The quadratic Clifford algebra elements−^12 γjγkforj < ksatisfy the com-
mutation relations ofso(4) =spin(4). These are explicitly
−
1
2
γ 0 γ 1 =−i
2(
σ 1 0
0 −σ 1)
, −
1
2
γ 2 γ 3 =−i
2(
σ 1 0
0 σ 1)
−
1
2
γ 0 γ 2 =−
i
2(
σ 2 0
0 −σ 2)
, −
1
2
γ 1 γ 3 =−
i
2(
σ 2 0
0 σ 2)
−
1
2
γ 0 γ 3 =−i
2(
σ 3 0
0 −σ 3)
, −
1
2
γ 1 γ 2 =−i
2(
σ 3 0
0 σ 3)
The Lie algebra spin representation is just matrix multiplication onS=C^4 ,
and it is obviously a reducible representation on two copies ofC^2 (the upper
and lower two components). One can also see that the Lie algebraspin(4) =
su(2)⊕su(2), with the twosu(2) Lie algebras having bases
−
1
4
(γ 0 γ 1 +γ 2 γ 3 ), −1
4
(γ 0 γ 2 +γ 1 γ 3 ), −1
4
(γ 0 γ 3 +γ 1 γ 2 )and
−1
4
(γ 0 γ 1 −γ 2 γ 3 ), −1
4
(γ 0 γ 2 −γ 1 γ 3 ), −1
4
(γ 0 γ 3 −γ 1 γ 2 )The irreducible spin representations ofSpin(4) are just the tensor product of
spin^12 representations of the two copies ofSU(2) (with each copy acting on a
different factor of the tensor product).
In the fermionic oscillator construction, we have
S=S++S−, S+= span{ 1 ,θ 1 θ 2 }, S−= span{θ 1 ,θ 2 }and the Clifford algebra action onSis given for the generators as (now indexing
dimensions from 1 to 4)
γ 1 =∂
∂θ 1
+θ 1 , γ 2 =i(
∂
∂θ 1
−θ 1)
γ 3 =∂
∂θ 2+θ 2 , γ 4 =i(
∂
∂θ 2−θ 2)
Note that in this construction there is a choice of complex structureJ=J 0.
This gives a distinguished vector| 0 〉= 1∈S+, as well as a distinguished sub-Lie
algebrau(2)⊂so(4) of transformations that act trivially on| 0 〉, given by linear
combinations of
θ 1∂
∂θ 1, θ 2∂
∂θ 2, θ 1∂
∂θ 2, θ 2∂
∂θ 1,
There is also a distinguished sub-Lie algebrau(1)⊂u(2) that has zero Lie
bracket with the rest, with basis element
θ 1∂
∂θ 1+θ 2∂
∂θ 2