PlanetPhysics/Clifford Algebra

A Noncommutative Quantum Observable Algebra is a Clifford Algebra

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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  , 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').