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