Chapter 9
Irreducible Tensors
Abstract At the begin of the more advanced part of the book, irreducible, i.e.
symmetric traceless tensors of any rank are treated in this chapter. Products of
irreducible tensors and contractions, a relation to Legendre polynomials as well as
spherical components of tensors are pointed out. Cubic tensors and cubic harmonics
associated with cubic symmetry are presented.
9.1 Definition and Examples
An arbitrary tensorAμ 1 μ 2 ...μ− 1 μof rankcan be reduced to a tensor of rank− 1
with the help of the epsilon-tensor:
εμμ− 1 μAμ 1 μ 2 ...μ− 1 μ. (9.1)Similarly, the contraction
Aμ 1 μ 2 ...μ− 2 μμ=δμ− 1 μAμ 1 μ 2 ...μ− 1 μ (9.2)yields a tensor of rank−2. A reduction of the rank of a tensor is not possible, when
the tensor is symmetric with respect to the interchange of any pair of subscripts and
when its partial trace vanishes, i.e. whenever two of its indices are equal and summed
over. Such a tensor, which cannot be reduced to a lower-rank tensor is anirreducible
tensor, sometimes it is also referred to assymmetric traceless. Just as in the case
of second rank tensors, cf. (3.3), the symbol...indicates the irreducible part of a
tensor, of rank:
Aμ 1 μ 2 ...μ− 1 μ. (9.3)Scalars and vectors, tensors of rank=0 and=1, are irreducible tensors, per se.
The irreducible part of a second rank tensorAμνis
Aμν =1
2
(
Aμν+Aνμ)
−
1
3
Aλλδμν,© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_9
155