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1 Gaussian integrals. Algebraic preliminaries |
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1 | (17) |
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1.1 Gaussian integrals: Wick's theorem |
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1 | (2) |
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1.2 Perturbative expansion. Connected contributions |
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3 | (1) |
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1.3 The steepest descent method |
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4 | (1) |
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1.4 Complex structures and Gaussian integrals |
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5 | (1) |
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1.5 Grassmann algebras. Differential forms |
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6 | (2) |
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1.6 Differentiation and integration in Grassmann algebras |
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8 | (5) |
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1.7 Gaussian integrals with Grassmann variables |
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13 | (3) |
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1.8 Legendre transformation |
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16 | (2) |
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2 Euclidean path integrals and quantum mechanics (QM) |
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18 | (24) |
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2.1 Markovian evolution and locality |
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19 | (1) |
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2.2 Statistical operator: Path integral representation |
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20 | (4) |
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2.3 Explicit evaluation of a path integral: The harmonic oscillator |
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24 | (1) |
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2.4 Partition function: Classical and quantum statistical physics |
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25 | (2) |
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2.5 Correlation functions. Generating functional |
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27 | (3) |
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2.6 Harmonic oscillator. Correlation functions and Wick's theorem |
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30 | (3) |
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2.7 Perturbed harmonic oscillator |
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33 | (2) |
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2.8 Semi-classical expansion |
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35 | (3) |
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38 | (4) |
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A2.1 A useful relation between determinant and trace |
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38 | (1) |
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A2.2 The two-point function: An integral representation |
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39 | (1) |
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A2.3 Time-ordered products of operators |
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40 | (2) |
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3 Quantum mechanics (QM): Path integrals in phase space |
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42 | (22) |
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3.1 General Hamiltonians: Phase-space path integral |
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42 | (3) |
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3.2 The harmonic oscillator. Perturbative expansion |
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45 | (2) |
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3.3 Hamiltonians quadratic in momentum variables |
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47 | (4) |
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3.4 The spectrum of the O(2)-symmetric rigid rotator |
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51 | (1) |
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3.5 The spectrum of the O(Ar)-symmetric rigid rotator |
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52 | (4) |
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A3 Quantization. Topological actions: Quantum spins, magnetic monopoles |
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56 | (8) |
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A3.1 Symplectic form and quantization: General remarks |
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56 | (2) |
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A3.2 Classical equations of motion and quantization |
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58 | (2) |
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60 | (4) |
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4 Quantum statistical physics: Functional integration formalism |
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64 | (26) |
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4.1 One-dimensional QM: Holomorphic representation |
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64 | (3) |
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4.2 Holomorphic path integral |
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67 | (4) |
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4.3 Several degrees of freedom. Boson interpretation |
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71 | (1) |
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4.4 The Bose gas. Field integral representation |
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72 | (8) |
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4.5 Fermion representation and complex Grassmann algebras |
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80 | (3) |
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4.6 Path integrals with fermions |
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83 | (4) |
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4.7 The Fermi gas. Field integral representation |
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87 | (3) |
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5 Quantum evolution: From particles to non-relativistic fields |
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90 | (15) |
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5.1 Time evolution and scattering matrix in quantum mechanics (QM) |
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90 | (2) |
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5.2 Path integral and 5-matrix: Perturbation theory |
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92 | (3) |
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5.3 Path integral and 5-matrix: Semi-classical expansions |
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95 | (4) |
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5.4 5-matrix and holomorphic formalism |
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99 | (3) |
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5.5 The Bose gas: Evolution operator |
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102 | (1) |
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5.6 Fermi gas: Evolution operator |
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103 | (1) |
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A5 Perturbation theory in the operator formalism |
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104 | (1) |
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6 The neutral relativistic scalar field |
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105 | (20) |
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6.1 The relativistic scalar field |
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105 | (5) |
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6.2 Quantum evolution and the 5-matrix |
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110 | (2) |
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6.3 5-matrix and field asymptotic conditions |
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112 | (4) |
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6.4 The non-relativistic limit: The Φ4 QFT |
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116 | (2) |
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6.5 Quantum statistical physics |
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118 | (4) |
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6.6 Kallen-Lehmann representation and field renormalization |
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122 | (3) |
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7 Perturbative quantum field theory (QFT): Algebraic methods |
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125 | (35) |
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7.1 Generating functional of correlation functions |
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126 | (1) |
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7.2 Perturbative expansion. Wick's theorem and Feynman diagrams |
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127 | (2) |
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7.3 Connected correlation functions: Generating functional |
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129 | (2) |
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7.4 The example of the Φ4 QFT |
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131 | (2) |
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7.5 Algebraic properties of field integrals. Quantum field equations |
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133 | (6) |
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7.6 Connected correlation functions. Cluster properties |
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139 | (2) |
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7.7 Legendre transformation. Vertex functions |
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141 | (3) |
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7.8 Momentum representation |
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144 | (2) |
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7.9 Loop or semi-classical expansion |
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146 | (5) |
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7.10 Vertex functions: One-line irreducibility |
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151 | (1) |
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7.11 Statistical and quantum interpretation of the vertex functional |
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152 | (3) |
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A7 Additional results and methods |
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155 | (5) |
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A7.1 Generating functional at two loops |
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155 | (1) |
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A7.2 The background field method |
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156 | (1) |
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A7.3 Connected Feynman diagrams: Cluster properties |
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157 | (3) |
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8 Ultraviolet divergences: Effective field theory (EFT) |
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160 | (25) |
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8.1 Gaussian expectation values and divergences: The scalar field |
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161 | (1) |
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8.2 Divergences of Feynman diagrams: Power counting |
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162 | (2) |
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8.3 Classification of interactions in scalar quamtum field theories |
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164 | (2) |
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8.4 Momentum regularization |
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166 | (3) |
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8.5 Example: The Φd=6 field theory at one-loop order |
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169 | (4) |
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8.6 Operator insertions: Generating functionals, power counting |
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173 | (2) |
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8.7 Lattice regularization. Classical statistical physics |
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175 | (1) |
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8.8 Effective QFT. The fine-tuning problem |
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176 | (3) |
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8.9 The emergence of renormalizable field theories |
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179 | (2) |
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181 | (4) |
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A8.1 Schwinger's proper-time representation |
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181 | (1) |
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A8.2 Regularization and one-loop divergences |
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181 | (3) |
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A8.3 More general momentum regularizations |
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184 | (1) |
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9 Introduction to renormalization theory and renormalization group (RG) |
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185 | (35) |
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9.1 Power counting. Dimensional analysis |
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186 | (1) |
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9.2 Regularization. Bare and renormalized QFT |
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187 | (4) |
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191 | (3) |
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9.4 Divergences beyond one-loop: Skeleton diagrams |
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194 | (2) |
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9.5 Callan-Symanzik equations |
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196 | (2) |
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9.6 Inductive proof of renormalizability |
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198 | (5) |
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9.7 The (Φ2Φ2) vertex function |
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203 | (1) |
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9.8 The renormalized action: General construction |
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204 | (1) |
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204 | (4) |
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9.10 Homogeneous RG equations: Massive QFT |
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208 | (2) |
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210 | (2) |
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9.12 Solution of bare RG equations: The triviality issue |
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212 | (2) |
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A9 Functional RG equations. Super-renormalizable QFTs. Normal order |
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214 | (6) |
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A9.1 Large-momentum mode integration and functional RG equations |
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214 | (2) |
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A9.2 The Φ4 QFT in three dimensions: Divergences |
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216 | (2) |
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A9.3 Super-renormalizable scalar QFTs in two dimensions: Normal order |
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218 | (2) |
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10 Dimensional continuation, regularization, minimal subtraction (MS). Renormalization group (RG) functions |
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220 | (20) |
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10.1 Dimensional continuation and dimensional regularization |
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220 | (4) |
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224 | (2) |
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10.3 The structure of renormalization constants |
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226 | (1) |
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227 | (3) |
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10.5 RG functions at two-loop order: The Φ4 QFT |
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230 | (5) |
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10.6 Generalization to N-component fields |
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235 | (4) |
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A10 Feynman parametrization |
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239 | (1) |
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11 Renormalization of local polynomials. Short-distance expansion (SDE) |
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240 | (18) |
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11.1 Renormalization of operator insertions |
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240 | (5) |
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11.2 Quantum field equations |
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245 | (3) |
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11.3 Short-distance expansion of operator products |
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248 | (5) |
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11.4 Large-momentum expansion of the SDE coefficients: CS equations |
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253 | (2) |
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11.5 SDE beyond leading order. General operator product |
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255 | (1) |
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11.6 Light-cone expansion of operator products |
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256 | (2) |
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12 Relativistic fermions: Introduction |
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258 | (34) |
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12.1 Massive Dirac fermions |
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258 | (5) |
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12.2 Self-interacting massive fermions: Non-relativistic limit |
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263 | (2) |
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12.3 Free Euclidean relativistic fermions |
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265 | (4) |
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12.4 Partition function. Correlations |
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269 | (1) |
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12.5 Generating functionals |
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270 | (2) |
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12.6 Connection between spin and statistics |
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272 | (2) |
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12.7 Divergences and momentum cut-off |
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274 | (2) |
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12.8 Dimensional regularization |
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276 | (1) |
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12.9 Lattice fermions and the doubling problem |
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276 | (4) |
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A12 Euclidean fermions, spin group and 7 matrices |
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280 | (12) |
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A12.1 Spin group. Dirac 7 matrices |
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280 | (8) |
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A12.2 The example of dimension 4 |
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288 | (1) |
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A12.3 The Fierz transformation |
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289 | (1) |
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A12.4 Traces of products of 7 matrices |
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290 | (2) |
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13 Symmetries, chiral symmetry breaking, and renormalization |
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292 | (32) |
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13.1 Lie groups and algebras: Preliminaries |
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293 | (2) |
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13.2 Linear global symmetries and WT identities |
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295 | (3) |
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13.3 Linear symmetry breaking |
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298 | (3) |
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13.4 Spontaneous symmetry breaking |
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301 | (3) |
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13.5 Chiral symmetry breaking in strong interactions: Effective theory |
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304 | (2) |
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306 | (4) |
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310 | (3) |
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13.8 Quadratic symmetry breaking |
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313 | (4) |
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A13 Currents and Noether's theorem |
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317 | (7) |
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A13.1 Currents in classical-field theory |
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317 | (1) |
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A13.2 The energy-momentum tensor |
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318 | (2) |
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A13.3 Euclidean theory: Dilatation and conformal invariance |
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320 | (2) |
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A13.4 QFT: Currents and correlation functions |
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322 | (1) |
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A13.5 Energy-momentum tensor and QFT |
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323 | (1) |
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14 Critical phenomena: General considerations. Mean-field theory (MFT) |
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324 | (33) |
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325 | (3) |
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14.2 The infinite transverse size limit: Ising-like systems |
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328 | (3) |
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14.3 Continuous symmetries |
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331 | (1) |
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14.4 Mean-field approximation |
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332 | (5) |
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14.5 Universality within mean-field approximation |
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337 | (5) |
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14.6 Beyond the mean-field approximation |
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342 | (3) |
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14.7 Power counting and the role of dimension 4 |
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345 | (1) |
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346 | (1) |
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A14 Additional considerations |
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347 | (10) |
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A14.1 High-temperature expansion |
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347 | (1) |
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A14.2 Mean-field approximation: General formalism |
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348 | (3) |
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A14.3 Mean-field expansion |
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351 | (1) |
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A14.4 High-, low-temperature, and mean-field expansions |
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352 | (2) |
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354 | (3) |
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15 The renormalization group (RG) approach: The critical theory near four dimensions |
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357 | (34) |
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15.1 RG: The general idea |
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358 | (5) |
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15.2 The Gaussian fixed point |
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363 | (3) |
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15.3 Critical behaviour: The effective φ4 field theory |
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366 | (2) |
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15.4 RG equations near four dimensions |
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368 | (2) |
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15.5 Solution of the RG equations: The ε-expansion |
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370 | (2) |
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15.6 Critical correlation functions with Φ2(x) insertions |
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372 | (4) |
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15.7 The O(N)-symmetric (Φ2)2 field theory |
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376 | (1) |
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15.8 Statistical properties of long self-repelling chains |
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377 | (5) |
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15.9 Liquid-vapour phase transition and Φ4 field theory |
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382 | (5) |
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15.10 Superfluid transition |
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387 | (4) |
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16 Critical domain: Universality, ε-expansion |
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391 | (30) |
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16.1 Strong scaling above Tc: The renormalized theory |
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392 | (4) |
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16.2 Critical domain: Homogeneous RG equations |
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396 | (1) |
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16.3 Scaling properties above Tc |
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396 | (3) |
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16.4 Correlation functions with Φ2 insertions |
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399 | (1) |
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16.5 Scaling properties in a magnetic field and below Tc |
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400 | (3) |
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403 | (2) |
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16.7 The general N-vector model |
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405 | (5) |
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16.8 Asymptotic expansion of the two-point function |
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410 | (2) |
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16.9 Some universal quantities as ε expansions |
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412 | (8) |
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16.10 Conformal bootstrap |
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420 | (1) |
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17 Critical phenomena: Corrections to scaling behaviour |
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421 | (15) |
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17.1 Corrections to scaling: Generic dimensions |
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421 | (2) |
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17.2 Logarithmic corrections at the upper-critical dimension |
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423 | (3) |
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17.3 Irrelevant operators and the question of universality |
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426 | (2) |
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17.4 Corrections coming from irrelevant operators. Improved action |
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428 | (3) |
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17.5 Application: Uniaxial systems with strong dipolar forces |
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431 | (5) |
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18 O(N)-symmetric vector models for N large |
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436 | (22) |
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436 | (2) |
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18.2 Large N limit: Saddle point equations, critical domain |
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438 | (7) |
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18.3 Renormalization group (RG) functions and leading corrections to scaling |
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445 | (2) |
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18.4 Small-coupling constant, large-momentum expansions for d > 4 |
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447 | (1) |
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18.5 Dimension 4: Triviality issue for N large |
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448 | (1) |
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18.6 The (Φ2)2 field theory and the non-linear σ-model for N large |
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449 | (4) |
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18.7 The 1/N-expansion: An alternative field theory |
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453 | (2) |
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18.8 Explicit calculations |
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455 | (3) |
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19 The non-linear σ-model near two dimensions: Phase structure |
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458 | (31) |
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19.1 The non-linear σ-model: Definition |
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459 | (2) |
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19.2 Perturbation theory. Power counting |
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461 | (2) |
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463 | (1) |
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464 | (2) |
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19.5 WT identities and master equation |
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466 | (3) |
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469 | (2) |
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19.7 The renormalized action: Solution to the master equation |
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471 | (3) |
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19.8 Renormalization of local functionals |
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474 | (1) |
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19.9 A linear representation |
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475 | (1) |
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19.10 (Φ2)2 field theory in the ordered phase and non-linear cr-model |
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476 | (3) |
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19.11 Renormalization, RG equations |
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479 | (1) |
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19.12 RG equations: Solutions (magnetic terminology) |
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480 | (6) |
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19.13 Results beyond one-loop order |
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486 | (2) |
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19.14 The dimension 2: Asymptotic freedom |
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488 | (1) |
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20 Gross-Neveu-Yukawa and Gross-Neveu models |
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489 | (18) |
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20.1 The GNY model: Spontaneous mass generation |
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489 | (5) |
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20.2 RG equations near four dimensions |
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494 | (4) |
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20.3 The GNY model in the large N limit |
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498 | (3) |
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20.4 The large N expansion |
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501 | (3) |
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504 | (3) |
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21 Abelian gauge theories: The framework of quantum electrodynamics (QED) |
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507 | (41) |
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21.1 The free massive vector field: Quantization |
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507 | (2) |
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21.2 The Euclidean free action. The two-point function |
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509 | (3) |
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512 | (2) |
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21.4 The massless limit: Gauge invariance |
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514 | (2) |
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21.5 Massless vector field, gauge invariance, and quantization |
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516 | (3) |
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21.6 Equivalence with covariant quantization |
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519 | (2) |
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21.7 Gauge symmetry and parallel transport |
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521 | (1) |
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21.8 Perturbation theory: Regularization |
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522 | (4) |
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21.9 WT identities and renormalization |
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526 | (2) |
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21.10 Gauge dependence: The fermion two-point function |
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528 | (3) |
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21.11 Renormalization and RG equations |
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531 | (1) |
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21.12 One-loop β function and the triviality issue |
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532 | (3) |
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21.13 The Abelian Landau-Ginzburg-Higgs model |
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535 | (2) |
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21.14 The Landau-Ginzburg-Higgs model: WT identities |
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537 | (1) |
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21.15 Spontaneous symmetry breaking: Decoupling gauge |
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538 | (1) |
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21.16 Physical observables. Unitarity of the 5-matrix |
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539 | (1) |
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21.17 Stochastic quantization: The example of gauge theories |
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540 | (2) |
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542 | (6) |
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A21.1 Vacuum energy and Casimir effect |
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542 | (3) |
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545 | (1) |
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A21.3 Divergences at one loop from Schwinger's representation |
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546 | (2) |
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22 Non-Abelian gauge theories: Introduction |
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548 | (19) |
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22.1 Geometric construction: Parallel transport |
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548 | (3) |
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22.2 Gauge-invariant actions |
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551 | (1) |
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22.3 Hamiltonian formalism. Quantization in the temporal gauge |
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551 | (3) |
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554 | (3) |
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22.5 Perturbation theory, regularization |
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557 | (2) |
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22.6 The non-Abelian Higgs mechanism |
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559 | (6) |
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A22 Massive Yang-Mills fields |
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565 | (2) |
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23 The Standard Model (SM) of fundamental interactions |
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567 | (26) |
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23.1 Weak and electromagnetic interactions: Gauge and scalar fields |
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568 | (2) |
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23.2 Leptons: Minimal SM extension with Dirac neutrinos |
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570 | (3) |
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23.3 Quarks and weak-electromagnetic interactions |
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573 | (3) |
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23.4 QCD. RG equations and β function |
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576 | (2) |
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23.5 General RG β-function at one-loop order: Asymptotic freedom |
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578 | (4) |
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23.6 Axial current, chiral gauge theories, and anomalies |
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582 | (9) |
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23.7 Anomalies: Applications in particle physics |
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591 | (2) |
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24 Large-momentum behaviour in quantum field theory (QFT) |
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593 | (14) |
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24.1 The (Φ2)2 Euclidean field theory: Large-momentum behaviour |
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593 | (5) |
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24.2 General Φ4-like field theories: d=4 |
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598 | (2) |
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24.3 Theories with scalar bosons and Dirac fermions |
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600 | (2) |
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602 | (2) |
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24.5 Applications: The theory of strong interactions |
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604 | (3) |
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25 Lattice gauge theories: Introduction |
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607 | (16) |
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25.1 Gauge invariance on the lattice: Parallel transport |
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607 | (2) |
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25.2 The matterless gauge theory |
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609 | (2) |
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25.3 Wilson's loop and confinement |
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611 | (6) |
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25.4 Mean-field approximation |
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617 | (4) |
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A25 Gauge theory and confinement in two dimensions |
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621 | (2) |
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26 Becchi-Rouet-Stora-Tyutin (BRST) symmetry. Gauge theories: Zinn-Justin equation (ZJ) and renormalization |
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623 | (33) |
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26.1 ST identities: The origin |
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624 | (2) |
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26.2 From ST symmetry to BRST symmetry |
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626 | (2) |
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26.3 BRST symmetry: More general coordinates. Group structure |
|
|
628 | (2) |
|
26.4 Stochastic equations |
|
|
630 | (2) |
|
26.5 BRST symmetry, Grassmann coordinates, and gradient equations |
|
|
632 | (3) |
|
26.6 Gauge theories: Notation and algebraic structure |
|
|
635 | (1) |
|
26.7 Gauge theories: Quantization |
|
|
636 | (3) |
|
26.8 WT identities and ZJ equation |
|
|
639 | (2) |
|
26.9 Renormalization: General considerations |
|
|
641 | (1) |
|
26.10 The renormalized gauge action |
|
|
642 | (5) |
|
26.11 Gauge independence: Physical observables |
|
|
647 | (2) |
|
A26 BRST symmetry and ZJ equation: Additional remarks |
|
|
649 | (7) |
|
A26.1 BRST symmetry and ZJ equation |
|
|
649 | (1) |
|
A26.2 Canonical invariance of the ZJ equation |
|
|
650 | (1) |
|
A26.3 Elements of BRST cohomology |
|
|
651 | (3) |
|
A26.4 From BRST symmetry to supersymmetry |
|
|
654 | (2) |
|
27 Supersymmetric quantum field theory (QFT): Introduction |
|
|
656 | (14) |
|
27.1 Scalar superfields in three dimensions |
|
|
656 | (5) |
|
27.2 The O(N) supersymmetric non-linear a model |
|
|
661 | (1) |
|
27.3 Supersymmetry in four dimensions |
|
|
662 | (4) |
|
27.4 Vector superfields and gauge invariance |
|
|
666 | (4) |
|
28 Elements of classical and quantum gravity |
|
|
670 | (22) |
|
28.1 Manifolds. Change of coordinates. Tensors |
|
|
671 | (2) |
|
28.2 Parallel transport: Connection, covariant derivative |
|
|
673 | (4) |
|
28.3 Riemannian manifold. The metric tensor |
|
|
677 | (1) |
|
28.4 The curvature (Riemann) tensor |
|
|
678 | (4) |
|
28.5 Fermions, vielbein, spin connection |
|
|
682 | (2) |
|
28.6 Classical GR. Equations of motion |
|
|
684 | (3) |
|
28.7 Quantization in the temporal gauge: Pure gravity |
|
|
687 | (3) |
|
28.8 Observational cosmology: A few comments |
|
|
690 | (2) |
|
29 Generalized non-linear a-models in two dimensions |
|
|
692 | (29) |
|
29.1 Homogeneous spaces and Goldstone modes |
|
|
692 | (3) |
|
29.2 WT identities and renormalization in linear coordinates |
|
|
695 | (4) |
|
29.3 Renormalization in general coordinates: BRST symmetry |
|
|
699 | (4) |
|
29.4 Symmetric spaces: Definition |
|
|
703 | (1) |
|
29.5 Classical field equations. Conservation laws |
|
|
704 | (2) |
|
29.6 QFT: Perturbative expansion and RG |
|
|
706 | (5) |
|
|
|
711 | (2) |
|
A29 Homogeneous spaces: A few algebraic properties |
|
|
713 | (8) |
|
A29.1 Pure gauge. Maurer-Cartan equations |
|
|
713 | (1) |
|
A29.2 Metric and curvature in homogeneous spaces |
|
|
714 | (1) |
|
A29.3 Explicit expressions for the metric |
|
|
715 | (2) |
|
A29.4 Symmetric spaces: Classification |
|
|
717 | (4) |
|
30 A few solvable two-dimensional quantum field theories (QFT) |
|
|
721 | (26) |
|
30.1 The free massless scalar field |
|
|
721 | (4) |
|
30.2 The free massless Dirac fermion |
|
|
725 | (3) |
|
30.3 The gauge-invariant fermion determinant and the anomaly |
|
|
728 | (3) |
|
|
|
731 | (1) |
|
|
|
732 | (4) |
|
30.6 The massive Thirring model |
|
|
736 | (3) |
|
30.7 A generalized Thirring model with two fermions |
|
|
739 | (3) |
|
30.8 The SU(N) Thirring model |
|
|
742 | (3) |
|
A30 Two-dimensional models: A few additional results |
|
|
745 | (2) |
|
A30.1 Four-fermion current interactions: RG β-function |
|
|
745 | (1) |
|
A30.2 The Schwinger model: The anomaly |
|
|
745 | (1) |
|
A30.3 Solitons in the sG model |
|
|
746 | (1) |
|
31 0(2) spin model and the Kosterlitz-Thouless's (KT) phase transition |
|
|
747 | (13) |
|
31.1 The spin correlation functions at low temperature |
|
|
748 | (1) |
|
31.2 Correlation functions in a field |
|
|
749 | (1) |
|
31.3 The Coulomb gas in two dimensions |
|
|
750 | (5) |
|
31.4 O(2) spin model and Coulomb gas |
|
|
755 | (1) |
|
31.5 The critical two-point function in the O(2) model |
|
|
756 | (2) |
|
31.6 The generalized Thirring model |
|
|
758 | (2) |
|
32 Finite-size effects in field theory. Scaling behaviour |
|
|
760 | (26) |
|
32.1 RG in finite geometries |
|
|
760 | (4) |
|
32.2 Momentum quantization |
|
|
764 | (2) |
|
32.3 The Φ4 field theory in a periodic hypercube |
|
|
766 | (6) |
|
32.4 The Φ4 field theory: Cylindrical geometry |
|
|
772 | (4) |
|
32.5 Finite size effects in the non-linear σ-model |
|
|
776 | (6) |
|
|
|
782 | (4) |
|
A32.1 Perturbation theory in a finite volume |
|
|
782 | (1) |
|
A32.2 Discrete symmetries and finite-size effects |
|
|
783 | (3) |
|
33 Quantum field theory (QFT) at finite temperature: Equilibrium properties |
|
|
786 | (45) |
|
33.1 Finite- (and high-) temperature field theory |
|
|
786 | (4) |
|
33.2 The example of the Φ41,d-1 field theory |
|
|
790 | (6) |
|
33.3 High temperature and critical limits |
|
|
796 | (3) |
|
33.4 The non-linear σ-model in the large N limit |
|
|
799 | (5) |
|
33.5 The perturbative non-linear σ-model at finite temperature |
|
|
804 | (6) |
|
33.6 The GN model in the large N expansion |
|
|
810 | (7) |
|
33.7 Abelian gauge theories: The QED framework |
|
|
817 | (7) |
|
33.8 Non-Abelian gauge theories |
|
|
824 | (4) |
|
A33 Feynman diagrams at finite temperature |
|
|
828 | (3) |
|
A33.1 One-loop calculations |
|
|
828 | (2) |
|
|
|
830 | (1) |
|
34 Stochastic differential equations: Langevin, Fokker-Planck (FP) equations |
|
|
831 | (26) |
|
34.1 The Langevin equation |
|
|
831 | (2) |
|
34.2 Time-dependent probability distribution and FP equation |
|
|
833 | (2) |
|
34.3 Equilibrium distribution. Correlation functions |
|
|
835 | (3) |
|
34.4 A special class: Dissipative Langevin equations |
|
|
838 | (1) |
|
34.5 The linear Langevin equation |
|
|
839 | (3) |
|
34.6 Path integral representation |
|
|
842 | (1) |
|
34.7 BRST and supersymmetry |
|
|
843 | (3) |
|
34.8 Gradient time-dependent force and Jarzynski's relation |
|
|
846 | (2) |
|
34.9 More general Langevin equations. Motion in Riemannian manifolds |
|
|
848 | (4) |
|
A34 Markov's stochastic processes: A few remarks |
|
|
852 | (5) |
|
A34.1 Discrete spaces: Markov's processes, phase transitions |
|
|
852 | (3) |
|
A34.2 Stochastic process with prescribed equilibrium distribution |
|
|
855 | (1) |
|
A34.3 Stochastic processes and phase transitions |
|
|
856 | (1) |
|
35 Langevin field equations: Properties and renormalization |
|
|
857 | (18) |
|
35.1 Langevin and Fokker-Planck (FP) equations |
|
|
857 | (1) |
|
35.2 Time-dependent correlation functions and equilibrium |
|
|
858 | (3) |
|
35.3 Renormalization and BRST symmetry |
|
|
861 | (3) |
|
35.4 Dissipative Langevin equation and supersymmetry |
|
|
864 | (3) |
|
35.5 Supersymmetry and equilibrium correlation functions |
|
|
867 | (1) |
|
35.6 Stochastic quantization of two-dimensional chiral models |
|
|
868 | (3) |
|
35.7 Langevin equation and Riemannian manifolds |
|
|
871 | (3) |
|
A35 The random field Ising model: Supersymmetry |
|
|
874 | (1) |
|
36 Critical dynamics and renormalization group (RG) |
|
|
875 | (24) |
|
36.1 Dissipative equation: RG equations in dimension d -- 4 -εe |
|
|
876 | (4) |
|
36.2 Dissipative dynamics: RG equations in dimension d = 2+ε |
|
|
880 | (2) |
|
36.3 Conserved order parameter |
|
|
882 | (1) |
|
36.4 Relaxational model with energy conservation |
|
|
883 | (3) |
|
36.5 A non-relaxational model |
|
|
886 | (2) |
|
36.6 Finite size effects and dynamics |
|
|
888 | (6) |
|
A36 RG functions at two loops |
|
|
894 | (5) |
|
A36.1 Supersymmetric perturbative calculations at two loops |
|
|
894 | (5) |
|
37 Instantons in quantum mechanics (QM) |
|
|
899 | (20) |
|
37.1 The quartic anharmonic oscillator for negative coupling |
|
|
899 | (2) |
|
37.2 A toy model: A simple integral |
|
|
901 | (1) |
|
|
|
902 | (2) |
|
37.4 Instanton contributions at leading order |
|
|
904 | (4) |
|
37.5 General analytic potentials: Instanton contributions |
|
|
908 | (1) |
|
37.6 Evaluation of the determinant: The shifting method |
|
|
909 | (6) |
|
37.7 Zero temperature limit: The ground state |
|
|
915 | (1) |
|
A37 Exact Jacobian. WKB method |
|
|
916 | (3) |
|
|
|
916 | (1) |
|
|
|
917 | (2) |
|
38 Metastable vacua in quantum field theory (QFT) |
|
|
919 | (23) |
|
38.1 The Φ4 QFT for negative coupling |
|
|
920 | (4) |
|
38.2 General potentials: Instanton contributions |
|
|
924 | (2) |
|
38.3 The Φ4 QFT in dimension 4 |
|
|
926 | (1) |
|
38.4 Instanton contributions at leading order |
|
|
927 | (4) |
|
38.5 Coupling constant renormalization |
|
|
931 | (1) |
|
38.6 The imaginary part of the n-point function |
|
|
932 | (1) |
|
|
|
933 | (1) |
|
38.8 Cosmology: The decay of the false vacuum |
|
|
934 | (2) |
|
A38 Instantons: Additional remarks |
|
|
936 | (6) |
|
|
|
936 | (1) |
|
A38.2 Sobolev inequalities |
|
|
937 | (2) |
|
A38.3 Instantons and RG equations |
|
|
939 | (1) |
|
A38.4 Conformal invariance |
|
|
940 | (2) |
|
39 Degenerate classical minima and instantons |
|
|
942 | (18) |
|
39.1 The quartic double-well potential |
|
|
942 | (3) |
|
39.2 The periodic cosine potential |
|
|
945 | (3) |
|
39.3 Instantons and stochastic dynamics |
|
|
948 | (3) |
|
39.4 Instantons in stable boson field theories: General remarks |
|
|
951 | (2) |
|
39.5 Instantons in CP(N - 1) models |
|
|
953 | (3) |
|
39.6 Instantons in the SU(2) gauge theory |
|
|
956 | (3) |
|
A39 Trace formula for periodic potentials |
|
|
959 | (1) |
|
40 Large order behaviour of perturbation theory |
|
|
960 | (15) |
|
|
|
960 | (3) |
|
40.2 Scalar field theories: The example of the Φ24 field theory |
|
|
963 | (1) |
|
40.3 The (Φ22)2 field theory in dimension 4 and 4-ε |
|
|
964 | (4) |
|
40.4 Field theories with fermions |
|
|
968 | (6) |
|
A40 large-order behaviour: Additional remarks |
|
|
974 | (1) |
|
41 Critical exponents and equation of state from series summation |
|
|
975 | (17) |
|
41.1 Divergent series: Borel summability, Borel summation |
|
|
975 | (3) |
|
41.2 Borel transformation: Series summation |
|
|
978 | (2) |
|
41.3 Summing the perturbative expansion of the (Φ2)2 field theory |
|
|
980 | (3) |
|
41.4 Summation method: Practical implementation |
|
|
983 | (2) |
|
41.5 Field theory estimates of critical exponents for the O(N) model |
|
|
985 | (1) |
|
41.6 Other three-dimensional theoretical estimates |
|
|
986 | (1) |
|
41.7 Critical exponents from experiments |
|
|
987 | (2) |
|
|
|
989 | (1) |
|
A41 Some other summation methods |
|
|
990 | (2) |
|
A41.1 Order-dependent mapping method (ODM) |
|
|
990 | (1) |
|
A41.2 Linear differential approximants |
|
|
991 | (1) |
|
42 Multi-instantons in quantum mechanics (QM) |
|
|
992 | (27) |
|
42.1 The quartic double-well potential |
|
|
993 | (7) |
|
42.2 The periodic cosine potential |
|
|
1000 | (4) |
|
42.3 General potentials with degenerate minima |
|
|
1004 | (3) |
|
42.4 The O(v)-symmetric anharmonic oscillator |
|
|
1007 | (2) |
|
42.5 Generalized Bohr-Sommerfeld quantization formula |
|
|
1009 | (2) |
|
|
|
1011 | (8) |
|
A42.1 Multi-instantons: The determinant |
|
|
1011 | (1) |
|
A42.2 The instanton interaction |
|
|
1012 | (2) |
|
A42.3 A simple example of non-Borel summability |
|
|
1014 | (2) |
|
A42.4 Multi-instantons and WKB approximation |
|
|
1016 | (3) |
| Bibliography |
|
1019 | (22) |
| Index |
|
1041 | |
