eiθC Cn2iR1
π′(iR)U(1) U(n)ππ(1)π(eiθ)π(eiθ) =
eiq^1 θ 0
..
.
0 eiqnθ
π′(iθ) =
iq 1 θ 0
..
.
0 iqnθ
Figure 2.2: Visualizing a representationπ:U(1)→U(n), along with its differ-
ential.
The spherical figure in the right-hand side of the picture is supposed to
indicate the spaceU(n)⊂GL(n,C) (GL(n,C) is thenbyncomplex matrices,
Cn
2
, minus the locus of matrices with zero determinant, which are those that
can’t be inverted). It has a distinguished point, the identity. The representation
πtakes the circleU(1) to a circle insideU(n). Its derivativeπ′is a linear map
taking the tangent spaceiRto the circle at the identity to a line in the tangent
space toU(n) at the identity.
In the very simple exampleG=U(1), this abstract picture is over-kill and
likely confusing. We will see the same picture though occurring in many other
much more complicated examples in later chapters. Just like in thisU(1) case,
for finite dimensional representations the linear mapsπ′will be matrices, and
the representation matricesπcan be found by exponentiating theπ′.
2.4 Conservation of charge andU(1) symmetry
The way we have defined observable operators in terms of a group representation
onH, the action of these operators has nothing to do with the dynamics. If
we start at timet= 0 in a state inHqj, with definite numerical valueqj for
the observable, there is no reason that time evolution should preserve this.