Mathematics for Physics

The following sections are included:

• Preface
• Notation
• Contents

The following sections are included:

• Classifying mathematical concepts
• Defining mathematical structures and mappings

The following sections are included:

• Generalizing numbers
  • Groups
  • Rings
• Generalizing vectors
  • Inner products of vectors
  • Norms of vectors
  • Multilinear forms on vectors
  • Orthogonality of vectors
  • Algebras: multiplication of vectors
  • Division algebras
• Combining algebraic objects
  • The direct product and direct sum
  • The free product
  • The tensor product
• Dividing algebraic objects
  • Quotient groups
  • Semidirect products
  • Quotient rings
  • Related constructions and facts
• Summary

The following sections are included:

• Constructing algebras from a vector space
  • The tensor algebra
  • The exterior algebra
  • Combinatorial notations
  • The Hodge star
  • Graded algebras
  • Clifford algebras
  • Geometric algebra
• Tensor algebras on the dual space
  • The structure of the dual space
  • Tensors
  • Tensors as multilinear mappings
  • Abstract index notation
  • Tensors as multi-dimensional arrays
• Exterior forms
  • Exterior forms as multilinear mappings
  • Exterior forms as completely anti-symmetric tensors
  • Exterior forms as anti-symmetric arrays
  • The Clifford algebra of the dual space
  • Algebra-valued exterior forms
  • Related constructions and facts

Using the algebraic tools we have developed, we can now move into geometry. Before launching into the main subject of this chapter, topology, we will examine the intuitive meanings of geometric objects in general, and the properties that define them.

Algebraic topology is concerned with characterizing spaces. The main tools used to do this, called homotopy groups and homology groups, measure the “holes” of a space, and so are invariant under homotopy equivalence. More precisely, these objects are functors from the category of spaces and continuous maps to that of groups and homomorphisms…

In studying spaces, we considered the locally Euclidean structure of topological manifolds as defining a subset of spaces that were “nicer,” meeting the minimum requirements of our idea of a geometrical “shape” such as integral dimension. By slightly narrowing our consideration to differentiable manifolds, we can essentially graft calculus onto our “rubber sheet.” The constructions of coordinates and tangent vectors enable us to define a family of derivatives associated with the concept of how vector fields change on the manifold. The challenge is in defining all these objects without an ambient space, which our intuitive picture normally depends upon…

The following sections are included:

• Combining algebra and geometry
  • Spaces with multiplication of points
  • Vector spaces with topology
• Lie groups and Lie algebras
  • The Lie algebra of a Lie group
  • The Lie groups of a Lie algebra
  • Relationships between Lie groups and Lie algebras
  • The universal cover of a Lie group
• Matrix groups
  • Lie algebras of matrix groups
  • Linear algebra
  • Matrix groups with real entries
  • Other matrix groups
  • Manifold properties of matrix groups
  • Matrix group terminology in physics
• Representations
  • Group actions
  • Group and algebra representations
  • Lie group and Lie algebra representations
  • Combining and decomposing representations
  • Other representations
• Classification of Lie groups
  • Compact Lie groups
  • Simple Lie algebras
  • Classifying representations

In this chapter we will go into some more detail on the structure of Clifford algebras, and then construct Lie groups within these algebras that are closely related to rotations and the concept of spin.

In this chapter we introduce two additional structures on a differentiable manifold. First we consider the “parallel transport” of a vector, which allows a vector at one point on the manifold to be “transported” along a path to another point, where it can then be compared to other vectors at the new point. This idea gives rise to a number of interdependent quantities, and is particularly important in physics, where it is generalized to gauge theories.

We then consider the introduction of a metric, an inner product in each tangent space that permits us to define lengths of vectors and angles between them. A metric determines a unique associated parallel transport, and is the fundamental quantity in general relativity. We then touch upon some other structures on manifolds that appear in physics.

A manifold includes a tangent space associated with each point. A frame defines a basis for the tangent space at each point, and a connection allows us to compare vectors at different points, leading to concepts including the covariant derivative and curvature. All of these concepts can be applied to an arbitrary vector space associated with each point in place of the tangent space. This is the idea behind gauge theories. Both manifolds with connection and gauge theories can then be described using the mathematical language of fiber bundles.

The following sections are included:

• Appendix A: Categories and functors
  • Generalizing sets and mappings
  • Mapping mappings
• Bibliography
• Index