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In [[mathematics]], a '''Clifford module''' is a [[representation of an algebra|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.
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The [[abstract algebra|abstract theory]] of Clifford modules was founded by a paper of [[Michael Atiyah|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]].
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==Matrix representations of real Clifford algebras==
We will need to study ''anticommuting'' [[matrix (mathematics)|matrices]] (''AB'' = −''BA'') 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 ''p'' + ''q'' 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> \begin{matrix}
\gamma_{a'} &=& S &\gamma_{a } &S^{-1}
\end{matrix}
</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 equation| 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 [[sign convention| 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]]
 
==References==
*{{citation|first1=Michael|last1=Atiyah|first2=Raoul|last2=Bott|first3=Arnold|last3=Shapiro|title=Clifford Modules|url=http://www.ma.utexas.edu/users/dafr/Index/ABS.pdf|journal=Topology|volume= 3|issue=(Suppl. 1)|year=1964|pages=3–38|doi=10.1016/0040-9383(64)90003-5}}
* {{citation|first=Pierre|last=Deligne|authorlink=Pierre Deligne|chapter=Notes on spinors|title= Quantum Fields and Strings: A Course for Mathematicians|editor= P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, E. Witten|publisher=American Mathematical Society|place= Providence|year=1999|pages=99–135}}. 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]]
 
[[nl:Clifford-algebra]]

Latest revision as of 15:36, 16 December 2014

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