PlanetPhysics/Clifford Algebra

A Noncommutative Quantum Observable Algebra is a Clifford Algebra

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

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

Failed to parse (unknown function "\xymatrix"): {\displaystyle \xymatrix{&&\hspace*{-1mm}\mbox{Cl}(V)\ar[ddrr]^{\phi}&&\\&&&&\\ V \ar[uurr]^{\mbox{Cl}} \ar[rrrr]_&&&& W}}

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