Gamma matrices

In mathematical physics, the gamma matrices, ${\displaystyle \ \left\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\right\}\ ,}$ also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra ${\displaystyle \ \mathrm {Cl} _{1,3}(\mathbb {R} )~.}$ It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin ${\displaystyle {\tfrac {\ 1\ }{2}}}$ particles. Gamma matrices were introduced by Paul Dirac in 1928.^[1][2]

In Dirac representation, the four contravariant gamma matrices are

${\displaystyle {\begin{aligned}\gamma ^{0}&={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},&\gamma ^{1}&={\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{pmatrix}},\\\\\gamma ^{2}&={\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{pmatrix}},&\gamma ^{3}&={\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}}~.\end{aligned}}}$

${\displaystyle \gamma ^{0}}$ is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices. More compactly, ${\displaystyle \ \gamma ^{0}=\sigma ^{3}\otimes I_{2}\ ,}$ and ${\displaystyle \ \gamma ^{j}=i\sigma ^{2}\otimes \sigma ^{j}\ ,}$ where ${\displaystyle \ \otimes \ }$ denotes the Kronecker product and the ${\displaystyle \ \sigma ^{j}\ }$ (for j = 1, 2, 3) denote the Pauli matrices.

In addition, for discussions of group theory the identity matrix (I) is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrices

${\displaystyle {\begin{aligned}\ I_{4}={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}\ ,\qquad \gamma ^{5}\equiv i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}={\begin{pmatrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{pmatrix}}~.\end{aligned}}}$

The "fifth matrix" ${\displaystyle \ \gamma ^{5}\ }$ is not a proper member of the main set of four; it is used for separating nominal left and right chiral representations.

The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacetime dimensions, the four gammas, above, together with the fifth gamma-matrix to be presented below generate the Clifford algebra.