Tensors for Physics

156 9 Irreducible Tensors

seeSects.3.1.2andChap. 6 .Thenumberofindependentcomponentsofanirreducible
tensor is of rankis

2 + 1.

This can be verified easily for the cases= 0 , 1 ,2, treated so far. In general, an
irreducible Cartesian tensorAμ 1 μ 2 ...μ− 1 μof rank, is isomorphic to a corresponding
spherical tensor with componentsAm. The integerm, with−≤m≤, has 2+ 1
possible values. The connection between Cartesian and spherical components is
discussed later, in Sect.9.4. Some examples for irreducible tensors of rank 3 and 4
are presented next.

Letaμbe a vector andAνλ= Aνλ an irreducible second rank tensor. The

irreducible third rank tensoraμAνλis explicitly given by

aμAνλ=

1

3

(aμAνλ+aνAμλ+aλAμν)−

2

5

aκ(δμνAλκ+δμλAνκ+δνλAμκ).(9.4)

The third rank irreducible tensor constructed from the components of the vectoraμis

aμaνaλ=aμaνaλ−

1

5

a^2 (aμδνλ+aνδμλ+aλδμν), (9.5)

witha^2 =aκaκ=a·a. Similarly, the explicit expression for the irreducible tensor
of rank four, constructed from the components of the vector, is

aμaνaλaκ =aμaνaλaκ

−

1

7

a^2 (aμaνδλκ+aμaλδνκ+aμaκδνλ+aνaλδμκ+aνaκδμλ+aλaκδμν)

+

1

35

a^4 (δμνδλκ+δμλδνκ+δμκδνλ). (9.6)

Notice: the symbol...is defined such that the factor in front of the termaμ 1 aμ 2 ...

aμin the irreducible tensoraμ 1 aμ 2 ...aμ is 1.
LetAμνbe an irreducible tensor. The irreducible fourth rank tensor constructed
from the product of the second rank tensor is

AμνAλκ =

1

3

(AμνAλκ+AμλAνκ+AμκAλν)

−

2

21

(AμτAτνδλκ+AμτAτλδνκ+AμτAτκδνλ+AντAτλδμκ+AντAτκδμλ+AλτAτκδμν)

• 2
  105

AτσAτσ(δμνδλκ+δμλδνκ+δμκδνλ). (9.7)