# PlanetPhysics/C Clifford Algebra

### Preliminary data for the definition of a C*-Clifford algebraEdit

Given a general Hilbert space , one can define an associated * -Clifford algebra* ,
**Failed to parse (unknown function "\Cl"): {\displaystyle \Cl[\mathcal{H}]}**
, which admits a canonical representation on **Failed to parse (unknown function "\bF"): {\displaystyle \mathcal L(\bF (\mathcal{H}))}**
the bounded linear operators on the Fock space **Failed to parse (unknown function "\bF"): {\displaystyle \bF (\mathcal{H})}**
of , (as in Plymen and Robinson, 1994), and hence one has a natural sequence of maps **Failed to parse (unknown function "\lra"): {\displaystyle \mathcal{H} \lra \Cl[\mathcal{H}] \lra \mathcal L(\bF (\mathcal{H}))~. }**

The details and notation related to the definition of a * -Clifford algebra* , are presented in the following
brief paragraph and diagram.

### A non--commutative quantum observable algebra (QOA) is a Clifford algebra.Edit

Let us briefly recall the notion of a *Clifford algebra* with the above notations and auxiliary concepts.
Consider first a pair , where denotes a real vector space and
is a quadratic form on ~. Then, the *Clifford algebra associated to * , 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').

Then, with the above notation, one has the precise definition of the * -Clifford algebra*
as **Failed to parse (unknown function "\Cl"): {\displaystyle \Cl[\mathcal{H}]}**
when
where is a real vector space, as specified above.

Also note that the Clifford algebra is sometimes denoted as .