Quantum Mechanics for Mathematicians

we have (for non-zero vectorsw)

w/−^1 =

w/

(w,w)

and the reflection transformation is just conjugation byw/times a minus sign

v/→/v′=v/−v/−w/v/w/−^1 =−w/v/w/−^1

Identifying vectors with Clifford algebra elements, the orthogonal transfor-
mation that is the result of one reflection is given by a conjugation (with a minus
sign). These reflections lie in the groupO(n), but not in the subgroupSO(n),
since they change orientation. The result of two reflections in hyperplanes or-
thogonal tow 1 ,w 2 will be a conjugation byw/ 2 w/ 1

v/→v/′=−w/ 2 (−w/ 1 v/w/− 11 )w/− 21 = (w/ 2 w/ 1 )/v(w/ 2 w/ 1 )−^1

This will be a rotation preserving the orientation, so of determinant one and in
the groupSO(n).
This construction not only gives an efficient way of representing rotations
(as conjugations in the Clifford algebra), but it also provides a construction of
the groupSpin(n) in arbitrary dimensionn. One can define:

Definition(Spin(n)).The groupSpin(n,R)is the group of invertible elements
of the Clifford algebra Cliff(n)of the form

w/ 1 w/ 2 ···w/k

where the vectorswjforj = 1,···,k(k≤n) are vectors inRnsatisfying
|wj|^2 = 1andkis even. Group multiplication is Clifford algebra multiplication.

The action ofSpin(n) on vectorsv∈Rnwill be given by conjugation

v/→(w/ 1 w/ 2 ···w/k)v/(w/ 1 w/ 2 ···w/k)−^1 (29.3)

and this will correspond to a rotation of the vectorv. This construction gen-
eralizes to arbitrarynthe one we gave in chapter 6 ofSpin(3) in terms of unit
length elements of the quaternion algebraH. One can see here the characteristic
fact that there are two elements of theSpin(n) group giving the same rotation
inSO(n) by noticing that changing the sign of the Clifford algebra element
w/ 1 w/ 2 ···w/kdoes not change the conjugation action, where signs cancel.

29.2.2 The Lie algebra of the rotation group and quadratic

elements of the Clifford algebra

For a second approach to understanding rotations in arbitrary dimension, one
can use the fact that these are generated by taking products of rotations in the
coordinate planes. A rotation by an angleθin thejkcoordinate plane (j < k)
will be given by
v→eθjkv