PlanetPhysics/Clifford Algebra
A Noncommutative Quantum Observable Algebra is a Clifford Algebra
editLet us briefly define the notion of a Clifford algebra . Thus, let us consider first a pair , where denotes a real vector space and is a quadratic form on ~. Then, the Clifford algebra associated to , is denoted here as , is the algebra over Failed to parse (unknown function "\bR"): {\displaystyle \bR} generated by , where for all , the relations: are satisfied; in particular, ~.
If is an algebra and Failed to parse (unknown function "\lra"): {\displaystyle c : V \lra W} is a linear map satisfying 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').