Clifford module explained

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.

Matrix representations of real Clifford algebras

We will need to study anticommuting matrices because in Clifford algebras orthogonal vectors anticommute

A ⋅ B = (1)/(2) ( AB + BA ) = 0 .

For the real Clifford algebra

R _{p,q} , we need mutually anticommuting matrices, of which p have +1 as square and q have −1 as square.\begin{matrix} \gamma ^{2} _{a} & = & +1 & if & 1 \le a \le p \\ \gamma ^{2} _{a} & = & -1 & if &p +1 \le a \le p + q\\ \gamma _{a} \gamma _{b} & = & - \gamma _{b} \gamma _{a} & 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.

See also

• Weyl–Brauer matrices
• Higher-dimensional gamma matrices
• Clifford module bundle

References

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