Preface Preface to the Third Edition General Sources and References PART Ⅰ BASICIDEAS AND TECHNIQUES 1 Pertinent concepts and ideas in the theory of critical phenomena 1—1 Description of critical phenomena 1—2 Scaling and homogeneity 1—3 Comparison of various results for critical exponents 1—4 Universality—dimensionality，symmetry Exercises 2 Formulation of the problem of phase transitions in terms of functional integrals 2—1 Introduction 2—2 Construction of the Lagrangian 2—2—1 The real scalar field 2—2—2 Complexfield 2—2—3 A hypercubic n—vector model 2—2—4 Two coupled fluctuating fields 2—3 The parameters appearing in ? 2—4 The partition function，or the generating functional 2—5 Representation of the Ising model in terms of functional integrals 2—5—1 Definition of the model and its thermodynamics 2—5—2 The Gaussian transformation 2—5—3 The free part 2—5—4 Some properties of the free theory—a free Euclidean field theory in less than four dimensions 2—6 Correlation functions including composite operators Exercises 3 Functional integrals in quanturn field theory 3—1 Introduction 3—2 Functionalintegrals for a quantum—mechanical system with one degree of freedom 3—2—1 Schwinger's transformation function 3—2—2 Matrix elements—Green functions 3—2—3 The generating functional 3—2—4 Analytic continuation in time—the Euclidean theory 3—3 Functional integrals for the scalar boson field theory 3—3—1 Introduction 3—3—2 The generating functional for Green functions 3—3—3 The generating functional as a functional integral 3—3—4 The S—matrix expressed in terms of the generating functional Exercises 4 Perturbation theory and Feynman graphs 4—1 Introduction 4—2 Perturbation expansionin coordinate space 4—3 The cancellation of vacuum graphs 4—4 Rules for the computation ofgraphs 4—5 More general cases 4—5—1 The M—vector theory 4—5—2 Comments on fields with higher spin 4—6 Diagrammatic expansion in momentum space 4—7 Perturbation expansion of Green functions with composite operators 4—7—1 In coordinate space
4—7—2 In momentum space 4—7—3 Insertion at zero momentum Exercises 5 Vertex functions and symmetry breaking 5—1 Introduction 5—2 Connected Green functions and their generating functional 5—3 The mass operator 5—4 The Legendre transform and vertex functions 5—5 The generating functional and the potential 5—6 Ward—Takahashi identities and Goldstone's theorem 5—7 Vertex parts for Green functions with composite operators Exercises 6 Expansions in the number of loops and in the number of components 6—1 Introduction 6—2 The expansion in the number of loops as a power series 6—3 The tree （Landau—Ginzburg）approximation 6—4 The one—loop approximation and the Ginzburg criterion 6—5 Mass and coupling constant renormalizationin the one—loop approximation 6—6 Composite field renormalization 6—7 Renormalization of the field at the two—loop level 6—8 The O（M）—symmetric theory in the limit of large M 6—8—1 Generalremarks 6—8—2 The origin of the M—dependence of the coupling constant 6—8—3 Faithful representation of graphs and the dominant terms inΓ（4） 6—8—4 Γ（2） in theinfinite M limit 6—8—5 Renormalization 6—8—6 Broken symmetry Appendix 6—1 The method of steepest descent and the loop expansion Exercises 7 Renormalization 7—1 Introduction 7—2 Some considerations concerning engineering dimensions 7—3 Power counting and primitive divergences 7—4 Renormalization of a cutoff φ4 theory 7—5 Normalization conditions for massive and massless theories 7—6 Renormalization constants for a massless theory to order two loops 7—7 Renormalization away from the critical point 7—8 Counterterms 7—9 Relevant and irrelevant operators 7—10 Renormalization of a φ4 theory with an O（M） symmetry 7—11 Ward identities and renormalization 7—12 Iterative construction of counterterms Exercises 8 The renormalization group and scaling in the critical region 8—1 Introduction 8—2 The renormalization group for the critical （massless） theory 8—3 Regularization by continuation in the number of dimensions 8—4 Massless theory below four dimensions—the emergence of ε 8—5 The solution of the renormalization group equation 8—6 Fixed points， scaling， and anomalous dimensions
8—7 The approach to the fixed point—asymptotic freedom 8—8 Renormalization group equation above Tc—identification of v 8—9 Below the critical temperature—the scaling form of the equation of state 8—10 The specific heat—renormalization group equation for an additively renormalized vertex 8—11 The Callan—Symanzik equations 8—12 Renormalization group equations for the bare theory 8—13 Renormalization group equations and scaling in the infinite M limit Appendix 8—1 General formulas for calculating Feynman integrals Exercises 9 The computation of the critical exponents 9—1 Introduction 9—2 The symbolic calculation of the renormalization constants and Wilson functions 9—3 The εexpansion of the critical exponents 9—4 The nature of the fixed points —universality 9—5 Scale invariance at finite cutoff 9—6 At the critical dimension —asymptotic infrared freedom 9—7 ε expansion for the Callan—Symanzik method 9—8 εexpansion of the renormalization group equations for the bare functions 9—9 Dimensional regularization and critical phenomena 9—10 Renormalization by minimal subtraction of dimensional poles 9—11 The calculation of exponents in minimal subtraction Appendix 9—1 Calculation of some integrals with cutoff 9—2 One—Ioop integrals in dimensional regularization 9—3 Two—Ioop integrals in dimensional regularization Exercises PART Ⅱ FURTHER APPLICATIONS AND DEVELOPMENTS
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