Dirac algebra

In mathematical physics, the Dirac algebra is the Clifford algebra Cℓ4(C), which may be thought of as Cℓ1,3(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
where formula_2 are the components of the Minkowski metric with signature (+ − − −) and formula_3 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
where
Higher powers.
The sigmas
only of which are non-zero due to antisymmetry of the bracket, span the six-dimensional representation space of the tensor -representation of the Lorentz algebra inside formula_7. Moreover, they have the commutation relations of the Lie algebra,
and hence constitute a representation of the Lorentz algebra (in addition to spanning a representation space) sitting inside formula_8 the spin representation.
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:
and the Klein–Gordon equation:
to be given, and requires that these equations lead to consistent results.
Derivation from consistency requirement (proof)
Multiplying the Dirac equation by its conjugate equation yields:
The demand of consistency with the Klein–Gordon equation leads immediately to:
where formula_13 is the anticommutator, formula_2 is the Minkowski metric with signature (+ − − −) and formula_15 is the 4x4 unit matrix.
Cℓ1,3() and Cℓ1,3().
The Dirac algebra can be regarded as a complexification of the real spacetime algebra Cℓ1,3():
Cℓ1,3() differs from Cℓ1,3(): in Cℓ1,3() 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.
In the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Cℓp,q() for arbitrary dimensions ; the anti-commutation of the Weyl spinors emerges naturally from the Clifford algebra. The Weyl spinors transform under the action of the spin group formula_17. The complexification of the spin group, called the spinc group formula_18, is a product formula_19 of the spin group with the circle formula_20 with the product formula_21 just a notational device to identify formula_22 with formula_23 The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the formula_24 component, which can be identified with the formula_24 fiber of the electromagnetic interaction. The formula_21 is entangling parity and charge conjugation in a manner suitable for relating the Dirac particle/anti-particle states (equivalently, the chiral states in the Weyl basis). The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field. This is in contrast to the Majorana spinor and the ELKO spinor, which cannot ("i.e." they are electrically neutral), as they explicitly constrain the spinor so as to not interact with the formula_27 part coming from the complexification.
Insofar as the presentation of charge and parity can be a confusing topic in conventional quantum field theory textbooks, the more careful dissection of these topics in a general geometric setting can be elucidating. Standard expositions of the Clifford algebra construct the Weyl spinors from first principles; that they "automatically" anti-commute is an elegant geometric by-product of the construction, completely by-passing any arguments that appeal to the Pauli exclusion principle (or the sometimes common sensation that Grassmann variables have been introduced via "ad hoc" argumentation.)
In contemporary physics practice, the Dirac algebra continues to be the standard environment the spinors of the Dirac equation "live" in, rather than the spacetime algebra.