Quantum Mechanics for Mathematicians

This isomorphism identifies basis vectors by

i

2

↔−i

σ 1

2

↔l 1

etc. The first of these identifications comes from the way we chose to identify
Hwith 2 by 2 complex matrices. The second identification is Φ′, the derivative
at the identity of the covering map Φ.
On each of these isomorphic Lie algebras we have adjoint Lie group (Ad)
and Lie algebra (ad) representations.Adis given by conjugation with the cor-
responding group elements inSp(1),SU(2) andSO(3). adis given by taking
commutators in the respective Lie algebras of pure imaginary quaternions, skew-
Hermitian trace-zero 2 by 2 complex matrices and 3 by 3 real antisymmetric
matrices.
Note that these three Lie algebras are all three dimensional real vector
spaces, so these are real representations. To get a complex representation, take
complex linear combinations of elements. This is less confusing in the case of
su(2) than forsp(1) since taking complex linear combinations of skew-Hermitian
trace-zero 2 by 2 complex matrices gives all trace-zero 2 by 2 matrices (the Lie
algebrasl(2,C)).
In addition, recall that there is a fourth isomorphic version of this repre-
sentation, the representation ofSO(3) on column vectors. This is also a real
representation, but can straightforwardly be complexified. Sinceso(3) andsu(2)
are isomorphic Lie algebras, their complexificationsso(3)Candsl(2,C) will also
be isomorphic.
In terms of 2 by 2 complex matrices, Lie algebra elements can be exponen-
tiated to get group elements inSU(2) and define

Ω(θ,w) =eθ(w^1 X^1 +w^2 X^2 +w^3 X^3 )=e−i

θ 2 w·σ

(6.3)

= 1 cos

θ

2

−i(w·σ) sin

θ

2

(6.4)

Transposing the argument of theorem 6.1 fromHto complex matrices, one finds
that, identifying

v↔v·σ=

(

v 3 v 1 −iv 2

v 1 +iv 2 −v 3

)

one has
Φ(Ω(θ,w)) =R(θ,w)

with Ω(θ,w) acting by conjugation, taking

v·σ→Ω(θ,w)(v·σ)Ω(θ,w)−^1 = (R(θ,w)v)·σ (6.5)

Note that in changing from the quaternionic to complex case, we are treating
the factor of 2 differently, since in the future we will want to use Ω(θ,w) to
perform rotations by an angleθ. In terms of the identificationSU(2) =Sp(1),
we have Ω(θ,w) =u(θ 2 ,w).