Quantum Mechanics for Mathematicians

vectors that we saw earlier. It is a representations ofSO(3,1) as well as

SL(2,C).

• (^12 ,0)⊕(0,^12 ): This reducible 4 complex dimensional representation is
  known as the representation on “Dirac spinors”.

One can manipulate these Weyl spinor representations (^12 ,0) and (0,^12 ) in
a similar way to the treatment of tangent vectors and their duals in tensor
analysis. Just like in that formalism, one can distinguish between a represen-
tation space and its dual by upper and lower indices, in this case using not
the metric but theSL(2,C) invariant bilinear formto raise and lower indices.
With complex conjugates and duals, there are four kinds of irreducibleSL(2,C)
representations onC^2 to keep track of:

• SL: This is the standard defining representation ofSL(2,C) onC^2 , with
  Ω∈SL(2,C) acting onψL∈SLby

ψL→ΩψL

A standard index notation for such things is called the “van der Waer-

den notation”. It uses a lower indexAtaking values 1,2 to label the

components with respect to a basis ofSLas

ψL=

(

ψ 1

ψ 2

)

=ψA

and in this notation Ω acts by

ψA→ΩBAψB

For instance, the element

Ω =e−i

θ 2 σ 3

corresponding to anSO(3) rotation by an angleθaround thez-axis acts

onSLby (

ψ 1

ψ 2

)

→e−i

θ 2 σ 3

(

ψ 1

ψ 2

)

• S∗L: This is the dual of the defining representation, with Ω∈SL(2,C)
  acting onψ∗L∈SL∗by
  ψL∗→(Ω−^1 )TψL∗
  This is a general property of representations: given any finite dimensional
  representation (π(g),V), the pairing betweenV and its dualV∗is pre-
  served by acting onV∗by matrices (π(g)−^1 )T, and these provide a repre-
  sentation ((π(g)−^1 )T,V∗). In van der Waerden notation, one uses upper
  indices and writes
  ψA→((Ω−^1 )T)ABψB