Note that both Ω and−Ω give the same linear transformation when they act
by conjugation like this. One can show that all elements ofSO(3,1) arise as
such conjugation maps, by finding appropriate Ω that give rotations or boosts
in theμνplanes, since these generate the group.
Recall that the double covering map
Φ :SU(2)→SO(3)was given for Ω∈SU(2) by taking Φ(Ω) to be the linear transformation in
SO(3) (
x 3 x 1 −ix 2
x 1 +ix 2 −x 3
)
→Ω
(
x 3 x 1 −ix 2
x 1 +ix 2 −x 3)
Ω−^1
We have found an extension of this map to a double covering map fromSL(2,C)
toSO(3,1). This restricts to Φ on the subgroupSU(2) ofSL(2,C) matrices
satisfying Ω†= Ω−^1.
Digression(The complex groupSpin(4,C) and its real forms). Recall from
chapter 6 that we found thatSpin(4) =Sp(1)×Sp(1), with the corresponding
SO(4)transformation given by identifyingR^4 with the quaternionsHand taking
not just conjugations by unit quaternions, but both left and right multiplication
by distinct unit quaternions. Rewriting this in terms of complex matrices instead
of quaternions, we haveSpin(4) =SU(2)×SU(2), and a pairΩ 1 ,Ω 2 ofSU(2)
matrices acts as anSO(4)rotation by
(
x 0 −ix 3 −x 2 −ix 1
x 2 −ix 1 x 0 +ix 3
)
→Ω 1
(
x 0 −ix 3 −x 2 −ix 1
x 2 −ix 1 x 0 +ix 3)
Ω 2
preserving the determinantx^20 +x^21 +x^22 +x^23.
For another example, consider the identification ofR^4 with 2 by 2 real ma-
trices given by
(x 0 ,x 1 ,x 2 ,x 3 )↔(
x 0 +x 3 x 2 +x 1
x 2 −x 1 x 0 −x 3)
Given a pair of matricesΩ 1 ,Ω 2 inSL(2,R), the linear transformation
(
x 0 +x 3 x 2 +x 1
x 2 −x 1 x 0 −x 3)
→Ω 1
(
x 0 +x 3 x 2 +x 1
x 2 −x 1 x 0 −x 3)
Ω 2
preserves the reality condition on the matrix, and preserves
det(
x 0 +x 3 x 2 +x 1
x 2 −x 1 x 0 −x 3)
=x^20 +x^21 −x^22 −x^23so gives an element ofSO(2,2), and we see thatSpin(2,2) = SL(2,R)×
SL(2,R).
These three different constructions for the cases
Spin(4) =SU(2)×SU(2), Spin(3,1) =SL(2,C)