Quantum Mechanics for Mathematicians

can instead be used. These satisfy the slightly simpler commutation relations

[Xj,Xk] =

∑^3

l=1

jklXl

or more explicitly

[X 1 ,X 2 ] =X 3 ,[X 2 ,X 3 ] =X 1 ,[X 3 ,X 1 ] =X 2 (3.5)

The non-triviality of the commutators reflects the non-commutativity of the
group. Group elementsU∈SU(2) near the identity satisfy

U' 1 + 1 X 1 + 2 X 2 + 3 X 3

forjsmall and real, just as group elementsz∈U(1) near the identity satisfy

z'1 +i

TheXjand their commutation relations can be thought of as an infinitesimal
version of the full group and its group multiplication law, valid near the identity.
In terms of the geometry of manifolds, recall thatSU(2) is the spaceS^3. The
Xjgive a basis of the tangent spaceR^3 to the identity ofSU(2), just asigives
a basis of the tangent space to the identity ofU(1).

iR

1

i

1

iσ 1

iσ 2

U(1) SU(2)

(one dimension suppressed)

Figure 3.1: Comparing the geometry ofU(1) asS^1 to the geometry ofSU(2)
asS^3.