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

[math]\displaystyle{ A \cdot B = \frac{1}{2}( AB + BA ) = 0. }[/math]

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

[math]\displaystyle{ \begin{matrix} \gamma_a^2 &=& +1 &\mbox{if} &1 \le a \le p \\ \gamma_a^2 &=& -1 &\mbox{if} &p+1 \le a \le p+q\\ \gamma_a \gamma_b &=& -\gamma_b \gamma_a &\mbox{if} &a \ne b. \ \\ \end{matrix} }[/math]

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.

[math]\displaystyle{ \gamma_{a'} = S \gamma_{a} S^{-1} , }[/math]

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

• Atiyah, Michael; Bott, Raoul; Shapiro, Arnold (1964), "Clifford Modules", Topology 3 (Suppl. 1): 3–38, doi:10.1016/0040-9383(64)90003-5, http://www.ma.utexas.edu/users/dafr/Index/ABS.pdf, retrieved 2011-07-28 
• Deligne, Pierre (1999), "Notes on spinors", in Deligne, P.; Etingof, P.; Freed, D.S. et al., 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 .