# Dirac algebra

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In mathematical physics, the Dirac algebra is the Clifford algebra C1,3(C). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation with the Dirac gamma matrices, which represent the generators of the algebra.

The gamma elements have the defining relation

$\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu }{\mathbf {1} }$ where $\eta ^{\mu \nu }\,$ are the components of the Minkowski metric with signature (+ − − −) and ${\mathbf {1} }$ is the identity element of the algebra (the identity matrix in the case of a matrix representation). This allows the definition of a scalar product

$\displaystyle \langle a,b\rangle =\sum _{\mu \nu }\eta ^{\mu \nu }a_{\mu }b_{\nu }^{\dagger }$ where

$\,a=\sum _{\mu }a_{\mu }\gamma ^{\mu }$ and $\,b=\sum _{\nu }b_{\nu }\gamma ^{\nu }$ .

## Derivation starting from the Dirac and Klein–Gordon equation

The defining form of the gamma elements can be derived if one assumes the covariant form of the Dirac equation:

$-i\hbar \gamma ^{\mu }\partial _{\mu }\psi +mc\psi =0\,.$ and the Klein–Gordon equation:

$-\partial _{t}^{2}\psi +\nabla ^{2}\psi =m^{2}\psi$ to be given, and requires that these equations lead to consistent results.

$Cl_{1,3}({\mathbb {C} })=Cl_{1,3}({\mathbb {R} })\otimes {\mathbb {C} }.$ 