Chapter 11
Isotropic Tensors
Abstract This chapter deals with isotropic Cartesian tensors. Firstly, the isotropic
Delta-tensors of rank 2are introduced which, when applied on a tensor of rank
, project onto the symmetric traceless part of that tensor. TheΔ-tensors can be
expressed in terms offold products of the second rank isotropic delta-tensor. Sec-
ondly, a generalized cross product between a vector and symmetric traceless tensor
of rankis defined via the-tensors. These isotropic tensors of rank 2+1 can
be expressed in terms of the product of epsilon-tensors and−1-fold products of
delta-tensors. They describe the action of the orbital angular momentum operator
on tensors. Furthermore, isotropic tensors are defined in connection with the cou-
pling of vectors and second rank tensors with tensors of rank. Scalar product of
three irreducible tensors and their relevance for the interaction potential of uniaxial
particles are discussed.
11.1 General Remarks on Isotropic Tensors.
A tensor which does not change its form when the Cartesian coordinate system is
replaced by a rotated one, is referred to asisotropic tensor. A scalar is isotropic, per
definition. Except for the zero-vector, no isotropic tensor of rank 1 exists. Isotropic
tensors of rank 2 and 3 are the delta- and epsilon-tensorsδμνandεμνλ, discussed
before.
Isotropic tensors of rank≥4 can be constructed as products ofδ-tensors and
theε-tensor. Notice, the product of twoε-tensors can be expressed byδ-tensors, cf.
Sect.4.1.2. Thus isotropic tensors of odd rank= 5 , 7 ,...can be expressed by one
ε-tensor and products ofδ-tensors. Isotropic tensors of even rank= 4 , 6 ,...are
expressed by products of/2 second rankδ-tensors. The projection tensors discussed
in Sect.3.1.5are isotropic tensors of rank 4.
© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_11
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