Pauli and Dirac matrices

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 form what 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, but a chiral basis requires \({\gamma_{5}}\) to take the form below (or its negative). 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 matrix representation \({C\mathbb{^{C}}(4)\cong\mathbb{C}(4)}\) acting on vectors (spinors) in \({\mathbb{C}^{4}}\). The covariant Dirac matrices are sometimes defined for a general metric as \({\gamma_{i}\equiv g_{ij}\gamma^{j}}\), in which case they are not necessarily orthonormal and according to our definition are not Dirac matrices.

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 \({su(2)\cong so(3)}\). Thus any element of the compact connected Lie groups \({SU(2)}\) and \({SO(3)}\) 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.

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