In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra C is a central simple algebra over some field extension L of the field K over which the quadratic form Q defining C is defined.

The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and Arnold S. Shapiro. A fundamental result on Clifford modules is that the Morita equivalence class of a Clifford algebra (the equivalence class of the category of Clifford modules over it) depends only on the signature p − q (mod 8). This is an algebraic form of Bott periodicity.

Matrix representations of real Clifford algebras

We will need to study anticommuting matrices (AB = −BA) because in Clifford algebras orthogonal vectors anticommute

${\displaystyle A\cdot B={\frac {1}{2))(AB+BA)=0.}$

For the real Clifford algebra ${\displaystyle \mathbb {R} _{p,q))$, we need p + q mutually anticommuting matrices, of which p have +1 as square and q have −1 as square.

${\displaystyle {\begin{matrix}\gamma _{a}^{2}&=&+1&{\mbox{if))&1\leq a\leq p\\\gamma _{a}^{2}&=&-1&{\mbox{if))&p+1\leq a\leq p+q\\\gamma _{a}\gamma _{b}&=&-\gamma _{b}\gamma _{a}&{\mbox{if))&a\neq b.\ \\\end{matrix))}$

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.

${\displaystyle \gamma _{a'}=S\gamma _{a}S^{-1},}$

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.