Quantum Mechanics for Mathematicians

For the adjoint representation on antisymmetric matrices

Ad(g)





0 −v 3 v 2

v 3 0 −v 1

−v 2 v 1 0



=g





0 −v 3 v 2

v 3 0 −v 1

−v 2 v 1 0



g−^1

The corresponding Lie algebra representation is given by

ad(X)





0 −v 3 v 2

v 3 0 −v 1

−v 2 v 1 0



= [X,





0 −v 3 v 2

v 3 0 −v 1

−v 2 v 1 0



]

whereXis a 3 by 3 antisymmetric matrix.
One can explicitly check that these representations are isomorphic, for in-
stance by calculating how basis elementslj∈so(3) act. On vectors, theseljact
by matrix multiplication, giving for instance, forj= 1

l 1 e 1 = 0, l 1 e 2 =e 3 , l 1 e 3 =−e 2

On antisymmetric matrices one has instead the isomorphic relations

(ad(l 1 ))(l 1 ) = 0,(ad(l 1 ))(l 2 ) =l 3 ,(ad(l 1 ))(l 3 ) =−l 2

6.2 Spin groups in three and four dimensions

A subtle and remarkable property of the orthogonal groupsSO(n) is that they
come with an associated group, calledSpin(n), with every element ofSO(n)
corresponding to two distinct elements ofSpin(n). There is a surjective group
homomorphism
Φ :Spin(n)→SO(n)

with the inverse image of each element ofSO(n) given by two distinct elements
ofSpin(n).

Digression.The topological reason for this is that, (forn > 2 ) the fundamental
group ofSO(n)is non-trivial, withπ 1 (SO(n)) =Z 2 (in particular there is a
non-contractible loop inSO(n), contractible if you go around it twice).Spin(n)
is topologically the simply-connected double cover ofSO(n), and the covering
mapΦ :Spin(n)→SO(n)can be chosen to be a group homomorphism.

Spin(n) is a Lie group of the same dimension asSO(n), with an isomorphic tan-
gent space at the identity, so the Lie algebras of the two groups are isomorphic:
so(n)'spin(n).
In chapter 29 we will explicitly construct the groupsSpin(n) for anyn, but
here we will only do this forn= 3 andn= 4, using methods specific to these two
cases. In the casesn= 5 (whereSpin(5) =Sp(2), the 2 by 2 norm-preserving
quaternionic matrices) andn= 6 (whereSpin(6) =SU(4)) special methods
can be used to identifySpin(n) with other matrix groups. Forn >6 the group
Spin(n) will be a matrix group, but distinct from other classes of such groups.