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

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 ${\displaystyle 16=2^{4}}$.

The algebra has a basis

where in each expression, each greek index is increasing as we move to the right. In particular, there is no repeated index in the expressions. By dimension counting, the dimension of the algebra is 16.

The algebra can be generated by taking products of the ${\displaystyle \gamma ^{\mu }}$ alone: the identity arises as

while the others are explicitly products of the ${\displaystyle \gamma ^{\mu }}$.