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, v2 = −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
commutes. (It is in this sense that Cl(V ) is considered to be ‘universal’).
