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 $$\bR$$ generated by $$V$$ , where for all $$v, w \in V$$, the relations: $$ v \cdot w + w \cdot v = -2 Q(v,w)~,$$ are satisfied; in particular, $$v^2 = -2Q(v,v)$$~.

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

$$\xymatrix{&&\hspace*{-1mm}\mbox{Cl}(V)\ar[ddrr]^{\phi}&&\\&&&&\\ V \ar[uurr]^{\mbox{Cl}} \ar[rrrr]_&&&& W}$$ commutes. (It is in this sense that Cl(V) is considered to be `universal').