The vector
1
√
2
(
(
1
0
)
⊗
(
0
1
)
−
(
0
1
)
⊗
(
1
0
)
)∈V^1 ⊗V^1
is clearly antisymmetric under permutation of the two factors ofV^1 ⊗V^1.
One can show that this vector is invariant underSU(2), by computing
either the action ofSU(2) or of its Lie algebrasu(2). So, this vector
is a basis for the componentV^0 in the decomposition ofV^1 ⊗V^1 into
irreducibles.
The other component,V^2 , is three dimensional, and has a basis
(
1
0
)
⊗
(
1
0
)
,
1
√
2
(
(
1
0
)
⊗
(
0
1
)
+
(
0
1
)
⊗
(
1
0
)
),
(
0
1
)
⊗
(
0
1
)
These three vectors span one dimensional complex subspaces of weights
q= 2, 0 ,−2 under theU(1)⊂SU(2) subgroup
(
eiθ 0
0 e−iθ
)
They are symmetric under permutation of the two factors ofV^1 ⊗V^1.
We see that if we take two identical quantum systems withH=V^1 =
C^2 and make a composite system out of them, if they were bosons we
would get a three dimensional state spaceV^2 =S^2 (V^1 ), transforming as
a vector (spin one) underSU(2). If they were fermions, we would get a
one dimensional state spaceV^0 = Λ^2 (V^1 ) of spin zero (invariant under
SU(2)). Note that in this second case we automatically get an entangled
state, one that cannot be written as a decomposable product.
- Tensor product of three or more spinors:
V^1 ⊗V^1 ⊗V^1 = (V^2 ⊕V^0 )⊗V^1 = (V^2 ⊗V^1 )⊕(V^0 ⊗V^1 ) =V^3 ⊕V^1 ⊕V^1
This says that the tensor product of three spinor representations decom-
poses as a four dimensional (“spin 3/2”) representation plus two copies of
the spinor representation.
This can be generalized by consideringN-fold tensor products (V^1 )⊗Nof
the spinor representation. This will be a sum of irreducible representa-
tions, including one copy of the irreducibleVN, giving an alternative to the
construction using homogeneous polynomials. Doing this however gives
the irreducible as just one component of something larger, and a method
is needed to project out the desired component. This can be done using
the action of the symmetric groupSNon (V^1 )⊗N and an understanding
of the irreducible representations ofSN. This relationship between irre-
ducible representations ofSU(2) and those ofSNcoming from looking at
how both groups act on (V^1 )⊗Nis known as “Schur-Weyl duality”. This
