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 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.

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.

\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.

References

Atiyah, Michael; Bott, Raoul; Shapiro, Arnold (1964), "Clifford Modules", Topology, 3 (Suppl. 1): 3–38, doi:10.1016/0040-9383(64)90003-5
Deligne, Pierre (1999), "Notes on spinors", in Deligne, P.; Etingof, P.; Freed, D.S.; Jeffrey, L.C.; Kazhdan, D.; Morgan, J.W.; Morrison, D.R.; Witten, E. (eds.), Quantum Fields and Strings: A Course for Mathematicians, Providence: American Mathematical Society, pp. 99–135, ISBN 978-0-8218-2012-4. See also the programme website for a preliminary version.
Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0-12-329650-4.
Lawson, H. Blaine; Michelsohn, Marie-Louise (1989), Spin Geometry, Princeton University Press, ISBN 0-691-08542-0.

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