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

Zdroj:https://en.wikipedia.org?pojem=Dirac_matrices
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---