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.
that satisfy where by convention, an identity matrix has been suppressed on the right-hand side.
The Dirac algebra is then the linear span of the identity, the gamma matrices
The algebra has a basis where in each expression, each greek index is increasing as we move to the right.
A general element of the algebra can be written with implicit summation.
, up to a scalar factor, using identities involving traces of gamma matrices (see here, and the anti-symmetry of the
For the theory in this section, there are many choices of conventions found in the literature, often corresponding to factors of
For clarity, here we will choose conventions to minimise the number of numerical factors needed, but may lead to generators being anti-Hermitian rather than Hermitian.
There is another common way to write the quadratic subspace of the Clifford algebra: with
, so that Hermitian conjugation (where transposing is done with respect to the spacetime greek indices) gives a 'Hermitian matrix' of sigma generators [1] only 6 of which are non-zero due to antisymmetry of the bracket, span the six-dimensional representation space of the tensor (1, 0) ⊕ (0, 1)-representation of the Lorentz algebra inside
Moreover, they have the commutation relations of the Lie algebra,[2] and hence constitute a representation of the Lorentz algebra (in addition to spanning a representation space) sitting inside
satisfy the Lorentz algebra, and turn out to exponentiate to a representation of the spin group
complex matrix are labelled by convention using greek letters from the start of the alphabet
Then a finite Lorentz transformation, parametrized by the components
as defined above satisfies This motivates the definition of Dirac adjoint for spinors
When considering the complex span, this basis element can alternatively be taken to be More details can be found here.
By total antisymmetry of the quartic element, it can be considered to be a volume form.
) only real linear combinations of the gamma matrices and their products are allowed.
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 p,q; the anti-commutation of the Weyl spinors emerges naturally from the Clifford algebra.
[4] The Weyl spinors transform under the action of the spin group
The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the
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.
[5]: 84 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.