Tensors for Physics

13.3 Irreducible Spin Tensors 245

For=2, (13.21) corresponds to

sμsν sμsν =

2

3

S^20 S^21 , S^20 =s(s+ 1 ), S 12 =S^20 −

3

4

. (13.23)

Clearly, one hasS 12 =0fors= 1 /2.
Some relations, where the contraction is just over one subscript, also follow from
the properties of the spin operator, viz.

sλsμsλ=(S 02 − 1 )sμ, sλsμsλ =sμsλsλ=

2

3

S 12 sμ, (13.24)

sλsμsνsλ=(S 02 − 3 )sμsν, (13.25)

sμsλ sλsν =

1

3

(

S 02 −

2

3

)

sμsν+

i

2

ενλκsμsλsκ+

i

3

S 12 εμνκsκ+

2

9

S 02 S^21 δμν.

(13.26)

13.4 Spin Traces

13.4.1 Traces of Products of Spin Tensors.

In the following, spin traces are denoted by the symbol tr. When spin operators are
represented by matrices, the tr-operation corresponds to the standard summation over
diagonal elements. For a spins, the trace of the relevant unit matrix is 2s+1, i.e. it
is equal to the number of magnetic sub-states. The expression

1

2 s+ 1

tr{...}

is equivalent to an orientational average. For classical variables, as discussed in
Chap. 12 , e.g. in connection with the integration over the unit sphere, cf. Sect.12.1,
or over the directions of the velocity, cf. Sect.12.3.1, a continuum of directions is
possible. In contradistinction, the spin allows a discrete set of directions only. As
demonstrated above, results of classical orientational averages are obtained without
performing explicit integrations over angle variables, when symmetry properties are
employed. In the same spirit, the traces of products of spin operators are found, in the
following, without using an explicit matrix representation. The trace operation is a
rotationally invariant process. Consequently, isotropic tensors come into play again.