Renormalized Quantum Field Theory

I. Elements of Quantum Field Theory.- 1. Quantum Free Fields.- 1.1. Fock Space.- 1.2. Free Real Scalar Field.- 1.3. Other Free Fields.- 2. The Chronological Products of Local Monomials of the Free Field.- 2.1. Wick Theorem.- 2.2. Wick Theorem for Chronological Products of Free Fields.- 2.3. Regularized T-Products.- 2.4. Ambiguity in the Choice of Chronological Products.- 3. Interacting Fields.- 3.1. Interpolating Heisenberg Field.- 3.2. Connection Between Two Systems of Axioms.- 3.3. T-Exponential, Lagrangian, Renormalization Constants.- 3.4. Green Functions, Functional Integral, Euclidean Quantum Field Theory.- 3.5. Interaction Lagrangians.- II. Parametric Representations for Feynman Diagrams. R-Operation.- 1. Regularized Feynman Diagrams.- 1.1. Intermediate Regularization. Divergency Index.- 1.2. Parametric Representation for Regularized Diagrams.- 1.3. The Proof of Statements (16)–(21).- 1.4. Parametric Representations in Other Dimensions and in Euclidean Theory. Coordinate Representation.- 2. Bogoliubov-Parasiuk R-Operation.- 2.1. Subtraction Operators M and Finite Renormalization Operators P. Definition of R-Operation.- 2.2. The Structure of the R-Operation.- 2.3. R-Operation with Non-Zero Subtraction Points or Other Subtraction Operators.- 3. Parametric Representations for Renormalized Diagrams.- 3.1. Renormalization over Forests.- 3.2. Non-Zero Subtraction Points.- 3.3. Renormalization over Nests.- 3.4. Renormalization by Means of Integral Operators.- III. Bogoliubov-Parasiuk Theorem. Other Renormalization Schemes.- 1. Existence of Renormalized Feynman Amplitudes.- 1.1. Division of the Integration Domain into Sectors. The Equivalence Classes of Nests.- 1.2. The Ultraviolet Convergence of Parametric Integrals.- 1.3. The Limit ? ? 0.- 2. Infrared Divergencies and Renormalization in Massless Theories.- 2.1. Infrared Convergence of Regularized Amplitudes.- 2.2. Illustrations and Heuristic Arguments.- 2.3. Classification of Theories.- 2.4. Ultraviolet Renormalization.- 2.5. More Refined Arguments.- 3. The Proof of Theorems 1 and 2.- 3.1. Preliminaries.- 3.2. Basic Lemma.- 3.3. Theorem 1. The Case of a Diagram without Massive Lines.- 3.4. Theorem 1. The Case of a Diagram with Massive Lines.- 3.5. The Scheme of the Proof for Theorem 2.- 3.6. The Structure of the Forms D, A, Bl, Kij.- 3.7. Transition from the Space S?(R4v \ {qµ = 0}) to the Space S?(R4v \ E).- 4. Analytic Renormalization and Dimensional Renormalization.- 4.1. Introductory Remarks.- 4.2. The Recipe for Analytic Renormalization.- 4.3. The Equivalence of R-Operation and Analytic Renormalization.- 4.4. Dimensional Renormalization.- 4.5. The Parametric Representation in the Case of Dimensional Renormalization.- 4.6. Equivalence of the Dimensional Renormalization and R-Operation.- 4.7. Modifications. Zero Mass Theories.- 4.8. Examples.- 5. Renormalization ‘without Subtraction’. Renormalization ‘over Asymptotes’.- 5.1. Intermediate Regularization and the Recipe of Renormalization ‘without Subtraction’.- 5.2. The Equivalence of the R-Operation and the Renormalization ‘without Subtractions’.- 5.3. Renormalization ‘over Asymptotes’.- IV. Composite Fields. Singularities of the Product of Currents at Short Distances and on the Light Cone.- 1. Renormalized Composite Fields.- 1.1. Basic Notions and Notations.- 1.2. The Subtraction Operator M.- 1.3. The Structure of Renormalization.- 1.4. Generalized Action Principle.- 1.5. Zimmermann Identities.- 2. Products of Fields at Short Distances.- 2.1. A Lowest Order Example.- 2.2. Wilson Expansions.- 2.3. A Massless Case.- 2.4. An Important Particular Case.- 3. Products of Currents at Short Distances.- 3.1. Short-Distance Expansions for Products of Currents.- 3.2. The Proof of the Lemma.- 3.3. The Structure of Renormalization with Incomplete Subgraphs. The Short-Distance Expansion in the Weinberg Renormalization Scheme.- 4. Products of Currents near the Light Cone.- 4.1. Lower Order Consideration.- 4.2. Subtraction Operator $$
{\bar m^{\left( {\rm{a}} \right)}}
$$. Light-Ray Fields.- 4.3. The Light-Cone Theorem.- 4.4. An Example. General Discussion. A Massless Case.- 5. Equations for Composite Fields.- 5.1. Equations of Motion for the Interpolating Field.- 5.2. Equations for Higher Composite Fields.- 5.3. The Proof of Relations (273) and (276).- 5.4. Renorm-Group Equations and Callan-Symanzik Equations.- 6. Equations for Regularized Green Functions.- 6.1. Relation of Renormalization Constants to Green Functions.- 6.2. Relations of Green Functions to Derivatives of the Renormalization Constants.- V. Renormalization of Yang-Mills Theories.- 1. Classical Theory and Quantization.- 1.1. Classical Yang-Mills Fields.- 1.2. Quantization.- 1.3. Fields of Matter. Abelian Theory.- 2. Gauge Invariance and Invariant Renormalizability.- 2.1. Abelian Theories. Ward Identities.- 2.2. Non-Abelian Yang-Mills Theories. BRST Symmetry. Slavnov Identities.- 2.3. A Linear Condition for the Gauge Invariance of Non-Abelian Yang-Mills Theories.- 2.4. The Structure of Subtractions.- 2.5. Invariant Renormalizability of the Yang-Mills Theory.- 3. Invariant Regularization and invariant Renormalization Schemes.- 3.1. Preliminary Discussion.- 3.2. Scalar Electrodynamics. Recipes for Regularization.- 3.3. Scalar Electrodynamics. Arguments in Favour of the Recipe.- 3.4. Spinor Electrodynamics. Recipes for Regularization.- 3.5. Spinor Electrodynamics. Argumentation.- 3.6. Examples and Remarks.- 3.7. Non-Abelian Yang-Mills Theories.- 3.8. An Example: Gluon Polarization Operator. Arguments.- 4. Anomalies.- 4.1. Is It Always Possible to Retain a Classical Symmetry in a Quantum Field Theory?.- 4.2. Main Statements.- 4.3. Heuristic Check of Ward Identities (Momentum Representation).- 4.4. The Triangle Diagram in the ?-Representation.- 4.5. Ward Identities.- Appendix. On Methods of Studying Deep-Inelastic Scattering.- A.1. Deep-Inelastic Scattering.- A.2. The Traditional Approach to Deep-Inelastic Scattering.- A.3. The Non-Local Light-Cone Expansion as the Basic Tool to Study Deep-Inelastic Scattering.- A Guide to Literature.- References.