Dirac algebra

Dirac algebra

In mathematical physics, the Dirac algebra is the Clifford algebra Cℓ1,3(C) which is generated by matrix multiplication and real and complex linear combination over the Dirac gamma matrices, introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles.

These matrices have the defining relation

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

where \eta^{\mu \nu} \, is the Minkowski metric with signature (+ − − −), allowing the definition of a scalar product

\displaystyle \langle a , b \rangle = \sum_{\mu\nu} \eta^{\mu\nu} a_{\mu} b^\dagger_\nu

where a = Σ aμγμ and b = Σ bνγν.

Cℓ1,3(C) and Cℓ1,3(R)

The Dirac algebra can be regarded as a complexification of the real algebra Cℓ1,3(R), called the space time algebra:

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

Cℓ1,3(R) differs from Cℓ1,3(C): in Cℓ1,3(R) only real linear combinations of the gamma matrices and their products are allowed.

Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible (and usually enlightening) to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in a real Clifford algebra that square to -1, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.

However, in contemporary practice, the Dirac algebra rather than the space time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.

References

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