Clifford module

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 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 $p-q$ 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

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

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

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

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.

${\begin{matrix}\gamma _{a'}&=&S&\gamma _{a}&S^{-1}\end{matrix}}$

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, 4x4 complex matrices or 8x8 real matrices are needed.

References

• Atiyah, Michael; Bott, Raoul; Shapiro, Michael (1964), "Clifford Modules", Topology, 3 ((Suppl. 1)): 3–38
• Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0123296504.
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989), Spin Geometry, Princeton University Press, ISBN 0-691-08542-0.