0.1 A Non–Commutative Quantum Observable Algebra is a Clifford Algebra

Definition 0.1. 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 Cl(V) = Cl(V, Q), is the algebra over ℝ generated by V, where for all v, w ∈ V, the relations: v · w + w · v = −2Q(v, w) , are satisfied; in particular, v^{2} = −2Q(v, v) .

If W is an algebra and c : V ⟶W is a linear map satisfying c(w)c(v)+c(v)c(w) = −2Q(v, w) , then there exists a unique algebra homomorphism ϕ : Cl(V)⟶W such that the diagram

\xymatrix{ & Cl(V) \ar[dr]^{ϕ} \\ V \ar[ur]^{Cl} \ar[rr]_{c} &  & W}

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