Editing Clifford module

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.

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]].

==Matrix representations of real Clifford algebras==
We will need to study ''anticommuting'' [[matrix (mathematics)|matrices]] ({{nowrap|1=''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 {{nowrap|''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>\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 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]]
* [[Clifford module bundle]]

==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]]