12 1. PROLOGUE

A → κ(A)|{0}×R3 of SU(2) on

R3

is essentially the adjoint action of SU(2) on its

Lie algebra.

The map κ has one more important property that is not quite obvious: it

respects adjoints, i.e.,

κ(A†)

=

κ(A)†.

This is most easily seen on the Lie algebra level: κ takes the Hermitian matrices

1

2

σj

to the symmetric matrices Xj0 and the skew-Hermitian matrices 1

2i

σj to the skew-

symmetric matrices Xkl. Thus κ (X†) = κ (X)†, so since κ(exp X) = exp κ (X)

and exp preserves adjoints, the same is true of κ.

One can form a group that doubly covers the whole Lorentz group O(1, 3): it

is a semidirect product of SL(2, C) with the group Z2 × Z2. We leave the details

to the reader.

The Poincar´ e group. The Poincar´ e group or inhomogeneous Lorentz group

P is the group of transformations of

R4

generated by O(1, 3) and the group of

translations (isomorphic to

R4

itself). That is, P is the semi-direct product of

R4

and O(1, 3),

P =

R4

O(1, 3),

whose underlying set is

R4

× O(1, 3) with group law given by

(a, S)(b, T ) = (a + Sb, ST ), (a,

S)−1

=

(−S−1a, S−1).

The action of (a, A) ∈ P on

R4

is

(a, S)x = Sx + a.

Like O(1, 3), P has four connected components, and the component of the identity

is P0 =

R4 SO↑(3,

1). The covering map κ of O(1, 3) by SL(2, C) induces a double

covering (a, A) → (a, κ(A)) of P0 by the group

R4

SL(2, C), whose group law is

(a, A)(b, B) = (a + κ(A)b, AB).