Tensors for Physics

13.4 Spin Traces 247

13.4.3 Multiple Products of Spin Tensors

The trace of the fourfold product of the spin is similar to the classical expression
(12.6). Yet it differs due to the non-commutativity of the spin components:

1

2 s+ 1

tr{sμsνsλsκ}=

S^20

30

[

2 S 22 δμλδνκ+( 2 S^20 + 1 )(δμνδλκ+δμκδνλ)

]

.

(13.34)

To check the numerical factor on the right hand side, putλequal toκand compare
with (13.29).
The trace of the triple product of second rank spin tensors, analogous to (12.7), is

1

2 s+ 1

tr{sμsνsλsκ sσsτ}=

8

105

S^20 S 12 S 32 Δμν,λκ,σ τ. (13.35)

Two trace formulas are listed which involve fourfold products of second rank spin
tensors, contracted such that the results are scalars:

1

2 s+ 1

tr{sμsνsλsκ sμsνsλsκ}=S^20 S 12

(

7 +

4

9

S 04 −

13

3

S 02

)

, (13.36)

1

2 s+ 1

tr{sμsν[sλsκ,sμsν]−sλsκ}=S^20 S 12

(

4 S 02 − 7

)

. (13.37)

13.5 Density Operator

13.5.1 Spin Averages

The orientation of the spins, in an ensemble, is described by thespin density operator
ρ.Justasthespinoperatorofaparticlewithspins,itcanberepresentedbyahermitian
( 2 s+ 1 )×( 2 s+ 1 )matrix. For this reason, theρis also calledspin density matrix.
In many applications, there is no need for an explicit matrix notation. Alternatively,
ρ=ρ(s)is considered as a function of the spin operator and its algebraic properties,
as given above, are used. In the following,ρis normalized to 1, viz. it obeys the
condition tr{ρ}=1.
The average〈Ψ〉of a functionΨ=Ψ(s), assumed to be a polynomial of the spin
operators, is determined by

〈Ψ〉=tr{Ψ(s)ρ(s)}, tr{ρ(s)}= 1. (13.38)