Tensors for Physics

184 11 Isotropic Tensors

11.2Δ-Tensors

11.2.1 Definition and Examples

Of special interest among the tensors of rank 2are those, which project onto the

symmetric traceless, i.e. irreducible partAof a tensorAof rank. By analogy to
theδ-tensor, these tensors are denoted asΔ()-tensors. They are defined by

Δ()μ 1 μ 2 ···μ,μ′

1 μ′ 2 ···μ′

Aμ′ 1 μ′ 2 ···μ′= Aμ 1 μ 2 ···μ. (11.1)

Examples for this type of isotropic tensors are

Δ(^0 )= 1 ,Δ(^1 )=δμν,

and

Δ(μν,μ^2 ) ′ν′≡Δμν,μ′ν′=

1

2

(δμμ′δνν′+δμν′δνμ′)−

1

3

δμνδμ′ν′. (11.2)

11.2.2 General Properties ofΔ-Tensors.

TheΔ()-tensors are symmetric against the exchange of any fore pair or hind pair of
subscript, as well as against the exchange of all fore against all hind indices, viz.

Δ()μ 1 μ 2 ···μ,μ′

1 μ

′

2 ···μ

′



=Δ()μ′

1 μ

′

2 ···μ

′

,μ^1 μ^2 ···μ

. (11.3)

TheΔ()-tensors obey the product rule

Δ()μ 1 μ 2 ···μ,ν 1 ν 2 ···νΔ()ν 1 ν 2 ···ν,μ′

1 μ′ 2 ···μ′

=Δ()μ 1 μ 2 ···μ,μ′

1 μ′ 2 ···μ′

, (11.4)

which is the relation typical for a projection tensor. On the other hand, the following
contraction of aΔ()-tensor with aΔ(−^1 )-tensor yields zero, more specifically:

Δ()μ 1 μ 2 ···μ,ν 1 ν 2 ···νΔ(ν−^1 )

1 ν 2 ···ν− 1 ,νμ′ 1 ···μ′− 2

= 0. (11.5)

TheΔ()-tensors are traceless in the sense that they vanish, whenever any pair of hind
indices or any pair of fore indices are equal and summed over. On the other hand,
when one fore subscript is equal to one hind subscript, and the standard summation
convention is used, a Delta-tensor with−1 is obtained: