Tensors for Physics

246 13 Spin Operators

By analogy to symmetry arguments which lead to (12.1), the trace of the product
of two irreducible spin tensors is given by

1

2 s+ 1

tr{sμ 1 sμ 2 ···sμ sν 1 sν 2 ···sν′}

=

!

( 2 + 1 )!!

S 02 S 12 ···S^2 l− 1 δ′Δ()μ 1 μ 2 ···μ,ν 1 ν 2 ···ν. (13.27)

ForSk^2 see (13.22). As a consequence, one has

tr{sμ 1 sμ 2 ···sμ}= 0 , (13.28)

for≥1.
Special cases of (13.27)are,for=′=1,

1

2 s+ 1

tr{sμsν}=

1

3

S 02 δμν, (13.29)

and for=′=2,

1

2 s+ 1

tr{sμsν sλsκ}=

2

15

S 02 S 12 Δμν,λκ. (13.30)

13.4.2 Triple Products of Spin Tensors

No classical analogue exists for the trace of the product of the spin vector components
and two irreducible spin tensors of rank. Due to symmetry considerations, the trace
must be proportional to the()-tensor. The result is

1

2 s+ 1

tr{sμ 1 sμ 2 ···sμsλsν 1 sν 2 ···sν}

=i



2

!

( 2 + 1 )!!

S^20 S^21 ···Sl^2 − 1 ()μ 1 μ 2 ···μ,λ,ν 1 ν 2 ···ν. (13.31)

Special cases of this expression for=1 and=2are

1

2 s+ 1

tr{sμsλsν}=

i

6

S 02 εμλν, (13.32)

and

1

2 s+ 1

tr{sμsνsλsκsσ}=

2 i

15

S 02 S 12 μν,λ,κσ. (13.33)