Dirac algebra

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In mathematical physics, the Dirac algebra is the Clifford algebra ${\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )}$. This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-⁠1/2⁠ particles with a matrix representation of the gamma matrices, which represent the generators of the algebra.

The gamma matrices are a set of four ${\displaystyle 4\times 4}$ matrices ${\displaystyle \{\gamma ^{\mu }\}=\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\}}$ with entries in ${\displaystyle \mathbb {C} }$, that is, elements of ${\displaystyle {\text{Mat}}_{4\times 4}(\mathbb {C} )}$ that satisfy

${\displaystyle \displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu },}$

where by convention, an identity matrix has been suppressed on the right-hand side. The numbers ${\displaystyle \eta ^{\mu \nu }\,}$ are the components of the Minkowski metric. For this article we fix the signature to be mostly minus, that is, ${\displaystyle (+,-,-,-)}$.

The Dirac algebra is then the linear span of the identity, the gamma matrices ${\displaystyle \gamma ^{\mu }}$ as well as any linearly independent products of the gamma matrices. This forms a finite-dimensional algebra over the field ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$, with dimension ${\displaystyle 16=2^{4}}$.