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).
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