1.3. MATHEMATICAL PROLOGUE: SOME LIE GROUPS AND LIE ALGEBRAS 11
We shall denote the triple (σ1,σ2,σ3) by σ. For any cyclic permutation (j, k, l) of
(1, 2, 3), we have
(1.13) σjσk = −σkσj = iσl, and hence [σj,σk] = 2iσl.
1
2i
σ1,
1
2i
σ2, and
1
2i
σ3 are a basis for the Lie algebra su(2) of skew-Hermitian 2 × 2
matrices of trace zero, and together with
1
2
σ1,
1
2
σ2, and
1
2
σ3, they are a basis (over
R) for the Lie algebra sl(2, C) of all 2 × 2 complex matrices of trace zero. In view
of (1.8) and (1.13), the linear map κ : sl(2, C) so(1, 3) defined on this basis by
(1.14)
κ (
1
2i
σj) = Xkl for (j, k, l) a cyclic permutation of (1, 2, 3), κ (
1
2
σk) = X0k
is an isomorphism of Lie algebras, and its restriction to su(2) is an isomorphism
from su(2) to so(3).
The corresponding homomorphism on the group level may be described as
follows. Let us add the “fourth Pauli matrix”
σ0 = I =
1 0
0 1
to obtain a basis σ0,...,σ3 for the space H of 2 × 2 Hermitian matrices. We then
identify
R4
with H by the correspondence
(1.15) x
R4
←→ M(x) =
xμσμ
=
x0
+
x3 x1

ix2
x1
+
ix2 x0

x3
.
The crucial feature of this correspondence is that
(1.16) det M(x) =
(x0)2

(x1)2

(x2)2

(x3)2
=
x2.
Every A SL(2, C) acts on H by the map X
AXA†,
and we denote the
corresponding action on
R4
by κ(A):
(1.17) M[κ(A)x] =
AM(x)A†.
Since det A = 1, we have det
AM(x)A†
= det M(x), and by (1.16) this means that
κ maps SL(2, C) into O(1, 3). It is easily verified that the differential of κ at the
identity, which takes S sl(2, C) to the map M(x)
SM(x)+M(x)S†,
is precisely
the map κ defined in (1.14). Thus κ is a local isomorphism, and since SL(2, C) is
connected, its image is the connected component of the identity in O(1, 3), namely,
SO↑(1,
3). Finally, it is easy to verify that the kernel of κ is ±I. In short, we have
proved:
The map κ is a double covering of
SO↑(1,
3) by SL(2, C).
As we observed earlier, SO(3) can be identified with the subgroup of SO↑(1, 3)
that fixes the point (1, 0, 0, 0). The inverse image of SO(3) in SL(2, C) is therefore
the set of all A SL(2, C) that fix the point I = M(1, 0, 0, 0), i.e., that satisfy
AA† = I. This is the group SU(2) of 2 × 2 unitary matrices. It is easily verified
that SU(2) is the set of matrices of the form a −b
b a
where |a|2 + |b|2 = 1, so
that SU(2) is homeomorphic to the unit sphere in
C2
and in particular is simply
connected. Hence:
SU(2) is the universal cover of SO(3), and the covering map is the restriction
of κ to SU(2).
There is another way to look at this. The map x iM(0, x) = ix · σ is
an isomorphism from
R3
to the space of 2 × 2 skew-Hermitian matrices of trace
zero, which is the Lie algebra su(2). Since
A†
=
A−1
for A SU(2), the action
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