Mathematics for Physics | An Illustrated Handbook
What this book is, and what it is not
Who this book is written for
Organization of the book
Standard notations
Defined notations
Notation conventions
Mathematical structures
Classifying mathematical concepts
Defining mathematical structures and mappings
Abstract algebra
Generalizing numbers
Generalizing vectors
Inner products of vectors
Norms and angles 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
Vector algebras
Constructing algebras from a vector space
The tensor algebra
The exterior algebra
Combinatorial notations
The Hodge star
Graded algebras
Clifford algebras
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
Topological spaces
Generalizing surfaces
Generalizing dimension
Generalizing tangent vectors
Existence and uniqueness of additional structure
Generalizing shapes
Defining spaces
Mapping spaces
Constructing spaces
Cell complexes
Projective spaces
Combining spaces
Classifying spaces
Algebraic topology
Constructing surfaces within a space
Simplices
Triangulations
Orientability
Chain complexes
Counting holes that aren't boundaries
The homology groups
Calculating homology groups
Counting the ways a sphere maps to a space
The fundamental group
The higher homotopy groups
Calculating the fundamental group
Calculating the higher homotopy groups
Defining coordinates and tangents
Tangent vectors and differential forms
Tangent vectors in terms of frames
Mapping manifolds
Diffeomorphisms
The differential and pullback
Immersions and embeddings
Critical points
Derivatives on manifolds
Derivations
The Lie derivative of a vector field
The Lie derivative of an exterior form
The exterior derivative of a 1-form
The exterior derivative of a k-form
Relationships between derivations
Homology on manifolds
The Poincaré lemma
de Rham cohomology
Poincaré duality
Lie groups
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
Matrix groups with real entries
Other matrix groups
Manifold properties of matrix groups
Matrix group terminology in physics
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
Clifford groups
Classification of Clifford algebras
Isomorphisms
Representations and spinors
Chiral decomposition
Pauli and Dirac matrices
Clifford groups and representations
Lie group properties
Lorentz transformations
Representations in spacetime
Spacetime and spinors in geometric algebra
Riemannian manifolds
Introducing parallel transport of vectors
Change of frame
The parallel transporter
The covariant derivative
The covariant derivative in terms of the connection
The parallel transporter in terms of the connection
Geodesics and normal coordinates
Manifolds with connection
The covariant derivative on the tensor algebra
The exterior covariant derivative of vector-valued forms
The exterior covariant derivative of algebra-valued forms
First Bianchi identity
Second Bianchi identity
The holonomy group
Introducing lengths and angles
The Riemannian metric
The Levi-Civita connection
Independent quantities and dependencies
The divergence and conserved quantities
Ricci and sectional curvature
Curvature and geodesics
Jacobi Fields and Volumes
Fiber bundles
Gauge theory
Matter fields and gauges
The gauge potential and field strength
Spinor fields
Defining bundles
G-bundles
Principal bundles
Generalizing tangent spaces
Associated bundles
Vector bundles
Frame bundles
Gauge transformations on frame bundles
Smooth bundles and jets
Vertical tangents and horizontal equivariant forms
Generalizing connections
Connections on bundles
Parallel transport on the frame bundle
The exterior covariant derivative on bundles
Curvature on principal bundles
The tangent bundle and solder form
Torsion on the tangent frame bundle
Spinor bundles
Characterizing bundles
Universal bundles
Characteristic classes
Appendix: Categories and functors
Generalizing sets and mappings
Mapping mappings
The matrix isomorphisms of Clifford algebras are often expressed in terms of Pauli matrices. We will follow the common convention of using \({\left\{ i,j,k\right\} }\) to represent matrix indices that are an even permutation of \({\left\{ 1,2,3\right\} }\); \({i}\) also represents the square root of negative one, but the distinction should be clear from context.
The Pauli matrices
\(\displaystyle \sigma_{1}\equiv\begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}\;\sigma_{2}\equiv\begin{pmatrix}0 & -i\\ i & 0 \end{pmatrix}\;\sigma_{3}\equiv\begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix} \)
are traceless, hermitian, unitary, determinant \({-1}\) matrices that satisfy the relations \({\sigma_{i}\sigma_{j}=i\sigma_{k}}\) and \({\sigma_{i}\sigma_{j}\sigma_{k}=i}\). They also all anti-commute and square to the identity \({\sigma_{0}\equiv I}\); therefore, if we take matrix multiplication as Clifford multiplication, they act as an orthonormal basis of the vector space that generates the Clifford algebra \({C(3,0)\cong\mathbb{C}(2)}\). In physics \({C(3,0)}\) is associated with space, and is sometimes called the Pauli algebra (AKA algebra of physical space).
We introduce the shorthand
\(\displaystyle \sigma_{13}\equiv\sigma_{1}\sigma_{3}=\begin{pmatrix}0 & -1\\ 1 & 0 \end{pmatrix} \)
so that \({\sigma_{2}=i\sigma_{13}}\). Since \({\left(\sigma_{13}\right)^{2}=-I}\), we can use it and \({\sigma_{0}}\) as a basis for \({\mathbb{C}\cong C(0,1)}\), allowing us to express complex numbers as real matrices via the isomorphism
\(\displaystyle a+ib\leftrightarrow a\sigma_{0}+b\sigma_{13}=\begin{pmatrix}a & -b\\ b & a \end{pmatrix}. \)
In physics \({C(3,1)}\) (or \({C(1,3)}\)) is associated with spacetime, but it turns out one is usually more interested in the complexified algebra \({C\mathbb{^{C}}(4)\cong\mathbb{C}(4)}\), which is sometimes called the Dirac algebra. Any four matrices in \({\mathbb{C}(4)}\) that act as an orthonormal basis of the vector space generating \({C(3,1)}\) or \({C(1,3)}\) (and via complexification \({C\mathbb{^{C}}(4)}\)) are called Dirac matrices (AKA gamma matrices), and denoted \({\gamma^{i}}\). A fifth related matrix is usually defined as \({\gamma_{5}\equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}}\). Many choices of Dirac matrices are in common use, a particular one being labeled the Dirac basis (AKA Dirac representation, standard basis). This is traditionally realized as a basis for \({C(1,3)}\):
\(\displaystyle \gamma^{0}=\begin{pmatrix}I & 0\\ 0 & -I \end{pmatrix},\;\gamma^{i}=\begin{pmatrix}0 & \sigma_{i}\\ -\sigma_{i} & 0 \end{pmatrix}\;\Rightarrow\gamma_{5}=\begin{pmatrix}0 & I\\ I & 0 \end{pmatrix} \)
Another common class of Dirac matrices requires \({\gamma_{5}}\) to be diagonal; this is called a chiral basis (AKA Weyl basis or chiral / Weyl representation). The meaning of \({\gamma_{5}}\) and "chiral" will be explained in the next section. A chiral basis for \({C(1,3)}\) is
\(\displaystyle \gamma^{0}=\begin{pmatrix}0 & I\\ I & 0 \end{pmatrix},\;\gamma^{i}=\begin{pmatrix}0 & \sigma_{i}\\ -\sigma_{i} & 0 \end{pmatrix}\;\Rightarrow\gamma_{5}=\begin{pmatrix}-I & 0\\ 0 & I \end{pmatrix}, \)
and a chiral basis for \({C(3,1)}\) is
\(\displaystyle \gamma^{0}=\begin{pmatrix}0 & I\\ -I & 0 \end{pmatrix},\;\gamma^{i}=\begin{pmatrix}0 & \sigma_{i}\\ \sigma_{i} & 0 \end{pmatrix}\;\Rightarrow\gamma_{5}=\begin{pmatrix}-I & 0\\ 0 & I \end{pmatrix}. \)
Finally, a Majorana basis generates the Majorana rep \({C(3,1)\cong\mathbb{R}(4)}\). We can find such a basis by applying the previous isomorphism for complex numbers as real matrices to the Pauli matrices themselves, obtaining anti-commuting matrices in \({\mathbb{R}(4)}\) that square to the identity; if we then include an initial anti-commuting matrix that squares to \({-I}\), we get:
\(\displaystyle \gamma^{0}=\begin{pmatrix}\sigma_{13} & 0\\ 0 & -\sigma_{13} \end{pmatrix},\;\gamma^{1}=\begin{pmatrix}\sigma_{1} & 0\\ 0 & \sigma_{1} \end{pmatrix},\;\gamma^{2}=\begin{pmatrix}0 & -\sigma_{13}\\ \sigma_{13} & 0 \end{pmatrix}, \)
\(\displaystyle \gamma^{3}=\begin{pmatrix}\sigma_{3} & 0\\ 0 & \sigma_{3} \end{pmatrix}\;\Rightarrow\gamma_{5}=\begin{pmatrix}0 & -\sigma_{2}\\ -\sigma_{2} & 0 \end{pmatrix}. \)
We know that these matrices act as a basis due to Pauli's fundamental theorem, whose extended form states that for even \({r+s=n}\), any two sets of \({n}\) anti-commuting matrices which square to \({\pm1}\) according to the signature are related by a similarity transformation; this means that any such elements can act as a basis for the vector space generating the Clifford algebra, since one of them must. This theorem also holds for \({C\mathbb{^{C}}(n)}\) for even \({n}\).
Note that all the above matrices are unitary, and those representing positive signature basis vectors are Hermitian, while those representing negative signature basis vectors are anti-Hermitian; these properties are sometimes required when (more restrictively) defining Dirac matrices. Dirac or gamma matrices can also be generalized to other dimensions and signatures; in this light the Pauli matrices are gamma matrices for \({C(3,0)}\). If the dimension is greater than 5, \({\gamma_{5}}\) can be confused with \({\gamma^{5}}\); this is made worse by the fact that one can also define the covariant Dirac matrices \({\gamma_{i}\equiv\eta_{ij}\gamma^{j}}\).
Δ The Dirac matrices and \({\gamma_{5}}\) are defined in various ways by different authors. Most differ from the above only by a factor of \({±1}\) or \({±i}\); however, there is not much standardization in this area. Sometimes the Clifford algebra definition itself is changed by a sign; in this case the matrices represent a basis with the wrong signature, and according to our definition are not Dirac matrices. This is sometimes done for example when working with Majorana spinors, which only exist in \({C(3,1)}\) spacetime, yet where an author works nevertheless in the \({C(1,3)}\) "mostly minuses" signature.
Δ It is important to remember that the Dirac matrices are matrix representations of an orthonormal basis of the underlying vector space used to generate a Clifford algebra. So the Dirac and chiral bases are different representations of the orthonormal basis which generates the same matrix representation of \({C\mathbb{^{C}}(4)\cong\mathbb{C}(4)}\) acting on vectors (spinors) in \({\mathbb{C}^{4}}\).
The standard basis for the quaternions \({\mathbb{H}\cong C(0,2)}\) can be obtained in terms of Pauli matrices via the association \({\left\{ 1,i,j,k\right\} }\) \({\leftrightarrow\left\{ \sigma_{0},-i\sigma_{1},-i\sigma_{2},-i\sigma_{3}\right\} }\). Thus a quaternion can be expressed as a complex matrix via the isomorphism
\(\displaystyle a+ib+jc+kd\leftrightarrow\begin{pmatrix}a-id & -c-ib\\ c-ib & a+id \end{pmatrix} \),
and composing this with the previous isomorphism for complex numbers as real matrices allows the quaternions to be expressed as a subalgebra of \({\mathbb{R}(4)}\).
The Pauli matrices also form a basis for the vector space of traceless hermitian \({2\times2}\) matrices, which means that \({i\sigma_{i}}\) is a basis for the vector space of traceless anti-hermitian matrices \({so(3)\cong su(2)}\). Thus any element of \({SU(2)}\) can be written \({\textrm{exp}\left(ia^{j}\sigma_{j}\right)}\) for real numbers \({a^{j}}\). A similar construction is the eight Gell-Mann matrices, which form a basis for the vector space of traceless hermitian \({3\times3}\) matrices and so multiplied by \({i}\) form a basis for \({su(3)}\).
Δ Since the Pauli matrices have so many potential roles, it is important to understand what use a particular author is making of them.
An Illustrated Handbook
By Adam Marsh © 2018 World Scientific Publishing