PlanetPhysics/Clifford Algebra

A Non--Commutative Quantum Observable Algebra is a Clifford AlgebraEdit

Let 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 Failed to parse (unknown function "\Cl"): {\displaystyle \Cl(V) = \Cl(V, Q)} , 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 "\Cl"): {\displaystyle \phi : \Cl(V) \lra W} such that the diagram

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

commutes. (It is in this sense that Failed to parse (unknown function "\Cl"): {\displaystyle \Cl(V)} is considered to be `universal').