- Title Pages
- Preface
- Acknowledgments
- General References
- 1 Algebraic Preliminaries
- 2 Euclidean Path Integrals In Quantum Mechanics
- 3 Path Integrals In Quantum Mechanics: Generalizations
- 4 Stochastic Differential Equations: Langevin, Fokker–Planck Equations
- 5 Path And Functional Integrals In Quantum Statistical Physics
- 6 Quantum Evolution: From Particles To Fields
- 7 Quantum Field Theory: Functional Methods and Perturbation Theory
- 8 Relativistic Fermions
- 9 Quantum Field Theory: Divergences and Regularization
- 10 Introduction to Renormalization Theory. Renormalization Group Equations
- 11 Dimensional Regularization, Minimal Subtraction: RG Functions
- 12 Renormalization Of Composite Operators. Short Distance Expansion
- 13 Symmetries And Renormalization
- 14 The Non-Linear σ-Model: An Example Of a Non-Linear Symmetry
- 15 General Non-Linear Models In Two Dimensions
- 16 St And Brs Symmetries, Stochastic Field Equations
- 17 From Langevin Equation To Supersymmetry
- 18 Abelian Gauge Theories
- 19 Non-Abelian Gauge Theories: Introduction
- 20 The Standard Model. Anomalies
- 21 Gauge Theories: Master Equation And Renormalization
- 22 Classical And Quantum Gravity. Riemannian Manifolds And Tensors
- 23 Critical Phenomena: General Considerations
- 24 Mean Field Theory For Ferromagnetic Systems
- 25 General Renormalization Group. The Critical Theory Near Dimension Four
- 26 Scaling Behaviour In The Critical Domain
- 27 Corrections to Scaling Behaviour
- 28 Non-Magnetic Systems and The (φ<sup>2</sup>)<sup>2</sup> Field Theory
- 29 Calculation Of Universal Quantities
- 30 The <i>O</i>(<i>N</i>) Vector Model For <i>N</i> Large
- 31 Phase Transitions Near Two Dimensions
- 32 Two-Dimensional Models and Bosonization Method
- 33 The <i>O</i>(2) Classical Spin Model In Two Dimensions
- 34 Critical Properties Of Gauge Theories
- 35 Uv Fixed Points In Quantum Field Theory
- 36 Critical Dynamics
- 37 Field Theory in a Finite Geometry: Finite Size Scaling
- 38 Quantum Field Theory At Finite Temperature: Equilibrium Properties
- 39 Instantons In Quantum Mechanics
- 40 Unstable Vacua In Quantum Field Theory
- 41 Degenerate Classical Minima And Instantons
- 42 Perturbation Series At Large Orders. Summation Methods
- 43 Multi-Instantons In Quantum Mechanics
- Index
Quantum Field Theory: Divergences and Regularization
Quantum Field Theory: Divergences and Regularization
- Chapter:
- (p.212) 9 QUANTUM FIELD THEORY: DIVERGENCES AND REGULARIZATION
- Source:
- Quantum Field Theory and Critical Phenomena
- Author(s):
JEAN ZINN-JUSTIN
- Publisher:
- Oxford University Press
This chapter considers specifically local relativistic field theories: the action is the integral of the lagrangian density that is a function of fields and their derivatives. It shows that as a consequence, infinities appear in perturbative calculations due to severe short distance singularities, or after Fourier transformation, to integrals diverging at large momenta: one speaks of UV divergences. These divergences are peculiar to local Quantum Field Theory: in contrast with Classical Mechanics or non-relativistic Quantum Mechanics with a finite number of particles, a straightforward construction of a quantum field theory of point-like objects is impossible. The chapter frst displays the problem, using the example of the φ3 field theory at one-loop order. It then systematically characterizes divergences by power counting: this leads to a classification of local field theories into three families: super-renormalizable, renormalizable, and non-renormalizable. It shows that divergences of correlation functions involving composite operators require a separate analysis.
Keywords: local relativistic field theories, quantum field theory, super-renormalizable, renormalizable, non-renormalizable
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- Title Pages
- Preface
- Acknowledgments
- General References
- 1 Algebraic Preliminaries
- 2 Euclidean Path Integrals In Quantum Mechanics
- 3 Path Integrals In Quantum Mechanics: Generalizations
- 4 Stochastic Differential Equations: Langevin, Fokker–Planck Equations
- 5 Path And Functional Integrals In Quantum Statistical Physics
- 6 Quantum Evolution: From Particles To Fields
- 7 Quantum Field Theory: Functional Methods and Perturbation Theory
- 8 Relativistic Fermions
- 9 Quantum Field Theory: Divergences and Regularization
- 10 Introduction to Renormalization Theory. Renormalization Group Equations
- 11 Dimensional Regularization, Minimal Subtraction: RG Functions
- 12 Renormalization Of Composite Operators. Short Distance Expansion
- 13 Symmetries And Renormalization
- 14 The Non-Linear σ-Model: An Example Of a Non-Linear Symmetry
- 15 General Non-Linear Models In Two Dimensions
- 16 St And Brs Symmetries, Stochastic Field Equations
- 17 From Langevin Equation To Supersymmetry
- 18 Abelian Gauge Theories
- 19 Non-Abelian Gauge Theories: Introduction
- 20 The Standard Model. Anomalies
- 21 Gauge Theories: Master Equation And Renormalization
- 22 Classical And Quantum Gravity. Riemannian Manifolds And Tensors
- 23 Critical Phenomena: General Considerations
- 24 Mean Field Theory For Ferromagnetic Systems
- 25 General Renormalization Group. The Critical Theory Near Dimension Four
- 26 Scaling Behaviour In The Critical Domain
- 27 Corrections to Scaling Behaviour
- 28 Non-Magnetic Systems and The (φ<sup>2</sup>)<sup>2</sup> Field Theory
- 29 Calculation Of Universal Quantities
- 30 The <i>O</i>(<i>N</i>) Vector Model For <i>N</i> Large
- 31 Phase Transitions Near Two Dimensions
- 32 Two-Dimensional Models and Bosonization Method
- 33 The <i>O</i>(2) Classical Spin Model In Two Dimensions
- 34 Critical Properties Of Gauge Theories
- 35 Uv Fixed Points In Quantum Field Theory
- 36 Critical Dynamics
- 37 Field Theory in a Finite Geometry: Finite Size Scaling
- 38 Quantum Field Theory At Finite Temperature: Equilibrium Properties
- 39 Instantons In Quantum Mechanics
- 40 Unstable Vacua In Quantum Field Theory
- 41 Degenerate Classical Minima And Instantons
- 42 Perturbation Series At Large Orders. Summation Methods
- 43 Multi-Instantons In Quantum Mechanics
- Index
