Master Dissertation

2.3 Spinor Representations of the Lorentz Group
We want to extend the spinor representation to four dimensions. 1 Doing
this we add the unit matrix
1 0
σ0 = , (2.6)
0 1
to the usual Pauli matrices:
σ1 =
0 1
1 0
, σ2 =
0 −i
i 0
1 0
, σ3 =
0 −1
. (2.7)
These form a basis of H(2), the vector space over R of all complex
Hermitian 2 × 2 matrices. Actually, given a 2 × 2 matrix H ∈ H(2) we get
by direct calculation that H can be represented by a vector x ∈ R4 in the
following way:
H := σ(x) = x µ σµ = 1
3
tr(Hσµ)σµ
(2.8)
2
The map x ↦→ σ(x) defined in this way is indeed an isomorphism of R 4
onto H(2).
Fixme: should this be written out?
Further we can define another isomorphism by
µ=0
x ↦→ σ ′ (x) = x 0 σ0 − x k σk =
3
xµσµ.
Now consider the transform of σ(x) with a complex 2 × 2 matrix A:
σ(x) ↦→ Aσ(y)A ∗ .
Clearly σ(y) := Aσ(x)A ∗ ∈ H(2). And from equation (2.8) we see that its
components in the basis of Pauli Matrices are given by
y ν = 1
2 tr(σ(y)σν) = 1
2 tr(Aσ(x)A∗σν) = 1
2
µ=0
3
µ=0
tr(AσµA ∗ σν)x µ .
In this way we get a linear map ΛA : x → y of R 4 into it self. Note that
Hence if det A = 1, then
σ(x) = x µ
x0 + x3 x1 − ix2 σµ =
x 1 + ix 2 x 0 − x 3
.
1 The two-component spinor formalism is treated in [8] chapter 3.2-3.3.
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