Tensors for Physics

13.3 Irreducible Spin Tensors 243

As stated before, for a spins, the irreducible tensors of ranks≥ 2 s+1 vanish. For
= 2 s+1, the proof is indicated as follows. Multiplication of the left-hand side

of the Hamilton-Cayley relation (13.7) by the irreducible tensorhμ 1 hμ 2 ···hμ( 2 s+ 1 )

and subsequent integration overd^2 hpicks out the highest order term

(h·s)(^2 s+^1 )=hν 1 sν 1 hν 2 sν 2 ···hν 2 s+ 1 sν( 2 s+ 1 )

in the product, because terms of lower order inhgive no contribution in the integral.
The only non-vanishing term is equivalent to

∫

hμ 1 hμ 2 ···hμ( 2 s+ 1 )hν 1 hν 2 ···hν( 2 s+ 1 )d^2 hsν 1 sν 2 ···sν( 2 s+ 1 ).

By analogy to (12.1), the integral is equal to the isotropic tensorΔ(···^2 s,+···^1 ), apart from
numerical factors. Multiplication of this tensor with the product of the Cartesian
components ofs,cf.(13.13) yields the irreducible spin tensor of rank 2s+1. Thus
the Hamilton-Cayley relation implies

sμ 1 sμ 2 ···sμ( 2 s+ 1 ) = 0. (13.15)

For particles with spins, the existing irreducible spin-tensors are of ranks 2sor
smaller.

13.2 Exercise: Verify a Relation Peculiar for Spin 1/ 2

For spins= 1 /2, the peculiar relation

sμsν=

−i

2

εμνλsλ+

1

4

δμν

holds true. To prove it, start fromsμsν=0, fors= 1 /2, and use the commutation
relation.

13.3.2 Commutation Relation for Spin Tensors

The angular momentum commutation relation (13.1) for spin vectors leads to a
generalization for irreducible spin tensors, which reads

[sμ 1 sμ 2 ···sμ,sλ]−=i()μ 1 μ 2 ···μ,λ,ν 1 ν 2 ···νsν 1 sν 2 ···sν. (13.16)

For=1, this relation is identical to (13.1). The case=2 corresponds to

[sμsν,sλ]−= 2 iμν,λ,αβsαsβ. (13.17)