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

:<math> A \cdot B = \frac{1}{2}( AB + BA ) = 0.</math>

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

:<math> \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>\gamma_{a'} = S \gamma_{a} S^{-1} ,</math>

where S is a non-singular matrix. The sets γ<sub>a′</sub> and γ<sub>a</sub> belong to the same equivalence class.

Real Clifford algebra R<sub>3,1</sub>

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

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