Turning now toSpin(3,1) =SL(2,C), one can use the same construction
using homogeneous polynomials as in theSU(2) case to get irreducible repre-
sentations of dimension 2s+ 1 fors= 0,^12 , 1 ,.... Instead of acting bySU(2) on
z 1 ,z 2 , one acts bySL(2,C), and then as before uses the induced action on poly-
nomials ofz 1 andz 2. This gives representations (πs,Vs) ofSL(2,C). Among
the things that are different though about these representations:
- They are not unitary (except in the case of the trivial representation). For
example, for the defining representationV
(^12)
onC^2 , the Hermitian inner
product 〈(
ψ
φ
)
,
(
ψ′
φ′)〉
=
(
ψ φ)
·
(
ψ′
φ′)
=ψψ′+φφ′is invariant underSU(2) transformations Ω since
〈
Ω(
ψ
φ)
,Ω
(
ψ′
φ′)〉
=
(
ψ φ)
Ω†·Ω
(
ψ′
φ′)
and Ω†Ω = 1 by unitarity. This is no longer true for Ω∈SL(2,C).
The representationV(^12)
ofSL(2,C) does have a non-degenerate bilinear
form, which we’ll denote by,
((
ψ
φ)
,
(
ψ′
φ′))
=
(
ψ φ)
(
0 1
−1 0
)(
ψ′
φ′)
=ψφ′−φψ′that is invariant under theSL(2,C) action onV1(^2) and can be used to
identify the representation and its dual. This is the complexification of the
symplectic form onR^2 studied in section 16.1.1, and the same calculation
there which showed that it wasSL(2,R) invariant here shows that the
complex version isSL(2,C) invariant.
- In the case ofSU(2) representations, the complex conjugate representa-
tion one gets by taking as representation matricesπ(g) instead ofπ(g) is
equivalent to the original representation (the same representation, with a
different basis choice, so matrices changed by a conjugation). To see this
for the spin^12 representation, note thatSU(2) matrices are of the form
Ω =
(
α β
−β α)
and one has
(
0 1
−1 0)(
α β
−β α)(
0 1
−1 0
)− 1
=
(
α β
−β α)
so the matrix (
0 1
−1 0