Contents:

Summary: The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.
Intro -- Preface -- Notation -- 0 Introduction -- I Symmetry Groups of Elementary Particles -- 1 Lorentz Group -- 1.1 Euclidean and Minkowski Spaces. Relativistic Notation -- 1.2 Homogeneous Lorentz Group -- 1.3 Inhomogeneous Lorentz Group-Poincaré Group -- 1.4 Complex Lorentz Transformations -- 1.5 Representations of the Lorentz and Poincaré Groups, Field Functions, and Physical States -- 1.5.1 Representation D(0,0) -- 1.5.2 Representations D(1/2,0) and D(0,1/2) -- 1.5.3 Representation D(1/2,1/2) -- 2 Groups of Internal Symmetries -- 2.1 Abelian Unitary Group U(1) -- 2.2 Charge Conjugation C -- 2.3 Special Unitary Group SU(n) -- 2.3.1 SU(2) Symmetry -- 2.3.2 SU(3) Symmetry -- 2.4 Groups of Local Transformations. Gauge Group -- 3 Problems to Part I -- II Classical Theory of the Free Fields -- 4 Lagrangian and Hamiltonian Formalisms of the Classical Field Theory -- 4.1 Variational Principle and Canonical Formalism of Classical Mechanics -- 4.1.1 Lagrangian Equations -- 4.1.2 Canonical Variables. Hamiltonian Equations -- 4.1.3 Poisson Brackets. Integrals of Motion -- 4.1.4 Canonical Formalism in the Presence of Constraints -- 4.2 From Classical to Quantum Mechanics. Primary Quantization -- 4.3 General Requirements to the Lagrangians of the Field Theory -- 4.4 Lagrange-Euler Equations -- 4.5 Noether's Theorem and Dynamic Invariants -- 4.6 Vector of Energy-Momentum -- 4.7 Tensors of Angular Momentum and Spin -- 4.8 Charge and the Vector of Current -- 4.9 Canonical Variables -- 5 Classical Theory of Free Scalar Fields -- 5.1 Klein-Fock-Gordon Equation -- 5.2 Relativistic Invariance of the Klein-Fock-Gordon Equation -- 5.3 Solutions of the Klein-Fock-Gordon Equation -- 5.4 Interpretation of Solutions. Hilbert Space of States -- 5.5 Ĉ, P̂, and T̂̂ Transformations -- 5.5.1 Transformation of Charge Conjugation Ĉ -- 5.5.2 Space Reflection P̂.

5.5.3 Time Reversal T̂̂ -- 5.5.4 ĈP̂T̂̂-Invariance -- 5.6 Representations of the Lorentz Group in the Space of States -- 5.7 Lagrangian Formalism of the Scalar Field. Dynamic Invariants -- 6 Spinor Field -- 6.1 Dirac Equation -- 6.1.1 Construction of the Dirac Equation -- 6.1.2 Properties of Dirac Matrices. Conjugate Equation -- 6.2 Relativistic Invariance -- 6.2.1 Transformation Properties of the Spinor Field -- 6.2.2 On Reducible and Irreducible Spinor Representations -- 6.2.3 Transformation Properties of Bilinear Forms ψ̄Oψ -- 6.3 Solutions of the Dirac Equation -- 6.3.1 Structure of Solutions in the Momentum Space -- 6.3.2 Classification of Solutions. Helicity -- 6.3.3 Relations Between Spinors -- 6.3.4 Wave Functions of the Electron and Positron. Charge Conjugation -- 6.3.5 ĈP̂T̂̂-Transformation -- 6.4 Lagrangian Formalism -- 6.5 Representations of the Lorentz Group -- 6.5.1 Hilbert Space of States -- 6.5.2 Representations of the Lorentz Group in the Space of States -- 6.6 Applications of the Dirac Equation -- 6.6.1 Dirac Equation in the Presence of External Fields -- 6.7 Massless Spinor Field -- 6.7.1 Two-component Massless Spinor Field -- 6.7.2 Relativistic Invariance -- 6.7.3 Are There Actual Particles Corresponding to the Massless Spinor Fields? Physical Interpretation of Solutions. Neutrino -- 6.7.4 Lagrangian and Dynamic Invariants -- 6.7.5 On the Mass of Neutrino and Majorana Spinors -- 7 Vector Fields -- 7.1 Lagrangian Formalism -- 7.2 Representations in the Momentum Space -- 7.3 Decomposition into the Longitudinal and Transverse Components -- 7.4 P̂, T̂̂, Ĉ-Transformations -- 8 Electromagnetic Field -- 8.1 Maxwell Equations -- 8.2 Potential of the Electromagnetic Field -- 8.3 Gradient Transformations and the Lorentz Condition: Transversality Condition -- 8.4 Lagrangian Formalism for Electromagnetic Fields.

8.5 Transversal, Longitudinal, and Time Components of the Electromagnetic Field -- 8.6 Quantum-Mechanical Characteristics of Photons -- 8.7 Ĉ, P̂, T̂̂-Transformations -- 8.8 Consistency of the Lorentz and Gauge Transformations. Various Types of Gauges -- 9 Equations for Fields with Higher Spins -- 9.1 Fields with Spin 3/2 -- 9.2 Particles with Spin 2 -- 10 Problems to Part II -- III Classical Theory of Interacting Fields -- 11 Gauge Theory of the Electromagnetic Interaction -- 11.1 Principle of Gauge Invariance in the Maxwell Theory -- 11.2 Schrödinger Equation and Gradient (Gauge) Invariance -- 11.3 Gauge Principle as the Dynamical Principle of Interaction between the Electromagnetic and Electron-Positron Fields -- 12 Classical Theory of Yang-Mills Fields -- 12.1 Gauge Principle and the Lagrangian of the Yang-Mills Fields -- 12.2 Equations of Motion for the Free Yang-Mills fields -- 12.3 Yang-Mills Fields for Arbitrary Representations of the Group SU(N) -- 13 Masses of Particles and Spontaneous Breaking of Symmetry -- 13.1 Spontaneous Breaking of Symmetry -- 13.2 Higgs Mechanism for the Local U(1) Symmetry -- 13.3 Higgs Mechanism for the Local SU(2) symmetry -- 13.4 Generation of the Masses of Fermions -- 14 On the Construction of the General Lagrangian of Interacting Fields -- 14.1 Lagrangian of the QCD -- 14.2 Lagrangian of Weak Interactions -- 14.3 On the Electroweak Interactions -- 14.4 On the Lagrangian of Great Unification -- 15 Solutions of the Equations for Classical Fields: Solitary Waves, Solitons, Instantons -- 16 Problems to Part III -- IV Second Quantization of Fields -- 17 Axioms and General Principles of Quantization -- 17.1 Why Do We Need the Procedure of Second Quantization? Operator Nature of the Field Functions -- 17.2 Schrödinger, Heisenberg, and Interaction Pictures -- 17.3 Axioms of Quantization.

17.4 Relativistic Heisenberg Equation for Quantized Fields -- 17.4.1 Heisenberg Equation for a Free Scalar Field -- 17.4.2 Heisenberg Equation for a Free Electron-Positron Field -- 17.5 Physical Content of Positive- and Negative-Frequency Solutions of Equations for Free-Field Operators -- 18 Quantization of the Free Scalar Field -- 18.1 Commutation Relations. Commutator Functions -- 18.2 Complex Scalar Field -- 18.3 Operator Relations for Dynamic Invariants -- 19 Quantization of the Free Spinor Field -- 19.1 Commutator Functions of Fermi Fields -- 19.2 Dynamic Invariants of a Free Spinor Field -- 20 Quantization of the Vector and Electromagnetic Fields. Specific Features of the Quantization of Gauge Fields -- 20.1 Quantization of the Complex Vector Field -- 20.2 Quantization of an Electromagnetic Field -- 20.2.1 Specific Features and Difficulties of the Quantization of an Electromagnetic Field -- 20.2.2 Gupta-Bleuler Formalism -- 20.2.3 Canonical Method of Quantization -- 20.3 On the Quantization of Gauge Fields -- 21 CPT. Spin and Statistics -- 21.1 The Transformation of Charge Conjugation -- 21.2 The Transformation of Space Reflection -- 21.3 The Transformation of Time Reversal -- 21.4 CPT-Theorem and the Connection of Spin and Statistics -- 21.5 Proof of the Pauli Theorem -- 22 Representations of Commutation and Anticommutation Relations -- 22.1 General Structure of the Fock Space -- 22.2 Representations of Commutation Relations for a Free Real Scalar Field -- 22.2.1 The Fock Space of Free Scalar Bosons -- 22.2.2 Operators of Creation and Annihilation in the Fock Space. Momentum Representation -- 22.2.3 Vacuum State of Free Particle. Cyclicity of Vacuum. Set of Exponential Vectors -- 22.2.4 Construction of Representations of Commutation Relations for a Complex Scalar Field.

22.2.5 Construction of Representations of Commutation Relations in the Configuration Space. Relativistic Invariance of a Free Field -- 22.3 Representation of Anticommutation Relations of Spinor Fields -- 22.3.1 Representation of Anticommutation Relations of the Operators of Creation and Annihilation of Fermions and Antifermions -- 22.3.2 Representation of Anticommutation Relations in the Configuration Space -- 22.4 Space of States of a Free Electromagnetic Field -- 22.5 Space of Occupation Numbers -- 23 Green Functions -- 23.1 Green Functions of the Scalar Field -- 23.2 The Green Functions of Spinor, Vector, and Electromagnetic Fields -- 23.3 Time-Ordered Product and Green Functions -- 23.4 Wick Theorems -- 23.4.1 Wick Theorem for Normal Products -- 23.4.2 Wick Theorem for a Time-Ordered Product -- 23.4.3 Generalized Wick Theorem -- 23.5 Operation of Multiplication and the Regularization of Distributions -- 23.6 N-Point Green Functions of Free Fields -- 24 Problems to Part IV -- V Quantum Theory of Interacting Fields. General Problems -- 25 Construction of Quantum Interacting Fields and Problems of This Construction -- 25.1 Formal Construction of a Quantum Field -- 25.2 Mathematical Problems of Construction of a Quantum Interacting Field -- 26 Scattering Theory. Scattering Matrix -- 26.1 Quantum Description of Scattering. Definition of Scattering Operator -- 26.2 Formal Construction of the Scattering Operator by the Method of Perturbation Theory -- 26.3 Main Properties of the S-Operator -- 26.3.1 Normal Form of the Operator S -- 26.3.2 Invariance of the Scattering Matrix under Lorentz Transformations and Transformations of Charge Conjugation -- 26.3.3 Unitarity of the Scattering Operator -- 26.3.4 Law of Conservation of Energy -- 26.3.5 Matrix Elements of the S-Operator and the Scattering Amplitude -- 26.4 Feynman Diagrams.

26.4.1 Feynman Diagrams for the S-Operator.

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Intro -- Preface -- Notation -- 0 Introduction -- I Symmetry Groups of Elementary Particles -- 1 Lorentz Group -- 1.1 Euclidean and Minkowski Spaces. Relativistic Notation -- 1.2 Homogeneous Lorentz Group -- 1.3 Inhomogeneous Lorentz Group-Poincaré Group -- 1.4 Complex Lorentz Transformations -- 1.5 Representations of the Lorentz and Poincaré Groups, Field Functions, and Physical States -- 1.5.1 Representation D(0,0) -- 1.5.2 Representations D(1/2,0) and D(0,1/2) -- 1.5.3 Representation D(1/2,1/2) -- 2 Groups of Internal Symmetries -- 2.1 Abelian Unitary Group U(1) -- 2.2 Charge Conjugation C -- 2.3 Special Unitary Group SU(n) -- 2.3.1 SU(2) Symmetry -- 2.3.2 SU(3) Symmetry -- 2.4 Groups of Local Transformations. Gauge Group -- 3 Problems to Part I -- II Classical Theory of the Free Fields -- 4 Lagrangian and Hamiltonian Formalisms of the Classical Field Theory -- 4.1 Variational Principle and Canonical Formalism of Classical Mechanics -- 4.1.1 Lagrangian Equations -- 4.1.2 Canonical Variables. Hamiltonian Equations -- 4.1.3 Poisson Brackets. Integrals of Motion -- 4.1.4 Canonical Formalism in the Presence of Constraints -- 4.2 From Classical to Quantum Mechanics. Primary Quantization -- 4.3 General Requirements to the Lagrangians of the Field Theory -- 4.4 Lagrange-Euler Equations -- 4.5 Noether's Theorem and Dynamic Invariants -- 4.6 Vector of Energy-Momentum -- 4.7 Tensors of Angular Momentum and Spin -- 4.8 Charge and the Vector of Current -- 4.9 Canonical Variables -- 5 Classical Theory of Free Scalar Fields -- 5.1 Klein-Fock-Gordon Equation -- 5.2 Relativistic Invariance of the Klein-Fock-Gordon Equation -- 5.3 Solutions of the Klein-Fock-Gordon Equation -- 5.4 Interpretation of Solutions. Hilbert Space of States -- 5.5 Ĉ, P̂, and T̂̂ Transformations -- 5.5.1 Transformation of Charge Conjugation Ĉ -- 5.5.2 Space Reflection P̂.

5.5.3 Time Reversal T̂̂ -- 5.5.4 ĈP̂T̂̂-Invariance -- 5.6 Representations of the Lorentz Group in the Space of States -- 5.7 Lagrangian Formalism of the Scalar Field. Dynamic Invariants -- 6 Spinor Field -- 6.1 Dirac Equation -- 6.1.1 Construction of the Dirac Equation -- 6.1.2 Properties of Dirac Matrices. Conjugate Equation -- 6.2 Relativistic Invariance -- 6.2.1 Transformation Properties of the Spinor Field -- 6.2.2 On Reducible and Irreducible Spinor Representations -- 6.2.3 Transformation Properties of Bilinear Forms ψ̄Oψ -- 6.3 Solutions of the Dirac Equation -- 6.3.1 Structure of Solutions in the Momentum Space -- 6.3.2 Classification of Solutions. Helicity -- 6.3.3 Relations Between Spinors -- 6.3.4 Wave Functions of the Electron and Positron. Charge Conjugation -- 6.3.5 ĈP̂T̂̂-Transformation -- 6.4 Lagrangian Formalism -- 6.5 Representations of the Lorentz Group -- 6.5.1 Hilbert Space of States -- 6.5.2 Representations of the Lorentz Group in the Space of States -- 6.6 Applications of the Dirac Equation -- 6.6.1 Dirac Equation in the Presence of External Fields -- 6.7 Massless Spinor Field -- 6.7.1 Two-component Massless Spinor Field -- 6.7.2 Relativistic Invariance -- 6.7.3 Are There Actual Particles Corresponding to the Massless Spinor Fields? Physical Interpretation of Solutions. Neutrino -- 6.7.4 Lagrangian and Dynamic Invariants -- 6.7.5 On the Mass of Neutrino and Majorana Spinors -- 7 Vector Fields -- 7.1 Lagrangian Formalism -- 7.2 Representations in the Momentum Space -- 7.3 Decomposition into the Longitudinal and Transverse Components -- 7.4 P̂, T̂̂, Ĉ-Transformations -- 8 Electromagnetic Field -- 8.1 Maxwell Equations -- 8.2 Potential of the Electromagnetic Field -- 8.3 Gradient Transformations and the Lorentz Condition: Transversality Condition -- 8.4 Lagrangian Formalism for Electromagnetic Fields.

8.5 Transversal, Longitudinal, and Time Components of the Electromagnetic Field -- 8.6 Quantum-Mechanical Characteristics of Photons -- 8.7 Ĉ, P̂, T̂̂-Transformations -- 8.8 Consistency of the Lorentz and Gauge Transformations. Various Types of Gauges -- 9 Equations for Fields with Higher Spins -- 9.1 Fields with Spin 3/2 -- 9.2 Particles with Spin 2 -- 10 Problems to Part II -- III Classical Theory of Interacting Fields -- 11 Gauge Theory of the Electromagnetic Interaction -- 11.1 Principle of Gauge Invariance in the Maxwell Theory -- 11.2 Schrödinger Equation and Gradient (Gauge) Invariance -- 11.3 Gauge Principle as the Dynamical Principle of Interaction between the Electromagnetic and Electron-Positron Fields -- 12 Classical Theory of Yang-Mills Fields -- 12.1 Gauge Principle and the Lagrangian of the Yang-Mills Fields -- 12.2 Equations of Motion for the Free Yang-Mills fields -- 12.3 Yang-Mills Fields for Arbitrary Representations of the Group SU(N) -- 13 Masses of Particles and Spontaneous Breaking of Symmetry -- 13.1 Spontaneous Breaking of Symmetry -- 13.2 Higgs Mechanism for the Local U(1) Symmetry -- 13.3 Higgs Mechanism for the Local SU(2) symmetry -- 13.4 Generation of the Masses of Fermions -- 14 On the Construction of the General Lagrangian of Interacting Fields -- 14.1 Lagrangian of the QCD -- 14.2 Lagrangian of Weak Interactions -- 14.3 On the Electroweak Interactions -- 14.4 On the Lagrangian of Great Unification -- 15 Solutions of the Equations for Classical Fields: Solitary Waves, Solitons, Instantons -- 16 Problems to Part III -- IV Second Quantization of Fields -- 17 Axioms and General Principles of Quantization -- 17.1 Why Do We Need the Procedure of Second Quantization? Operator Nature of the Field Functions -- 17.2 Schrödinger, Heisenberg, and Interaction Pictures -- 17.3 Axioms of Quantization.

17.4 Relativistic Heisenberg Equation for Quantized Fields -- 17.4.1 Heisenberg Equation for a Free Scalar Field -- 17.4.2 Heisenberg Equation for a Free Electron-Positron Field -- 17.5 Physical Content of Positive- and Negative-Frequency Solutions of Equations for Free-Field Operators -- 18 Quantization of the Free Scalar Field -- 18.1 Commutation Relations. Commutator Functions -- 18.2 Complex Scalar Field -- 18.3 Operator Relations for Dynamic Invariants -- 19 Quantization of the Free Spinor Field -- 19.1 Commutator Functions of Fermi Fields -- 19.2 Dynamic Invariants of a Free Spinor Field -- 20 Quantization of the Vector and Electromagnetic Fields. Specific Features of the Quantization of Gauge Fields -- 20.1 Quantization of the Complex Vector Field -- 20.2 Quantization of an Electromagnetic Field -- 20.2.1 Specific Features and Difficulties of the Quantization of an Electromagnetic Field -- 20.2.2 Gupta-Bleuler Formalism -- 20.2.3 Canonical Method of Quantization -- 20.3 On the Quantization of Gauge Fields -- 21 CPT. Spin and Statistics -- 21.1 The Transformation of Charge Conjugation -- 21.2 The Transformation of Space Reflection -- 21.3 The Transformation of Time Reversal -- 21.4 CPT-Theorem and the Connection of Spin and Statistics -- 21.5 Proof of the Pauli Theorem -- 22 Representations of Commutation and Anticommutation Relations -- 22.1 General Structure of the Fock Space -- 22.2 Representations of Commutation Relations for a Free Real Scalar Field -- 22.2.1 The Fock Space of Free Scalar Bosons -- 22.2.2 Operators of Creation and Annihilation in the Fock Space. Momentum Representation -- 22.2.3 Vacuum State of Free Particle. Cyclicity of Vacuum. Set of Exponential Vectors -- 22.2.4 Construction of Representations of Commutation Relations for a Complex Scalar Field.

22.2.5 Construction of Representations of Commutation Relations in the Configuration Space. Relativistic Invariance of a Free Field -- 22.3 Representation of Anticommutation Relations of Spinor Fields -- 22.3.1 Representation of Anticommutation Relations of the Operators of Creation and Annihilation of Fermions and Antifermions -- 22.3.2 Representation of Anticommutation Relations in the Configuration Space -- 22.4 Space of States of a Free Electromagnetic Field -- 22.5 Space of Occupation Numbers -- 23 Green Functions -- 23.1 Green Functions of the Scalar Field -- 23.2 The Green Functions of Spinor, Vector, and Electromagnetic Fields -- 23.3 Time-Ordered Product and Green Functions -- 23.4 Wick Theorems -- 23.4.1 Wick Theorem for Normal Products -- 23.4.2 Wick Theorem for a Time-Ordered Product -- 23.4.3 Generalized Wick Theorem -- 23.5 Operation of Multiplication and the Regularization of Distributions -- 23.6 N-Point Green Functions of Free Fields -- 24 Problems to Part IV -- V Quantum Theory of Interacting Fields. General Problems -- 25 Construction of Quantum Interacting Fields and Problems of This Construction -- 25.1 Formal Construction of a Quantum Field -- 25.2 Mathematical Problems of Construction of a Quantum Interacting Field -- 26 Scattering Theory. Scattering Matrix -- 26.1 Quantum Description of Scattering. Definition of Scattering Operator -- 26.2 Formal Construction of the Scattering Operator by the Method of Perturbation Theory -- 26.3 Main Properties of the S-Operator -- 26.3.1 Normal Form of the Operator S -- 26.3.2 Invariance of the Scattering Matrix under Lorentz Transformations and Transformations of Charge Conjugation -- 26.3.3 Unitarity of the Scattering Operator -- 26.3.4 Law of Conservation of Energy -- 26.3.5 Matrix Elements of the S-Operator and the Scattering Amplitude -- 26.4 Feynman Diagrams.

26.4.1 Feynman Diagrams for the S-Operator.

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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