Methods of Quantization: Lectures Held at the 39. Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria (Lecture Notes in Physics, 572)

The present set of lecture notes gives both a status report and a survey of recent advances for the most important quantization methods in the field theories for elementary particle physics. The first part of the book introduces light-cone quantization as an interesting alternative to the commonly used covariant perturbation theory and functional-integral methods. Next, a general formalism for quantizing systems with constraints, the projection-operator approach, is presented and structural aspects of the renormalization problem for gauge invariant field theories are discussed. Finally, the mathematics underlying the functional-integral quantizaion is reviewed.Suitable as a reference for researchers in the field, the book will prove particularly useful for lecturers and gradjuate students in search of additional reading beyond the standard texts on quantum field theory. Chapter 1 1 Introduction 2 The Poincaré Group 3 Forms of Relativistic Dynamics 3.1 Comparison of Instant Form,Front Form,and Point Form 4 Light-Front Dynamics 4.1 Relative Momentum, Invariant Mass 4.2 The Box Diagram 5 Poincaré Generators in Field Theory 5.1 Fermions Interacting with a Scalar Field 5.2 Instant Form 5.3 Front Form (LF) 5.4 Interacting and Non-interacting Generators on an Instant and on the Light Front 6 Light-Front Perturbation Theory 6.1 Connection of Covariant Amplitudes to Light-Front Amplitudes 6.2 Regularization 6.3 Minus Regularization 7 Triangle Diagram in Yukawa Theory 7.1 Covariant Calculation 7.2 Construction of the Current in LFD 7.3 Numerical Results 8.1 Covariant Calculation 8.2 Instant-Form Calculation 8.3 Calculation in Light-Front Coordinates 8.4 Front-Form Calculation 9 Dimensional Regularization: Basic Formulae 10 Four-Dimensional Integration 11 Some Useful Integrals References Chapter 2 1 Introduction 2 Relativistic Particle Dynamics 2.1 The Free Relativistic Point Particle 2.2 Dirac’s Forms of Relativistic Dynamics 2.3 The Front Form 3 Light-Cone Quantization of Fields 3.1 Construction of the Poincaré Generators 3.2 Schwinger’s (Quantum) Action Principle 3.3 Quantization as an Initial- and/or Boundary-Value Problem 3.4 DLCQ – Basics 3.5 DLCQ – Causality 3.6 The Functional Schrödinger Picture 3.7 The Light-Cone Vacuum 4 Light-Cone Wave Functions 4.1 Kinematics 4.2 Definition of Light-Cone Wave Functions 4.3 Properties of Light-Cone Wave Functions 4.4 Examples of Light-Cone Wave Functions 5 The Pion Wave Function in the NJL Model 5.1 A Primer on Spontaneous Chiral Symmetry Breaking 5.2 NJL Folklore 5.3 Schwinger–Dyson Approach 5.4 Observables 6 Conclusions Acknowledgments References Chapter 3 1 Introduction 1.1 Initial Comments 1.2 Classical Background 1.3 Quantization First: Standard Operator Quantization 1.4 Reduction First: Standard Path Integral Quantization 1.5 Quantization First =/= Reduction First 1.6 Outline of the Remaining Sections 2 Overview of the Projection Operator Approach to Constrained System Quantization 2.1 Coherent States 2.2 Constraints 2.3 Dynamics for First-Class Systems 2.4 Zero in the Continuous Spectrum 2.5 Alternative View of Continuous Zeros 3 Coherent State Path Integrals Without Gauge Fixing 3.1 Enforcing the Quantum Constraints 3.2 Reproducing Kernel Hilbert Spaces 3.3 Reduction of the Reproducing Kernel 3.4 Single Regularized Constraints 3.5 Basic First-Class Constraint Example 4 Application to General Constraints 4.1 Classical Considerations 4.2 Quantum Considerations 4.3 Universal Procedure to Generate Single Regularized Constraints 4.4 Basic Second-Class Constraint Example 4.5 Conversion Method 4.6 Equivalent Representations 4.7 Equivalence of Criteria for Second-Class Constraints 5 Selected Examples of First-Class Constraints 5.1 General Configuration Space Geometry 5.2 Finite-Dimensional Hilbert Space Examples 5.3 Helix Model 5.4 Reparameterization Invariant Dynamics 5.5 Elevating the Lagrange Multiplier to an Additional Dynamical Variable 6 Special Applications 6.1 Algebraically Inequivalent Constraints 6.2 Irregular Constraints 7 Some Other Applications of the Projection Operator Approach Acknowledgements References Chapter 4 1 Generalities 1.1 Renormalization Schemes 1.2 The Action Principle 1.3 Green Functions and Operators 2 The Quantization of Gauge Theories 2.1 The Abelian Case 2.2 BRS Transformations 2.3 The Slavnov–Taylor Identity 3 Applications 3.1 The Electroweak Standard Model 3.2 Supersymmetry in Non-linear Realization 3.3 SUSY Gauge Theories References Chapter 5 1 Introduction 2 White Noise Analysis 2.1 Smooth and Generalized Functionals 2.2 Characterization of Generalized Functionals Phi in (S)* 2.3 Calculus 3 Quantum Field Theory 3.1 The Vacuum Density 3.2 Dynamics in Terms of the Vacuum 4 Feynman Integrals 4.1 The Interactions 4.2 The Morse Potential References Most of our present understanding of the elementary building blocks of matter and the forces between them is based on the quantized version of the field theories which are locally symmetric under gauge transformations. The present set of lecture notes gives both a status report and a survey of recent advances for the most important quantization methods in the field theories for elementary particle physics. The first part of the book introduces light-cone quantization as an interesting alternative to the commonly used covariant perturbation theory and functional-integral methods. Next, a general formalism for quantizing systems with constraints, the projection-operator approach, is presented and structural aspects of the renormalization problem for gauge invariant field theories are discussed. Finally, the mathematics underlying the functional-integral quantization is reviewed. Suitable as a reference for researchers in the field, the book will prove particularly useful for lecturers and graduate students in search of additional reading beyond the standard texts on quantum field theory. Annotation The present set of lecture notes gives both a status report and a survey of recent advances for the most important quantization methods in the field theories for elementary particle physics. The first part of the book introduces light-cone quantization as an interesting alternative to the commonly used covariant perturbation theory and functional-integral methods. Next, a general formalism for quantizing systems with constraints, the projection-operator approach, is presented and structural aspects of the renormalization problem for gauge invariant field theories are discussed. Finally, the mathematics underlying the functional-integral quantizaion is reviewed. Suitable as a reference for researchers in the field, the book will prove particularly useful for lecturers and gradjuate students in search of additional reading beyond the standard texts on quantum field theory The two fundamental revolutions in physics of the twentieth century: relativity theory and quantum mechanics, force us to formulate questions about the smallest building blocks of matter in a language that accounts for the quantum nature of those systems, yet respects the fundamental space-time symmetries.