar fields
8 Path integrals in quantum and in statistical mechanics
8.1 Quantum spin and Ising model
8.2 Particle in a potential
8.3 Euclidean continuation, and comments
Problems
Further reading
9 Quantization of the Klein-Gordon field
9.1 The quantization of elastic vibrations
9.2 Quantization of the Klein-Gordon field
9.3 Coupling to a classical source, and Wick's theorem
Problems
Further reading
10 Green's functions and the S matrix
10.1 Perturbation expansion of the Green's functions
10.2 Path integrals and the Euclidean theory
10.3 Cross-sections and the S-matrix
10.4 The unitarity of the S-matrix
10.5 Generalizations
Problems
Further reading
Part IV - Gauge theories
11 Quantization of the Dirac field and of the electromagnetic field
11.1 Quantization of the Dirac field
11.2 Wick's theorem for fermions
11.3 The Lagrangean formalism for the classical electromagnetic field
11.4 Quantization of the electromagnetic field
Problems
Further reading
12 Quantum electrodynamics
12.1 The Feynman rules for quantum electrodynamics
12.2 Applications
12.3 One-loop diagrams in electrodynamics
12.4 Ward identities, unitarity, and renormalization
Problems
Further reading
13 Non-Abelian gauge theories
13.1 Non-Abelian gauge fields: the classical theory
13.2 The quantization of non-Abelian gauge theories
13.3 The Glashow-Salam-Weinberg model of electroweak interactions
13.4 Quantum chromodynamics
13.5 Lattice gauge theories
Problems
Further reading
Appendices
A Fourier transforms and Gaussian integration
A.1 Fourier transforms
A.2 Gaussian integrals
A.3 Integrals in D dimensions
B Feynman integrals with dimensional regularization (Euclidean case)
C Noether's theorem, conserved currents and Ward identities
C.1 Noether's theorem and conserved currents
C.2 Energy-momentum tensor
C.3 Ward identities
D Formulary
D.1 The Lorentz group
D.2 Dirac matrices
D.3 Cross-sections
D.4 Feynman rules
References
Index
