Writing elements of the dual as row vectors, our example above of a par-
ticular Ω acts by (
ψ^1 ψ^2
)
→
(
ψ^1 ψ^2
)
ei
θ 2 σ 3
Note that the bilinear formgives an isomorphism of representations
betweenSLandSL∗, written in index notation as
ψA=ABψB
where
AB=
(
0 1
−1 0
)
- SR: This is the complex conjugate representation toSL, with Ω∈SL(2,C)
acting onψR∈SRby
ψR→ΩψR
The van der Waerden notation uses a separate set of dotted indices for
these, writing this as
ψA ̇→Ω
B ̇
A ̇ψB ̇
Another common notation among physicists puts a bar over theψ to
denote that the vector is in this representation, but we’ll reserve that
notation for complex conjugation. The Ω corresponding to a rotation
about thez-axis acts as
(
ψ 1 ̇
ψ 2 ̇
)
→ei
θ 2 σ 3
(
ψ 1 ̇
ψ 2 ̇
)
- S∗R: This is the dual representation toSR, with Ω∈SL(2,C) acting on
ψR∗∈SR∗by
ψ∗R→(Ω
− 1
)TψR∗
and the index notation uses raised dotted indices
ψ
A ̇
→((Ω
− 1
)T)
A ̇
B ̇ψ
B ̇
Our standard example of a Ω acts by
(
ψ^1 ̇ ψ^2 ̇
)
→
(
ψ^1 ̇ ψ^2 ̇
)
e−i
θ
2 σ^3
Another copy of
A ̇B ̇
=
(
0 1
−1 0
)
gives the isomorphism ofSRandSR∗as representations, by
ψ
A ̇
=
A ̇B ̇
ψB ̇
