Classical Theory of Gauge Fields.

By: Rubakov, ValeryContributor(s): Wilson, Stephen S | Wilson, Stephen S. SPublisher: Princeton : Princeton University Press, 2002Copyright date: ©2002Description: 1 online resource (457 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781400825097Subject(s): Gauge fields (Physics)Genre/Form: Electronic books. Additional physical formats: Print version:: Classical Theory of Gauge FieldsDDC classification: 530.14/35 LOC classification: QC793.3.G38 -- R8313 2002ebOnline resources: Click to View
Contents:
Contents -- Preface -- Part I -- 1 Gauge Principle in Electrodynamics -- 1.1 Electromagnetic-field action in vacuum -- 1.2 Gauge invariance -- 1.3 General solution of Maxwell's equations in vacuum -- 1.4 Choice of gauge -- 2 Scalar and Vector Fields -- 2.1 System of units h = c=1 -- 2.2 Scalar field action -- 2.3 Massive vector field -- 2.4 Complex scalar field -- 2.5 Degrees of freedom -- 2.6 Interaction of fields with external sources -- 2.7 Interacting fields. Gauge-invariant interaction in scalar electrodynamics -- 2.8 Noether's theorem -- 3 Elements of the Theory of Lie Groups and Algebras -- 3.1 Groups -- 3.2 Lie groups and algebras -- 3.3 Representations of Lie groups and Lie algebras -- 3.4 Compact Lie groups and algebras -- 4 Non-Abelian Gauge Fields -- 4.1 Non-Abelian global symmetries -- 4.2 Non-Abelian gauge invariance and gauge fields: the group SU(2) -- 4.3 Generalization to other groups -- 4.4 Field equations -- 4.5 Cauchy problem and gauge conditions -- 5 Spontaneous Breaking of Global Symmetry -- 5.1 Spontaneous breaking of discrete symmetry -- 5.2 Spontaneous breaking of global U(1) symmetry. Nambu-Goldstone bosons -- 5.3 Partial symmetry breaking: the SO(3)model -- 5.4 General case. Goldstone's theorem -- 6 Higgs Mechanism -- 6.1 Example of an Abelian model -- 6.2 Non-Abelian case: model with complete breaking of SU(2) symmetry -- 6.3 Example of partial breaking of gauge symmetry: bosonic sector of standard electroweak theory -- Supplementary Problems for Part I -- Part II -- 7 The Simplest Topological Solitons -- 7.1 Kink -- 7.2 Scale transformations and theorems on the absence of solitons -- 7.3 The vortex -- 7.4 Soliton in a model of n-field in (2 + 1)-dimensional space-time -- 8 Elements of Homotopy Theory -- 8.1 Homotopy of mappings -- 8.2 The fundamental group -- 8.3 Homotopy groups.
8.4 Fiber bundles and homotopy groups -- 8.5 Summary of the results -- 9 Magnetic Monopoles -- 9.1 The soliton in a model with gauge group SU(2) -- 9.2 Magnetic charge -- 9.3 Generalization to other models -- 9.4 The limit mH/mV → 0 -- 9.5 Dyons -- 10 Non-Topological Solitons -- 11 Tunneling and Euclidean Classical Solutions in Quantum Mechanics -- 11.1 Decay of a metastable state in quantum mechanics of one variable -- 11.2 Generalization to the case of many variables -- 11.3 Tunneling in potentials with classical degeneracy -- 12 Decay of a False Vacuum in Scalar Field Theory -- 12.1 Preliminary considerations -- 12.2 Decay probability: Euclidean bubble (bounce) -- 12.3 Thin-wall approximation -- 13 Instantons and Sphalerons in Gauge Theories -- 13.1 Euclidean gauge theories -- 13.2 Instantons in Yang-Mills theory -- 13.3 Classical vacua and θ-vacua -- 13.4 Sphalerons in four-dimensional models with the Higgs mechanism -- Supplementary Problems for Part II -- Part III -- 14 Fermions in Background Fields -- 14.1 Free Dirac equation -- 14.2 Solutions of the free Dirac equation. Dirac sea -- 14.3 Fermions in background bosonic fields -- 14.4 Fermionic sector of the Standard Model -- 15 Fermions and Topological External Fields in Two-dimensional Models -- 15.1 Charge fractionalization -- 15.2 Level crossing and non-conservation of fermion quantum numbers -- 16 Fermions in Background Fields of Solitons and Strings in Four-Dimensional Space-Time -- 16.1 Fermions in a monopole background field: integer angular momentum and fermion number fractionalization -- 16.2 Scattering of fermions off a monopole: non-conservation of fermion numbers -- 16.3 Zero modes in a background field of a vortex: superconducting strings -- 17 Non-Conservation of Fermion Quantum Numbers in Four-dimensional Non-Abelian Theories.
17.1 Level crossing and Euclidean fermion zero modes -- 17.2 Fermion zero mode in an instanton field -- 17.3 Selection rules -- 17.4 Electroweak non-conservation of baryon and lepton numbers at high temperatures -- Supplementary Problems for Part III -- Appendix. Classical Solutions and the Functional Integral -- A.1 Decay of the false vacuum in the functional integral formalism -- A.2 Instanton contributions to the fermion Green's functions -- A.3 Instantons in theories with the Higgs mechanism. Integration along valleys -- A.4 Growing instanton cross sections -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- P -- Q -- R -- S -- T -- W -- Y -- Z.
Summary: Based on a highly regarded lecture course at Moscow State University, this is a clear and systematic introduction to gauge field theory. It is unique in providing the means to master gauge field theory prior to the advanced study of quantum mechanics. Though gauge field theory is typically included in courses on quantum field theory, many of its ideas and results can be understood at the classical or semi-classical level. Accordingly, this book is organized so that its early chapters require no special knowledge of quantum mechanics. Aspects of gauge field theory relying on quantum mechanics are introduced only later and in a graduated fashion--making the text ideal for students studying gauge field theory and quantum mechanics simultaneously.The book begins with the basic concepts on which gauge field theory is built. It introduces gauge-invariant Lagrangians and describes the spectra of linear perturbations, including perturbations above nontrivial ground states. The second part focuses on the construction and interpretation of classical solutions that exist entirely due to the nonlinearity of field equations: solitons, bounces, instantons, and sphalerons. The third section considers some of the interesting effects that appear due to interactions of fermions with topological scalar and gauge fields. Mathematical digressions and numerous problems are included throughout. An appendix sketches the role of instantons as saddle points of Euclidean functional integral and related topics.Perfectly suited as an advanced undergraduate or beginning graduate text, this book is an excellent starting point for anyone seeking to understand gauge fields.
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Contents -- Preface -- Part I -- 1 Gauge Principle in Electrodynamics -- 1.1 Electromagnetic-field action in vacuum -- 1.2 Gauge invariance -- 1.3 General solution of Maxwell's equations in vacuum -- 1.4 Choice of gauge -- 2 Scalar and Vector Fields -- 2.1 System of units h = c=1 -- 2.2 Scalar field action -- 2.3 Massive vector field -- 2.4 Complex scalar field -- 2.5 Degrees of freedom -- 2.6 Interaction of fields with external sources -- 2.7 Interacting fields. Gauge-invariant interaction in scalar electrodynamics -- 2.8 Noether's theorem -- 3 Elements of the Theory of Lie Groups and Algebras -- 3.1 Groups -- 3.2 Lie groups and algebras -- 3.3 Representations of Lie groups and Lie algebras -- 3.4 Compact Lie groups and algebras -- 4 Non-Abelian Gauge Fields -- 4.1 Non-Abelian global symmetries -- 4.2 Non-Abelian gauge invariance and gauge fields: the group SU(2) -- 4.3 Generalization to other groups -- 4.4 Field equations -- 4.5 Cauchy problem and gauge conditions -- 5 Spontaneous Breaking of Global Symmetry -- 5.1 Spontaneous breaking of discrete symmetry -- 5.2 Spontaneous breaking of global U(1) symmetry. Nambu-Goldstone bosons -- 5.3 Partial symmetry breaking: the SO(3)model -- 5.4 General case. Goldstone's theorem -- 6 Higgs Mechanism -- 6.1 Example of an Abelian model -- 6.2 Non-Abelian case: model with complete breaking of SU(2) symmetry -- 6.3 Example of partial breaking of gauge symmetry: bosonic sector of standard electroweak theory -- Supplementary Problems for Part I -- Part II -- 7 The Simplest Topological Solitons -- 7.1 Kink -- 7.2 Scale transformations and theorems on the absence of solitons -- 7.3 The vortex -- 7.4 Soliton in a model of n-field in (2 + 1)-dimensional space-time -- 8 Elements of Homotopy Theory -- 8.1 Homotopy of mappings -- 8.2 The fundamental group -- 8.3 Homotopy groups.

8.4 Fiber bundles and homotopy groups -- 8.5 Summary of the results -- 9 Magnetic Monopoles -- 9.1 The soliton in a model with gauge group SU(2) -- 9.2 Magnetic charge -- 9.3 Generalization to other models -- 9.4 The limit mH/mV → 0 -- 9.5 Dyons -- 10 Non-Topological Solitons -- 11 Tunneling and Euclidean Classical Solutions in Quantum Mechanics -- 11.1 Decay of a metastable state in quantum mechanics of one variable -- 11.2 Generalization to the case of many variables -- 11.3 Tunneling in potentials with classical degeneracy -- 12 Decay of a False Vacuum in Scalar Field Theory -- 12.1 Preliminary considerations -- 12.2 Decay probability: Euclidean bubble (bounce) -- 12.3 Thin-wall approximation -- 13 Instantons and Sphalerons in Gauge Theories -- 13.1 Euclidean gauge theories -- 13.2 Instantons in Yang-Mills theory -- 13.3 Classical vacua and θ-vacua -- 13.4 Sphalerons in four-dimensional models with the Higgs mechanism -- Supplementary Problems for Part II -- Part III -- 14 Fermions in Background Fields -- 14.1 Free Dirac equation -- 14.2 Solutions of the free Dirac equation. Dirac sea -- 14.3 Fermions in background bosonic fields -- 14.4 Fermionic sector of the Standard Model -- 15 Fermions and Topological External Fields in Two-dimensional Models -- 15.1 Charge fractionalization -- 15.2 Level crossing and non-conservation of fermion quantum numbers -- 16 Fermions in Background Fields of Solitons and Strings in Four-Dimensional Space-Time -- 16.1 Fermions in a monopole background field: integer angular momentum and fermion number fractionalization -- 16.2 Scattering of fermions off a monopole: non-conservation of fermion numbers -- 16.3 Zero modes in a background field of a vortex: superconducting strings -- 17 Non-Conservation of Fermion Quantum Numbers in Four-dimensional Non-Abelian Theories.

17.1 Level crossing and Euclidean fermion zero modes -- 17.2 Fermion zero mode in an instanton field -- 17.3 Selection rules -- 17.4 Electroweak non-conservation of baryon and lepton numbers at high temperatures -- Supplementary Problems for Part III -- Appendix. Classical Solutions and the Functional Integral -- A.1 Decay of the false vacuum in the functional integral formalism -- A.2 Instanton contributions to the fermion Green's functions -- A.3 Instantons in theories with the Higgs mechanism. Integration along valleys -- A.4 Growing instanton cross sections -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- P -- Q -- R -- S -- T -- W -- Y -- Z.

Based on a highly regarded lecture course at Moscow State University, this is a clear and systematic introduction to gauge field theory. It is unique in providing the means to master gauge field theory prior to the advanced study of quantum mechanics. Though gauge field theory is typically included in courses on quantum field theory, many of its ideas and results can be understood at the classical or semi-classical level. Accordingly, this book is organized so that its early chapters require no special knowledge of quantum mechanics. Aspects of gauge field theory relying on quantum mechanics are introduced only later and in a graduated fashion--making the text ideal for students studying gauge field theory and quantum mechanics simultaneously.The book begins with the basic concepts on which gauge field theory is built. It introduces gauge-invariant Lagrangians and describes the spectra of linear perturbations, including perturbations above nontrivial ground states. The second part focuses on the construction and interpretation of classical solutions that exist entirely due to the nonlinearity of field equations: solitons, bounces, instantons, and sphalerons. The third section considers some of the interesting effects that appear due to interactions of fermions with topological scalar and gauge fields. Mathematical digressions and numerous problems are included throughout. An appendix sketches the role of instantons as saddle points of Euclidean functional integral and related topics.Perfectly suited as an advanced undergraduate or beginning graduate text, this book is an excellent starting point for anyone seeking to understand gauge fields.

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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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