Clifford module

Matrix representations of real Clifford algebras

We will need to study anticommuting matrices () because in Clifford algebras orthogonal vectors anticommute : A \cdot B = \frac{1}{2}( AB + BA ) = 0.

For the real Clifford algebra \mathbb{R}_{p,q}, we need p + q mutually anticommuting matrices, of which p have +1 as square and q have −1 as square. : \begin{matrix} \gamma_a^2 &=& +1 &\mbox{if} &1 \le a \le p \ \gamma_a^2 &=& -1 &\mbox{if} &p+1 \le a \le p+q\ \gamma_a \gamma_b &=& -\gamma_b \gamma_a &\mbox{if} &a \ne b. \ \ \end{matrix}

Such a basis of gamma matrices is not unique. One can always obtain another set of gamma matrices satisfying the same Clifford algebra by means of a similarity transformation.

:\gamma_{a'} = S \gamma_{a} S^{-1} ,

where S is a non-singular matrix. The sets γ**a′ and γ**a belong to the same equivalence class.

Real Clifford algebra R_3,1

Developed by Ettore Majorana, this Clifford module enables the construction of a Dirac-like equation without complex numbers, and its elements are called Majorana spinors.

The four basis vectors are the three Pauli matrices and a fourth antihermitian matrix. The signature is (+++−). For the signatures (+−−−) and (−−−+) often used in physics, 4×4 complex matrices or 8×8 real matrices are needed.

References

• . See also the programme website for a preliminary version.
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