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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 {{nowrap|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 ({{nowrap|1=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 {{nowrap|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 \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}

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 γa′ and γa 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

References

  • {{citation|first1=Michael|last1=Atiyah|first2=Raoul|last2=Bott|first3=Arnold|last3=Shapiro|title=Clifford Modules|journal=Topology|volume=3|issue=Suppl. 1|year=1964|pages=3–38|doi=10.1016/0040-9383(64)90003-5|doi-access=free}}
  • {{citation|first=Pierre|last=Deligne|authorlink=Pierre Deligne|chapter=Notes on spinors |title= Quantum Fields and Strings: A Course for Mathematicians |editor-first=P. |editor-last=Deligne |editor2-first=P. |editor2-last=Etingof |editor3-first=D.S. |editor3-last=Freed |editor4-first=L.C. |editor4-last=Jeffrey |editor5-first=D. |editor5-last=Kazhdan |editor6-first=J.W. |editor6-last=Morgan |editor7-first=D.R. |editor7-last=Morrison |editor8-first=E. |editor8-last=Witten |publisher=American Mathematical Society|place= Providence|year=1999|pages=99–135 |isbn=978-0-8218-2012-4}}. See also [http://www.math.ias.edu/QFT the programme website] for a preliminary version.
  • {{citation|title=Spinors and Calibrations|last=Harvey|first= F. Reese|publisher=Academic Press|year=1990|isbn=978-0-12-329650-4}}.
  • {{citation|last1=Lawson|first1= H. Blaine|last2=Michelsohn|first2=Marie-Louise|author2-link=Marie-Louise Michelsohn|title=Spin Geometry|publisher= Princeton University Press|year=1989|isbn= 0-691-08542-0}}.

Category:Representation theory

Category:Clifford algebras