Effective field theory for spontaneously broken symmetry
This open access book is about spontaneous symmetry breaking, which is a classic area of theoretical physics that lies at the core of many fascinating phenomena such as ferromagnetism, superfluidity, superconductivity, or the Higgs mechanism. The book brings an up-to-date overview of spontaneous sym...
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| Format: | eBook Book |
| Language: | English |
| Published: |
Cham
Springer
2024
Springer Nature Springer International Publishing AG |
| Edition: | 1 |
| Series: | Lecture Notes in Physics |
| Subjects: | |
| ISBN: | 3031483774, 9783031483776, 3031483782, 9783031483783 |
| Online Access: | Get full text |
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Table of Contents:
- 9 Applications to Particle and Condensed-Matter Physics -- 9.1 Chiral Perturbation Theory of Mesons -- 9.1.1 Power Counting -- 9.1.2 Effective Lagrangian -- 9.1.3 Interaction with External Fields -- 9.1.4 Effects of the Chiral Anomaly -- 9.2 Spin Waves in Ferro- and Antiferromagnets -- 9.2.1 Power Counting and Effective Lagrangian -- 9.2.1.1 Ferromagnets -- 9.2.1.2 Antiferromagnets -- 9.2.2 Equation of Motion and Magnon Spectrum -- 9.2.3 Effects of Symmetry-Breaking Perturbations -- 9.2.4 Some Topological Aspects of Ferromagnets -- References -- 10 Scattering of Nambu-Goldstone Bosons -- 10.1 Adler Zero Revisited -- 10.1.1 Generalized Soft Theorem -- 10.1.2 Application to Coset Effective Theories -- 10.2 Geometric Framework for Scattering Amplitudes -- 10.2.1 Geometric Soft Theorem for Nambu-GoldstoneBosons -- 10.2.2 Adler Zero or Not? -- 10.2.3 Symmetric Coset Spaces -- 10.3 Beyond Adler Zero -- 10.3.1 Dirac-Born-Infeld Theory -- 10.3.2 Galileon and Special Galileon Theory -- 10.3.3 Effective Theories with Enhanced Soft Limit from Symmetry -- 10.4 Soft Recursion -- 10.4.1 Complexified Kinematics -- 10.4.2 Recursion Relation for On-Shell Amplitudes -- 10.4.3 Soft Bootstrap -- References -- Part IV Spontaneously Broken Spacetime Symmetry -- 11 Locally Equivalent Symmetries -- 11.1 Relations Between Noether Currents -- 11.2 Noether Currents from Background Gauging -- 11.3 Examples -- 11.3.1 Galileon Symmetry -- 11.3.2 Spacetime Translations and Rotations -- 11.3.3 Galilei Invariance -- 11.3.4 Changing the Background: Magnetic Translations -- 11.4 Application to Scattering of Nambu-Goldstone Bosons -- 11.4.1 Galileon Theory -- 11.4.2 Theories with Generalized Shift Symmetry -- References -- 12 Nonlinear Realization of Spacetime Symmetry -- 12.1 Reminder of Nonlinear Realization of Internal Symmetry -- 12.2 Spacetime Symmetry as a Point Transformation
- 12.3 Standard Nonlinear Realization -- 12.3.1 Summary of the Construction -- 12.3.2 Relation to Physics of Broken Spacetime Symmetry -- 12.4 Examples -- 12.4.1 Lorentz Scalars with Internal Symmetry -- 12.4.2 Lorentz Scalars with Scale Invariance -- 12.4.3 Lorentz Vector with(out) Lorentz Scalar -- 12.4.4 Schrödinger Scalars with Galilei Symmetry -- References -- 13 Broken Spacetime Symmetry in Quantum Matter -- 13.1 Building Blocks for Construction of Effective Actions -- 13.1.1 Maurer-Cartan Form -- 13.1.2 Covariant Derivatives of Fields -- 13.2 Twisting Order Parameter for Internal Symmetry -- 13.2.1 New Features of the Old Setup -- 13.2.2 Case Study: Relativistic Superfluids -- 13.3 Vector Modes: The Relevant, the Irrelevant and the Unphysical -- 13.3.1 The Relevant: Helimagnets -- 13.3.2 The Irrelevant: Smectic Liquid Crystals -- 13.3.3 The Unphysical: Nonrelativistic Superfluids -- 13.3.4 Inverse Higgs Constraints -- 13.4 Genuine Breaking of Translation Invariance -- 13.4.1 One-Dimensional Modulation of the Order Parameter -- 13.4.2 Case Study: Fluctuations of a Domain Wall -- 13.4.3 Effective Action from Background Gauge Invariance -- 13.4.4 Further Possible Applications -- References -- 14 Broken Spacetime Symmetry in Classical Matter -- 14.1 Emergent Symmetry of Classical Matter -- 14.1.1 Introduction: Spring Model of Elasticity -- 14.1.2 Emergent Symmetries of Solids and Fluids -- 14.2 Nonlinear Realization of Emergent Symmetry -- 14.2.1 Field Variables and Unbroken Symmetry -- 14.2.2 Building Blocks for Construction of Effective Actions -- 14.3 Effective Field Theory of Classical Matter -- 14.3.1 Relativistic Solids -- 14.3.2 Nonrelativistic Supersolids -- 14.3.3 Nonrelativistic Solids -- 14.3.4 Perfect Fluids -- 14.4 Coupling Nambu-Goldstone Bosons to Classical Matter -- References -- Part V Epilogue -- 15 Topics Not Covered in This Book
- Intro -- Preface -- Contents -- Notation and Conventions -- List of Acronyms -- Mathematical Conventions -- Part I Prologue -- 1 Introduction -- 1.1 What Is Effective Field Theory? -- 1.2 Broken Symmetry Zoo -- 1.3 Structure of This Book -- 1.3.1 Further Reading -- References -- 2 Our First Model -- 2.1 Spontaneous Symmetry Breaking -- 2.2 Nambu-Goldstone Boson and Its Interactions -- 2.2.1 Linear Parameterization -- 2.2.2 Scattering of Nambu-Goldstone Bosons -- 2.2.3 Nonlinear Parameterization -- 2.3 Low-Energy Effective Field Theory -- 2.3.1 Matching -- 2.3.2 Eliminating the Heavy Modes -- 2.4 Moral Lessons -- References -- 3 Generalizations of the Model -- 3.1 Relativistic Models with Non-Abelian Symmetry -- 3.1.1 Spectrum of Nambu-Goldstone Bosons -- 3.1.2 Low-Energy Effective Field Theory -- 3.2 Going Nonrelativistic -- 3.2.1 Single Schrödinger Field -- 3.2.2 Multiple Nambu-Goldstone Fields -- 3.3 Moral Lessons -- References -- Part II Foundations -- 4 Symmetry and Conservation Laws -- 4.1 What Is Symmetry? -- 4.1.1 Symmetry Transformations -- 4.1.2 Object of Symmetry -- 4.2 Lagrangian Approach to Symmetry -- 4.2.1 Noether's Theorem -- 4.2.2 Tensor Conservation Laws -- 4.3 Symmetry and Conservation Laws in Hamiltonian Formalism -- 4.3.1 Symplectic Formulation of Hamiltonian Dynamics -- 4.3.2 Symmetry in Quantum Physics -- References -- 5 Spontaneous Symmetry Breaking -- 5.1 Physical State and Its Symmetry -- 5.1.1 Broken and Unbroken Symmetry -- 5.1.2 Symmetrization by Group Averaging -- 5.2 Effect of External Perturbations -- 5.2.1 Taking the Thermodynamic Limit -- 5.2.2 Order Parameter and the Vacuum Manifold -- 5.2.3 Intermediate Summary -- 5.3 Some Subtle Features of Spontaneous Symmetry Breaking -- 5.3.1 Free Schrödinger Field in Finite Volume -- 5.3.2 Pathologies of the Infinite-Volume Limit
- 5.3.3 Uniqueness of the Finite-Volume Ground State -- References -- 6 Nambu-Goldstone Bosons -- 6.1 Intuitive Picture -- 6.1.1 Redundancy of Order Parameter Fluctuations -- 6.1.2 Canonical Conjugation of Nambu-Goldstone Fields -- 6.1.3 The Big Picture -- 6.2 Goldstone Theorem -- 6.2.1 Operator Proof -- 6.2.2 Effective Action Proof -- 6.3 Classification and Counting -- 6.3.1 Independent Order Parameter Fluctuations -- 6.3.2 Type-A and Type-B Nambu-Goldstone Bosons -- 6.4 Nambu-Goldstone-Like Modes -- 6.4.1 Pseudo-Nambu-Goldstone Bosons -- 6.4.2 Quasi-Nambu-Goldstone Bosons -- 6.4.3 Massive Nambu-Goldstone Bosons -- References -- Part III Spontaneously Broken Internal Symmetry -- 7 Nonlinear Realization of Symmetry -- 7.1 Group Action on Manifolds -- 7.2 Classification of Nonlinear Realizations -- 7.2.1 Linearization of Group Action -- 7.2.2 From Linear Representation to Nonlinear Realization -- 7.3 Standard Realization of Symmetry -- 7.3.1 Nonlinear Realization on Coset Spaces -- 7.3.2 Symmetric Coset Spaces -- 7.4 Geometry of the Coset Space -- 7.4.1 Canonical and Torsion-Free Connection -- 7.4.2 Riemannian Metric -- References -- 8 Low-Energy Effective Field Theory -- 8.1 Structure of the Effective Lagrangian -- 8.1.1 Reminder of the Standard Nonlinear Realization -- 8.1.2 Lagrangians with Two Spatial or Two Temporal Derivatives -- 8.1.3 Lagrangians with One Temporal Derivative -- 8.1.4 Overview of the Lowest-Order Effective Lagrangian -- 8.2 Effective Lagrangians from Background Gauge Invariance -- 8.2.1 Methodology of Construction of Effective Actions -- 8.2.2 Lagrangians Up to Order Two in DerivativeExpansion -- 8.2.3 Effects of Explicit Symmetry Breaking -- 8.2.4 Coupling to Matter Fields -- 8.3 Equation of Motion -- 8.3.1 Spectrum of Nambu-Goldstone Bosons Revisited -- 8.3.2 More on the Geometry of the Coset Space -- References
- 15.1 Effects of Nonzero Temperature -- 15.2 No-Go Theorems for Spontaneous Symmetry Breaking -- 15.3 Topological Aspects of Spontaneous Symmetry Breaking -- 15.4 Generalized Symmetries -- References -- 16 Some Open Questions -- References -- Correction to: Effective Field Theory for Spontaneously Broken Symmetry -- A Elements of Differential Geometry -- A.1 Smooth Manifolds -- A.2 Linear Structures on Manifolds -- A.2.1 Tangent Vectors and Vector Fields -- A.2.2 Tensors and Tensor Fields -- A.3 Maps Between and on Manifolds -- A.3.1 Push-Forward and Pull-Back -- A.3.2 Flow of Vector Fields -- A.3.3 Lie Derivative -- A.4 Exterior Derivative -- A.5 Affine Connection -- A.5.1 Covariant Derivative -- A.5.2 Curvature and Torsion -- A.6 Riemannian Geometry -- A.6.1 Riemannian Metric -- A.6.2 Isometries of Riemannian Metric -- A.6.3 Symmetries of Curvature Tensor -- A.6.4 Geodesic Normal Coordinates -- A.6.5 Hodge Star -- A.7 Integration on Manifolds -- A.7.1 Orientable Manifolds -- A.7.2 Riemannian Manifolds -- A.8 Homology and Cohomology -- A.8.1 Singular Homology -- A.8.2 De Rham Cohomology -- References