Quantum Mechanics for Mathematicians

i.e., rotations. To see the relation between Clifford algebras and geometry,
consider first the positive definite case Cliff(n,R). To an arbitrary vector

v= (v 1 ,v 2 ,...,vn)∈Rn

we associate the Clifford algebra elementv/=γ(v) whereγis the map

v∈Rn→γ(v) =v 1 γ 1 +v 2 γ 2 +···+vnγn∈Cliff(n,R) (29.1)

Using the Clifford algebra relations for theγj, given two vectorsv, w the
product of their associated Clifford algebra elements satisfies

v/w/+w/v/= [v 1 γ 1 +v 2 γ 2 +···+vnγn, w 1 γ 1 +w 2 γ 2 +···+wnγn]+

= 2(v 1 w 1 +v 2 w 2 +···+vnwn)

= 2(v,w) (29.2)

where (·,·) is the symmetric bilinear form onRncorresponding to the standard
inner product of vectors. Note that takingv=wone has

/v^2 = (v,v) =||v||^2

The Clifford algebra Cliff(n,R) thus containsRnas the subspace of linear
combinations of the generatorsγj. It can be thought of as a sort of enhancement
of the vector spaceRnthat encodes information about the inner product, and
it will sometimes be written Cliff(Rn,(·,·)). In this larger structure vectors can
be multiplied as well as added, with the multiplication determined by the inner
product and given by equation 29.2. Note that different people use different
conventions, with
v/w/+w//v=−2(v,w)

another common choice. One also sees variants without the factor of 2.
Forndimensional vector spaces overC, we have seen that for any non-
degenerate symmetric bilinear form a basis can be found such that (·,·) has the
standard form
(z,w) =z 1 w 1 +z 2 w 2 +···+znwn

As a result, up to isomorphism, there is just one complex Clifford algebra in
dimensionn, the one we defined as Cliff(n,C). Forndimensional vector spaces
overRwith a non-degenerate symmetric bilinear form of typer,ssuch that
r+s=n, the corresponding Clifford algebras Cliff(r,s,R) are the ones defined
in terms of generators in section 28.2.
In special relativity, space-time is a real four dimensional vector space with
an indefinite inner product corresponding to (depending on one’s choice of con-
vention) either the caser= 1,s= 3 or the caser= 3,s= 1. The group of
linear transformations preserving this inner product is called the Lorentz group,
and its orientation preserving component is written asSO(3,1) orSO(1,3) de-
pending on the choice of convention. In later chapters we will consider what
happens to quantum mechanics in the relativistic case, and there encounter the