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M. Group Theory
M.7 The relation between SU(2) and SO(3)
We have seen (sections M.4.1 and M.4.2) that the algebras of these two groups
are identical. So the groups are isomorphic in the vicinity of their respective
identity elements. Furthermore, matrix representations of one algebra automatically provide representations of the other. Since exponentiating these
infinitesimal matrix transformations produces matrices representing group
elements corresponding to finite transformations in both cases, it might appear that the groups are fully isomorphic. But actually they are not, as we
shall now discuss.
We begin by re-considering the parameters used to characterize elements
of SO(3) and SU(2). A general 3-D rotation is described by the SO(3) matrix
R(n ˆ , θ), where n ˆ is the axis of the rotation and θ is the angle of rotation. For
example,
(
)
cos θ
sin θ 0
(
)
R(ˆ z, θ) =
− sin θ cos θ 0
.
(M.101)
0
0
1
On the other hand, we can write the general SU(2) matrix V in the form
(
)
a
b
V =
,
(M.102)
∗
−b
∗ a
where |a|
2 +|b|
2 = 1 from the unit determinant condition. It therefore depends
on three real parameters, the choice of which we are now going to examine
in more detail than previously. In (12.32) we wrote V as exp(iα · τ /2),
which certainly involves three real parameters α 1 , α 2 , α 3 ; and below (12.35)
we proposed, further, to write α = n ˆ θ, where θ is an angle and n ˆ is a unit
vector. Then, since (as the reader may verify)
exp(iθτ · n ˆ /2) = cos θ/2 + iτ · n ˆ sin θ/2,
(M.103)
it follows that this latter parametrization corresponds to writing, in (M.102),
a = cos θ/2 + in z sin θ/2, b = (n y + in x ) sin θ/2,
(M.104)
2
2
2
with n +
+
= 1. Clearly the condition |a|
2 + |b|
2 = 1 is satisfied, and
one can convince oneself that the full range of a and b is covered if θ/2 lies
n
n
x
y
z
between 0 and π (in particular, it is not necessary to extend the range of θ/2
so as to include the interval π to 2π, since the corresponding region of a, b can
be covered by changing the orientation of n ˆ , which has not been constrained
in any way). It follows that the parameters α satisfy α
2
≤ 4π
2 ; that is, the
space of the α’s is the interior, and surface, of a sphere of radius 2π, as shown
in figure M.1.
What about the parameter space of SO(3)? In this case, the same parameters n ˆ and θ specify a rotation, but now θ (rather than θ/2) runs from 0 to π.
M. Group Theory
M.7 The relation between SU(2) and SO(3)
We have seen (sections M.4.1 and M.4.2) that the algebras of these two groups
are identical. So the groups are isomorphic in the vicinity of their respective
identity elements. Furthermore, matrix representations of one algebra automatically provide representations of the other. Since exponentiating these
infinitesimal matrix transformations produces matrices representing group
elements corresponding to finite transformations in both cases, it might appear that the groups are fully isomorphic. But actually they are not, as we
shall now discuss.
We begin by re-considering the parameters used to characterize elements
of SO(3) and SU(2). A general 3-D rotation is described by the SO(3) matrix
R(n ˆ , θ), where n ˆ is the axis of the rotation and θ is the angle of rotation. For
example,
(
)
cos θ
sin θ 0
(
)
R(ˆ z, θ) =
− sin θ cos θ 0
.
(M.101)
0
0
1
On the other hand, we can write the general SU(2) matrix V in the form
(
)
a
b
V =
,
(M.102)
∗
−b
∗ a
where |a|
2 +|b|
2 = 1 from the unit determinant condition. It therefore depends
on three real parameters, the choice of which we are now going to examine
in more detail than previously. In (12.32) we wrote V as exp(iα · τ /2),
which certainly involves three real parameters α 1 , α 2 , α 3 ; and below (12.35)
we proposed, further, to write α = n ˆ θ, where θ is an angle and n ˆ is a unit
vector. Then, since (as the reader may verify)
exp(iθτ · n ˆ /2) = cos θ/2 + iτ · n ˆ sin θ/2,
(M.103)
it follows that this latter parametrization corresponds to writing, in (M.102),
a = cos θ/2 + in z sin θ/2, b = (n y + in x ) sin θ/2,
(M.104)
2
2
2
with n +
+
= 1. Clearly the condition |a|
2 + |b|
2 = 1 is satisfied, and
one can convince oneself that the full range of a and b is covered if θ/2 lies
n
n
x
y
z
between 0 and π (in particular, it is not necessary to extend the range of θ/2
so as to include the interval π to 2π, since the corresponding region of a, b can
be covered by changing the orientation of n ˆ , which has not been constrained
in any way). It follows that the parameters α satisfy α
2
≤ 4π
2 ; that is, the
space of the α’s is the interior, and surface, of a sphere of radius 2π, as shown
in figure M.1.
What about the parameter space of SO(3)? In this case, the same parameters n ˆ and θ specify a rotation, but now θ (rather than θ/2) runs from 0 to π.