Quantum Mechanics for Mathematicians

wherejkis annbynmatrix with only two non-zero entries:jkentry−1 and
kjentry +1 (see equation 5.2.1). Restricting attention to thejkplane,eθjk
acts as the standard rotation matrix in the plane
(
vj
vk

)

→

(

cosθ −sinθ

sinθ cosθ

)(

vj

vk

)

In theSO(3) case we saw that there were three of these matrices

 23 =l 1 ,  13 =−l 2 ,  12 =l 3

providing a basis of the Lie algebraso(3). Inndimensions there will be^12 (n^2 −n)
of them, providing a basis of the Lie algebraso(n).
Just as in the case ofSO(3) where unit length quaternions were used, in
dimensionnwe can use elements of the Clifford algebra to get these same
rotation transformations, but as conjugations in the Clifford algebra. To see
how this works, consider the quadratic Clifford algebra elementγjγkforj 6 =k
and notice that
(γjγk)^2 =γjγkγjγk=−γjγjγkγk=− 1

so one has

e

θ

2 γjγk=

(

1 −

(θ/2)^2

2!

+···

)

+γjγk

(

θ/ 2 −

(θ/2)^3

3!

+···

)

= cos

(

θ

2

)

+γjγksin

(

θ

2

)

Conjugating a vectorvjγj+vkγkin thejkplane by this, one can show that

e−

θ 2 γjγk

(vjγj+vkγk)e

θ 2 γjγk

= (vjcosθ−vksinθ)γj+ (vjsinθ+vkcosθ)γk

which is a rotation byθin thejkplane. Such a conjugation will also leave
invariant theγlforl 6 =j,k. Thus one has

e−

θ 2 γjγk

γ(v)e

θ 2 γjγk

=γ(eθjkv) (29.4)

and, taking the derivative atθ= 0, the infinitesimal version
[
−

1

2

γjγk,γ(v)

]

=γ(jkv) (29.5)

Note that these relations are closely analogous to what happens in the symplectic
case, where the symplectic groupSp(2d,R) acts on linear combinations of the
Qj,Pjby conjugation by the exponential of an operator quadratic in theQj,Pj.
We will examine this analogy in greater detail in chapter 31.
One can also see that, just as in our earlier calculations in three dimensions,
one gets a double cover of the group of rotations, with here the elementse
θ 2 γjγk

of the Clifford algebra giving a double cover of the group of rotations in the
jkplane (asθgoes from 0 to 2π). General elements of the spin group can