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Concept
Clifford module
Summary
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.
We will need to study anticommuting matrices (AB = −BA) because in Clifford algebras orthogonal vectors anticommute
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 γa′ and γa 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 (+++−). For the signatures (+−−−) and (−−−+) often used in physics, 4×4 complex matrices or 8×8 real matrices are needed.
Official source
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