Gauge field theory in natural geometric language : a revisitation of mathematical notions of quantum physics / Daniel Canarutto.

Format:

Book

Author/Creator:

Canarutto, Daniel, author.

Contributor:

Emma Louise McClellan Fund.

Language:

English

Subjects (All):

Gauge fields (Physics).

Particles (Nuclear physics).

Physical Description:

xxix, 331 pages : illustrations ; 24 cm

Edition:

First edition.

Place of Publication:

Oxford ; New York : Oxford University Press, 2020.

Summary:

"Gauge Field theory in Natural Geometric Language addresses the need to clarify basic mathematical concepts at the crossroad between gravitation and quantum physics. Selected mathematical and theoretical topics are exposed within a brief, integrated approach that exploits standard and non-standard notions, as well as recent advances, in a natural geometric language in which the role of structure groups can be regarded as secondary even in the treatment of the gauge fields themselves. In proposing an original bridge between physics and mathematics, this text will appeal not only to mathematicians who wish to understand some of the basic ideas involved in quantum particle physics, but also to physicists who are not satisfied with the usual mathematical presentations of their field."- publisher.

Contents:

Machine generated contents note: pt. I CLASSICAL GEOMETRY

1. Bundle Prolongations and Connections

1.1. Classical Manifolds and Bundles

1.2. Tangent Prolongations and Differential Operators

1.3. Vertical and Jet Prolongations

1.4. Higher-order Jets and Multi-indices

1.5. The Frolicher-Nijenhuis Bracket

1.6. Connections

1.7. Linear Connections

1.8. Tensor-product Connection and Dual Connection

1.9. Tangent-bundle Connections

2. Special Algebraic Notions

2.1. Unit Spaces and Physical Scales

2.2. Complex Spaces and Conjugate Spaces

2.3. Hermitian Tensors

2.4. Anti-Hermitian Lie Algebra

2.5. Clifford Algebra

3. Spinors and Minkowski Space

3.1. Two-spinor Space

3.2. Two-spinor-generated Minkowski Space

3.3. Dirac Spinors

3.4. Charge Conjugation

3.5. Observers and Positive Hermitian Structures

3.6. Algebraic Dirac Equation

3.7. Further Observer-dependent Objects

3.8. Decompositions of Endomorphisms

4. Spinor Bundles and Spacetime Geometry

4.1. Two-spinor Bundles and their Connections

4.2. Two-spinor Soldering Form

4.3. Complementary Soldering Form

4.4. Soldering Form and Connections

4.5. Fermi Transport of Spinors

4.6. Two-spinors and Lorentzian Distance

pt. II PRE-QUANTUM FIELD THEORY

5. Classical Gauge Field Theory

5.1. Lagrangian Field Theory on Jet Bundles

5.2. Fields as Infinite-dimensional Mechanical Systems

5.3. Gauge Fields

5.4. Covariant Prolongation Bundle in Gauge Field Theory

5.5. Coordinate Expressions of Covariant Prolongations

5.6. Covariant Differential and Lagrangian Density

5.7. Field Equations in the Covariant-differential Approach

5.8. Field Equations: Adaptations of the Basic Scheme

5.9. Canonical Energy Tensor and Currents

5.10. Generalized Replacement Principle

6. Gauge Field Theory and Gravitation

6.1. Tetrad-affine Setting

6.2. Fields with Spin on a Tetrad-affine Background

6.3. Basic Gauge Field Theory Examples

6.4. Dynamical Gravitational Field in Vacuum

6.5. Interacting Gravitational Field

6.6. `Minimal Geometric Data' ECMD Theory

7. Optical Geometry

7.1. Optical Algebra

7.2. Two-spinors and Optical Algebra

7.3. Helicity of Photons

7.4. Optical Bundles

7.5. Complexified Electromagnetic Field

7.6. Null Electromagnetic Field in Curved Spacetime

7.7. Electromagnetic Radiation in Two-spinor Form

8. Electroweak Geometry and Fields

8.1. Spin One-half and Gauge Fields

8.2. Symmetry Breaking

8.3. Electroweak Fields

8.4. Electroweak Symmetry Breaking

8.5. Covariant Differentials of Electroweak Fields

8.6. Electroweak Lagrangian

9. First-order Theory of Fields with Arbitrary Spin

9.1. Higher-spin Extensions of the Dirac Map

9.2. Symmetric Spinors

9.3. Generalized Algebraic Dirac Equation

9.4. Generalized Dirac Equation and Plane Waves

9.5. First-order Higher-spin Lagrangian

9.6. Interactions of Higher-spin Fields and Gauge Fields

9.7. Further Spinor Field Types

10. Infinitesimal Deformations of ECD Fields

10.1. Lie Derivative of Spinors

10.2. Lie Derivative of a Soldering Form

10.3. Lie Derivative of a Spinor Connection

10.4. Deformed Tetrad Gravity

pt. III QUANTUM GEOMETRY

11. Generalized Maps

11.1. Spaces of Generalized Sections

11.2. Tensor Products of Distributional Spaces

11.3. Special Distributions

11.4. Division

11.5. Elementary Solutions of Field Equations

11.6. Fourier Transforms

11.7. Fourier Transforms: Basic Properties and Examples

12. Special Generalized Densities on Minkowski Spacetime

12.1. Minkowskian Framework

12.2. Mass-shell Leray Densities

12.3. Fourier Transforms of Mass-shell Leray Densities

12.4. Mass-shell Principal Values

12.5. Elementary Solutions of the Klein

Gordon Equation

12.6. Klein

Gordon Propagators

12.7. Massless Case and Wave Equation

12.8. Spinor Propagators

13. Multi-particle Spaces

13.1. Freely Generated Vector Spaces

13.2. Multi-particle Algebra

13.3. Multi-particle Bases

13.4. Several Types of Particles

13.5. Operator Algebra

13.6. Conjugation and the Role of Hermitian Structure

13.7. Freely Generated Spaces versus Distributional Spaces

13.8. Generalized Bases

13.9. Multi-particle Spaces of Generalized Semi-densities

14. Bundles of Quantum States

14.1. Frolicher-smooth Spaces

14.2. Frolicher-smooth Distributional Bundles

14.3. Geometry of Distributional Bundles

14.4. Bundles over Classical Mass-shell Bundles

14.5. Bundles of Multi-particle States over Momenta

14.6. Quantum Frames

15. Quantum Bundles

15.1. Graded Generalized Scaling

15.2. F-smooth Geometry of Quantum Bundles

15.3. Quantum Functions

15.4. Partial Fibre Derivatives in Quantum Bundles

15.5. Batalin

Vilkovisky Algebra

pt. IV QUANTUM FIELDS

16. Quantum Fields

16.1. Horizontal Forms and Lagrangian Quantum Field Theory

16.2. Infinitesimal Vertical Symmetries and Currents

16.3. BRST Symmetry in Lagrangian Field Theory

16.4. Fields of an Essential Gauge Theory with Ghosts

16.5. Lagrangian Field Theory with Ghosts

16.6. BRST Symmetry: The Basic Example

16.7. Alternative Ghost Lagrangian

17. Detectors

17.1. Detectors, Synchronizations, Observers

17.2. Quantum Configuration Space

17.3. Detectors and Quantum Fields

18. Free Quantum Fields

18.1. The Notion of Free Field

18.2. Conjugate Free Fields

18.3. Super-commutators of Free Fields

18.4. Free-field Hamiltonian

18.5. Free Fields of an Essential Gauge Theory

18.6. Functionals in Terms of Free Fields

18.7. Canonical Super-commutation Relations

19. Electroweak Extensions

19.1. Further Scalar Invariants from Higgs Geometry

19.2. Higgs Potential Revisited

19.3. Possible Interactions of the Extended Higgs Sector

pt. V QUANTUM PHYSICS

20. Basic Notions in Particle Physics

20.1. Quantum `Pictures'

20.2. Dyson Series and Scattering Operator

20.3. Interaction Semi-density

20.4. Generalized Index Types of the Interaction

20.5. Interaction Morphism for Scalar Particles

20.6. Quantum Interactions and Internal Structure

21. Scattering Matrix Computations

21.1. Technical Preliminaries

21.2. One-point Interaction

21.3. Propagator of a Scalar Particle

21.4. Self-interaction

21.5. Computations with an Internal Structure

21.6. Feynman Rules

22. Quantum Electrodynamics

22.1. Free QED States

22.2. QED Interactions

22.3. One-point Interaction in QED

22.4. Electron Propagator

22.5. Positron Propagator

22.6. Photon Propagator

23. On Gauge Freedom and Interactions

23.1. Two-spinors and Gauge Freedom

23.2. Interactions in Extended Electrodynamics

23.3. Electroweak Interactions among Gauge Bosons

23.4. Concluding Remarks.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Emma Louise McClellan Fund.

ISBN:

0198861494

9780198861492

OCLC:

1152441827

Publisher Number:

99987391875