Clifford module

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

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

γ a 2 = + 1 if 1 ≤ a ≤ p γ a 2 = − 1 if p + 1 ≤ a ≤ p + q γ a γ b = − γ b γ a if a ≠ b .   {\displaystyle {\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}}} {\displaystyle {\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.

γ a ′ = S γ a S − 1 , {\displaystyle \gamma _{a'}=S\gamma _{a}S^{-1},} {\displaystyle \gamma _{a'}=S\gamma _{a}S^{-1},}

where S is a non-singular matrix. The sets γa′ and γa belong to the same equivalence class.

Real Clifford algebra R3,1

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

References