Quantum Mechanics for Mathematicians

in the form

(u,u′) =u 1 u′ 1 +u 2 u′ 2 +···uru′r−ur+1u′r+1−···−unu′n

=

(

u 1 ... un

)













1 0 ... 0 0

0 1 ... 0 0

0 0 ... − 1 0

0 0 ... 0 − 1













︸ ︷︷ ︸

r+ signs,s- signs













u′ 1

u′ 2

u′n− 1

u′n













For a proof by Gram-Schmidt orthogonalization, see [8].

We can thus extend our definition of the orthogonal group as the group of
transformationsgpreserving an inner product

(gu,gu′) = (u,u′)

to the caser,sarbitrary by:

Definition(Orthogonal groupO(r,s,R)).The groupO(r,s,R)is the group of
realr+sbyr+smatricesgthat satisfy

gT

      

1 0 ... 0 0

0 1 ... 0 0

0 0 ... − 1 0

0 0 ... 0 − 1

      

︸ ︷︷ ︸

r+ signs,s- signs

g=

      

1 0 ... 0 0

0 1 ... 0 0

0 0 ... − 1 0

0 0 ... 0 − 1

      

︸ ︷︷ ︸

r+ signs,s- signs

SO(r,s,R)⊂O(r,s,R)is the subgroup of matrices of determinant+1.

If one complexifies, taking components of vectors to be inCn, using the
same formula for (·,·), one can change basis by multiplying thesbasis elements
by a factor ofi, and in this new basis all basis vectorsejsatisfy (ej,ej) = 1.
One thus sees that onCn, as in the symplectic case, up to change of basis there
is only one non-degenerate symmetric bilinear form. The group preserving this
is calledO(n,C). Note that onCn(·,·) is not the Hermitian inner product
(which is antilinear on the first variable), and it is not positive definite.

29.2 Clifford algebras and geometry

As defined by generators in the last chapter, Clifford algebras have no obvious
geometrical significance. It turns out however that they are powerful tools in
the study of the geometry of linear spaces with an inner product, including
especially the study of linear transformations that preserve the inner product,