Tensors for Physics

18.3 The 4D-Epsilon Tensor 379

18.3.2 Products of Two Epsilon Tensors

Formulas for the product of two 4D epsilon tensors, similar to those presented for
the 3D tensor in Sect.4.1.2, follow from the definition (18.36). The product of two
epsilon-tensors is a tensor of rank 8 which can be expressed in terms of fourfold
products of the unit second rank tensor. Contractions yield tensors of ranks 6, 4 , 2 ,0,
in analogy the formulas valid for the 3D epsilon tensor, cf. Sect.4.1.2. In particular,
the first contraction of the product is

εkmnεk

′′m′n

=−

∣ ∣ ∣ ∣ ∣ ∣

δk

′

k δ

′

k δ

m′

k

δk

′

 δ

′

 δ

m′



δk

′

mδ

′

mδ

m′

m

∣ ∣ ∣ ∣ ∣ ∣

. (18.38)

In many applications, the two-fold contracted version of the product of two epsilon-
tensors is needed. Form=m′,(18.38) reduces to

εkmnεk

′′mn

=− 2

∣

∣

∣

∣

∣

δk

′

k δ

′

k

δk

′

 δ

′



∣

∣

∣

∣

∣

=− 2 (δk

′

kδ

′

−δ

′

kδ

k′

). (18.39)

The further contraction of (18.39), with=′, yields

εkmnεk

′mn

=− 6 δk

′

k. (18.40)

The total contraction of two epsilon tensors is equal to−24, viz.:

εkmnεkmn=− 24. (18.41)

This numerical value 24= 4 !is equal to the number of non-zero elements of the
epsilon tensor in 4D.

18.3.3 Dual Tensor, Determinant

In 3D, the antisymmetric part of a second rank tensor has 3 independent components
which can be related to a vector. In 4D, the antisymmetric part of a second rank
tensor has 6 independent components. Here a similarduality relationexists, which,
however, links an antisymmetric second rank tensor with another second rank tensor
referred to as itsdual tensor. More specifically, letAbe an antisymmetric tensor
withAk=−Ak, then its dualA ̃is defied by

A ̃k=^1

2

εkmnAmn. (18.42)