have the same Lie algebra asSO(3,1), and we will sometimes refer to either
group as the “Lorentz group”.
Recall from chapter 6 that forSO(3) the spin double coverSpin(3) can be
identified with eitherSp(1) (the unit quaternions) orSU(2), and then the action
ofSpin(3) asSO(3) rotations ofR^3 was given by conjugation of imaginary
quaternions (usingSp(1)) or certain 2 by 2 complex matrices (usingSU(2)). In
theSU(2) case this was done explicitly by identifying
(x 1 ,x 2 ,x 3 )↔(
x 3 x 1 −ix 2
x 1 +ix 2 −x 3)
and then showing that conjugating this matrix by an element ofSU(2) was a
linear map leaving invariant
det(
x 3 x 1 −ix 2
x 1 +ix 2 −x 3)
=−(x^21 +x^22 +x^23 )and thus a rotation inSO(3).
The same sort of thing works for the Lorentz group case. Now we identify
R^4 with the space of 2 by 2 complex self-adjoint matrices by
(x 0 ,x 1 ,x 2 ,x 3 )↔(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3)
and observe that
det(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3)
=x^20 −x^21 −x^22 −x^23This provides a very useful way to think of Minkowski space: as complex self-
adjoint 2 by 2 matrices, with norm-squared minus the determinant of the matrix.
The linear transformation
(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3
)
→Ω
(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3)
Ω† (40.4)
for Ω∈SL(2,C) preserves the determinant and thus the inner-product, since
det(Ω(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3)
Ω†) =(det Ω) det(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3)
(det Ω†)=x^20 −x^21 −x^22 −x^23It also takes self-adjoint matrices to self-adjoints, and thusR^4 toR^4 , since
(Ω
(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3)
Ω†)†=(Ω†)†
(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3)†
Ω†
=Ω
(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3