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 . This is an algebraic form of Bott periodicity.
We will need to study anticommuting matrices () because in Clifford algebras orthogonal vectors anticommute
For the real Clifford algebra , we need mutually anticommuting matrices, of which p have +1 as square and q have âÂÂ1 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 ó<sub>aâ²</sub> and ó<sub>a</sub> belong to the same equivalence class.
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 (+++âÂÂ). For the signatures (+âÂÂâÂÂâÂÂ) and (âÂÂâÂÂâÂÂ+) often used in physics, 4ÃÂ4 complex matrices or 8ÃÂ8 real matrices are needed.