This unique book complements traditional textbooks by providing a visual yet rigorous survey of the mathematics used in theoretical physics beyond that typically covered in undergraduate math and physics courses. The exposition is pedagogical but compact, and the emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints. Certain topics which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations, are avoided. The primary physical models targeted are general relativity, spinors, and gauge theories, with notable chapters on Riemannian geometry, Clifford algebras, and fiber bundles.
Contents:
Mathematical Structures
Abstract Algebra
Vector Algebras
Topological Spaces
Algebraic Topology
Manifolds
Lie Groups
Clifford Groups
Riemannian Manifolds
Fiber Bundles
Categories and Functors
Readership: Students in mathematics and physics who want to explore a level deeper into actual mathematical content.
Author(s): Adam Marsh
Edition: 1
Publisher: World Scientific Publishing Co Pte Ltd
Year: 2018
Language: English
Pages: 300
Tags: mathematics; theoretical physics; general relativity; spinors; gauge theories; Riemann geometry; Clifford algebras; fiber bundles; Mathematical Structures; Abstract Algebra; Vector Algebras; Topological Spaces; Algebraic Topology; Manifolds; Lie Groups; Clifford Groups; Riemann Manifolds
Contents
Preface
What this book is, and what it is not
Who this book is written for
Organization of the book
Notation
Standard notations
Defined notations
Notation conventions
Formatting
1. Mathematical structures
1.1 Classifying mathematical concepts
1.2 Defining mathematical structures and mappings
2. Abstract algebra
2.1 Generalizing numbers
2.1.1 Groups
2.1.2 Rings
2.2 Generalizing vectors
2.2.1 Inner products of vectors
2.2.2 Norms of vectors
2.2.3 Multilinear forms on vectors
2.2.4 Orthogonality of vectors
2.2.5 Algebras: multiplication of vectors
2.2.6 Division algebras
2.3 Combining algebraic objects
2.3.1 The direct product and direct sum
2.3.2 The free product
2.3.3 The tensor product
2.4 Dividing algebraic objects
2.4.1 Quotient groups
2.4.2 Semidirect products
2.4.3 Quotient rings
2.4.4 Related constructions and facts
2.5 Summary
3. Vector algebras
3.1 Constructing algebras from a vector space
3.1.1 The tensor algebra
3.1.2 The exterior algebra
3.1.3 Combinatorial notations
3.1.4 The Hodge star
3.1.5 Graded algebras
3.1.6 Clifford algebras
3.1.7 Geometric algebra
3.2 Tensor algebras on the dual space
3.2.1 The structure of the dual space
3.2.2 Tensors
3.2.3 Tensors as multilinear mappings
3.2.4 Abstract index notation
3.2.5 Tensors as multi-dimensional arrays
3.3 Exterior forms
3.3.1 Exterior forms as multilinear mappings
3.3.2 Exterior forms as completely anti-symmetric tensors
3.3.3 Exterior forms as anti-symmetric arrays
3.3.4 The Clifford algebra of the dual space
3.3.5 Algebra-valued exterior forms
3.3.6 Related constructions and facts
4. Topological spaces
4.1 Generalizing surfaces
4.1.1 Spaces
4.1.2 Generalizing dimension
4.1.3 Generalizing tangent vectors
4.1.4 Existence and uniqueness of additional structure
4.1.5 Summary
4.2 Generalizing shapes
4.2.1 Defining spaces
4.2.2 Mapping spaces
4.3 Constructing spaces
4.3.1 Cell complexes
4.3.2 Projective spaces
4.3.3 Combining spaces
4.3.4 Classifying spaces
5. Algebraic topology
5.1 Constructing surfaces within a space
5.1.1 Simplices
5.1.2 Triangulations
5.1.3 Orientability
5.1.4 Chain complexes
5.2 Counting holes that aren’t boundaries
5.2.1 The homology groups
5.2.2 Examples
5.2.3 Calculating homology groups
5.2.4 Related constructions and facts
5.3 Counting the ways a sphere maps to a space
5.3.1 The fundamental group
5.3.2 The higher homotopy groups
5.3.3 Calculating the fundamental group
5.3.4 Calculating the higher homotopy groups
5.3.5 Related constructions and facts
6. Manifolds
6.1 Defining coordinates and tangents
6.1.1 Coordinates
6.1.2 Tangent vectors and differential forms
6.1.3 Frames
6.1.4 Tangent vectors in terms of frames
6.2 Mapping manifolds
6.2.1 Diffeomorphisms
6.2.2 The differential and pullback
6.2.3 Immersions and embeddings
6.2.4 Critical points
6.3 Derivatives on manifolds
6.3.1 Derivations
6.3.2 The Lie derivative of a vector field
6.3.3 The Lie derivative of an exterior form
6.3.4 The exterior derivative of a 1-form
6.3.5 The exterior derivative of a k-form
6.3.6 Relationships between derivations
6.4 Homology on manifolds
6.4.1 The Poincare lemma
6.4.2 de Rham cohomology
6.4.3 Poincare duality
7. Lie groups
7.1 Combining algebra and geometry
7.1.1 Spaces with multiplication of points
7.1.2 Vector spaces with topology
7.2 Lie groups and Lie algebras
7.2.1 The Lie algebra of a Lie group
7.2.2 The Lie groups of a Lie algebra
7.2.3 Relationships between Lie groups and Lie algebras
7.2.4 The universal cover of a Lie group
7.3 Matrix groups
7.3.1 Lie algebras of matrix groups
7.3.2 Linear algebra
7.3.3 Matrix groups with real entries
7.3.4 Other matrix groups
7.3.5 Manifold properties of matrix groups
7.3.6 Matrix group terminology in physics
7.4 Representations
7.4.1 Group actions
7.4.2 Group and algebra representations
7.4.3 Lie group and Lie algebra representations
7.4.4 Combining and decomposing representations
7.4.5 Other representations
7.5 Classification of Lie groups
7.5.1 Compact Lie groups
7.5.2 Simple Lie algebras
7.5.3 Classifying representations
8. Clifford groups
8.1 Classification of Clifford algebras
8.1.1 Isomorphisms
8.1.2 Representations and spinors
8.1.3 Pauli and Dirac matrices
8.1.4 Chiral decomposition
8.2 Clifford groups and representations
8.2.1 Reflections
8.2.2 Rotations
8.2.3 Lie group properties
8.2.4 Lorentz transformations
8.2.5 Representations in spacetime
8.2.6 Spacetime and spinors in geometric algebra
9. Riemannian manifolds
9.1 Introducing parallel transport of vectors
9.1.1 Change of frame
9.1.2 The parallel transporter
9.1.3 The covariant derivative
9.1.4 The connection
9.1.5 The covariant derivative in terms of the connection
9.1.6 The parallel transporter in terms of the connection
9.1.7 Geodesics and normal coordinates
9.1.8 Summary
9.2 Manifolds with connection
9.2.1 The covariant derivative on the tensor algebra
9.2.2 The exterior covariant derivative of vector-valued forms
9.2.3 The exterior covariant derivative of algebra-valued forms
9.2.4 Torsion
9.2.5 Curvature
9.2.6 First Bianchi identity
9.2.7 Second Bianchi identity
9.2.8 The holonomy group
9.3 Introducing lengths and angles
9.3.1 The Riemannian metric
9.3.2 The Levi-Civita connection
9.3.3 Independent quantities and dependencies
9.3.4 The divergence and conserved quantities
9.3.5 Ricci and sectional curvature
9.3.6 Curvature and geodesics
9.3.7 Jacobi fields and volumes
9.3.8 Summary
9.3.9 Related constructions and facts
10. Fiber bundles
10.1 Gauge theory
10.1.1 Matter fields and gauges
10.1.2 The gauge potential and field strength
10.1.3 Spinor fields
10.2 Defining bundles
10.2.1 Fiber bundles
10.2.2 G-bundles
10.2.3 Principal bundles
10.3 Generalizing tangent spaces
10.3.1 Associated bundles
10.3.2 Vector bundles
10.3.3 Frame bundles
10.3.4 Gauge transformations on frame bundles
10.3.5 Smooth bundles and jets
10.3.6 Vertical tangents and horizontal equivariant forms
10.4 Generalizing connections
10.4.1 Connections on bundles
10.4.2 Parallel transport on the frame bundle
10.4.3 The exterior covariant derivative on bundles
10.4.4 Curvature on principal bundles
10.4.5 The tangent bundle and solder form
10.4.6 Torsion on the tangent frame bundle
10.4.7 Spinor bundles
10.5 Characterizing bundles
10.5.1 Universal bundles
10.5.2 Characteristic classes
10.5.3 Related constructions and facts
Appendix A Categories and functors
A.1 Generalizing sets and mappings
A.2 Mapping mappings
Bibliography
Index
