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Concept
Dirac algebra
Résumé
In mathematical physics, the Dirac algebra is the Clifford algebra . 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 matrices with entries in , that is, elements of , satisfying
where by convention, an identity matrix has been suppressed on the right-hand side. The numbers are the components of the Minkowski metric.
For this article we fix the signature to be mostly minus, that is, .
The Dirac algebra is then the linear span of the identity, the gamma matrices as well as any linearly independent products of the gamma matrices. This forms a finite-dimensional algebra over the field or , with dimension .
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 alone: the identity arises as
while the others are explicitly products of the .
These elements span the space generated by . We conclude that we really do have a basis of the Clifford algebra generated by 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 . Note .
There is another way to write this which holds even when :
This form can be used to show that the form a representation of the Lorentz algebra (with real conventions)
It is common convention in physics to include a factor of , so that Hermitian conjugation (where transposing is done with respect to the spacetime greek indices) gives a 'Hermitian matrix' of sigma generators
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 .
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