Quantum Mechanics for Mathematicians

so
det(L) =± 1

O(n) is a continuous Lie group, with two components distinguished by the sign
of the determinant: SO(n), the subgroup of orientation-preserving transfor-
mations, which include the identity, and a component of orientation-changing
transformations.
The simplest non-trivial example is forn= 2, where all elements ofSO(2)
are given by matrices of the form
(
cosθ −sinθ
sinθ cosθ

)

These matrices give counter-clockwise rotations inR^2 by an angleθ. The other
component ofO(2) will be given by matrices of the form

(

cosθ sinθ

sinθ −cosθ

)

which describe a reflection followed by a rotation. Note that the groupSO(2)
is isomorphic to the groupU(1) by

(

cosθ −sinθ

sinθ cosθ

)

↔eiθ

so the representation theory ofSO(2) is just as forU(1), with irreducible com-
plex representations one dimensional and classified by an integer.
In chapter 6 we will consider in detail the case ofSO(3), which is crucial for
physical applications because it is the group of rotations in the physical three
dimensional space.

4.6.2 Unitary groups

In the complex case, groups of invertible transformations preserving the Hermi-
tian inner product are called unitary groups:

Definition(Unitary group).The unitary groupU(n)inndimensions is the
group of invertible transformations preserving a Hermitian inner product on a
complexndimensional vector spaceV. This is isomorphic to the group ofnby
ncomplex invertible matrices satisfying

L−^1 =L

T

=L†

The subgroup ofU(n)of matrices with determinant 1 is calledSU(n).

In the unitary case, the dual of a representationπhas representation matrices
that are transpose-inverses of those forπ, but

(π(g)T)−^1 =π(g)