The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and A. 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 mod 8. This is an algebraic form of Bott periodicity.
For the real Clifford algebra , we need p + q mutually anticommuting matrices, of which p have +1 as square and q have −1 as square.
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.
where S is a non-singular matrix. The sets γ a' and γ a belong to the same equivalence class.
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, 4x4 complex matrices or 8x8 real matrices are needed.