# 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').