# 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 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 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.