# PlanetPhysics/Clifford Algebra

### A Non--Commutative Quantum Observable Algebra is a Clifford Algebra

Let us briefly define the notion of a Clifford algebra . Thus, let us consider first a pair ${\displaystyle (V,Q)}$ , where ${\displaystyle V}$  denotes a real vector space and ${\displaystyle Q}$  is a quadratic form on ${\displaystyle V}$ ~. Then, the Clifford algebra associated to ${\displaystyle V}$  , is denoted here as $\displaystyle \Cl(V) = \Cl(V, Q)$ , is the algebra over $\displaystyle \bR$ generated by ${\displaystyle V}$  , where for all ${\displaystyle v,w\in V}$ , the relations: ${\displaystyle v\cdot w+w\cdot v=-2Q(v,w)~,}$  are satisfied; in particular, ${\displaystyle v^{2}=-2Q(v,v)}$ ~.

If ${\displaystyle W}$  is an algebra and $\displaystyle c : V \lra W$ is a linear map satisfying ${\displaystyle c(w)c(v)+c(v)c(w)=-2Q(v,w)~,}$  then there exists a unique algebra homomorphism $\displaystyle \phi : \Cl(V) \lra W$ such that the diagram

$\displaystyle \xymatrix{&&\hspace*{-1mm}\Cl(V)\ar[ddrr]^{\phi}&&\\&&&&\\ V \ar[uurr]^{\Cl} \ar[rrrr]_&&&& W}$

commutes. (It is in this sense that $\displaystyle \Cl(V)$ is considered to be `universal').