Quantum Mechanics for Mathematicians

corresponding Clifford algebras Cliff(3, 1 ,R) or Cliff(1, 3 ,R). The generators
γjof such a Clifford algebra are well known in the subject as the “Diracγ-
matrices”.
For now though, we will restrict attention to the positive definite case, so
just will be considering Cliff(n,R) and seeing how it is used to study the group
O(n) ofndimensional rotations inRn.

29.2.1 Rotations as iterated orthogonal reflections

We’ll consider two different ways of seeing the relationship between the Clifford
algebra Cliff(n,R) and the groupO(n) of rotations inRn. The first is based
upon the geometrical fact (known as the Cartan-Dieudonn ́e theorem) that one
can get any rotation by doing at mostnorthogonal reflections in different hy-
perplanes. Orthogonal reflection in the hyperplane perpendicular to a vectorw
takes a vectorvto the vector

v′=v− 2

(v,w)

(w,w)

w

something that can easily be seen from the following picture

v

w

v′

(v,w)

(w,w)

− 2

(v,w)

(w,w)

w

Figure 29.1: Orthogonal reflection in the hyperplane perpendicular tow.

From now on we identify vectorsv,v′,wwith the corresponding Clifford
algebra elements by the mapγof equation 29.1. The linear transformation
given by reflection inwis

/v→v/′=/v−^2

(v,w)

(w,w)

w/

=/v−(v/w/+w/v/)

w/

(w,w)

Since

w/

w/

(w,w)

=

(w,w)

(w,w)

= 1