11.6 Coupling of Second Rank Tensors with Irreducible Tensors 195
bνbλ bμ 1 bμ 2 ···bμ =bνbλbμ 1 bμ 2 ···bμ+2
2 + 3
b^2 Δ(,μ 12 μ,) 2 ···μ,νλ,ν 1 ν 2 ···νbν 1 bν 2 ···bν (11.57)+
(− 1 )
( 2 + 1 )( 2 − 1 )
b^4 Δ()μ 1 μ 2 ···μ,νλν 1 ν 2 ···ν− 2 bν 1 bν 2 ···bν− 2.For=1, the (11.54) is recovered. Of special interest is the case=2. Here (11.57)
is equivalent to
bμbν bλbκ =bμbνbλbκ+4
7
b^2 Δμν,λκ,σ τbσbτ +2
15
b^4 Δμν,λκ. (11.58)Multiplication of this equation with symmetric tensorsaμνandcλκyields
aμνbμbνcλκbλbκ =aμνcλκbμbνbλbκ +4
7
b^2 aμνbνbλcλμ+2
15
b^4 aμνcμν.
(11.59)
Applications of these relations involving the products of irreducible tensors are found
in the following sections.
11.7 Scalar Product of Three Irreducible Tensors
11.7.1 Scalar Invariants
Consider three symmetric irreducible tensorsa,b,c,viz.aμ 1 μ 2 ·μ 1 ,bν 1 ν 2 ·ν 2 and
cκ 1 κ 2 ·κ. Provided that= 1 + 2 , their product and total contraction yields the
scalar
aμ 1 μ 2 ·μ 1 bν 1 ν 2 ·ν 2 cμ 1 μ 2 ···μ 1 ν 1 ν 2 ···ν 2.However, using in the product of the tensorsa,btwo subscripts which are equal
and summed over, one also obtains a scalar, when= 1 + 2 −2 applies and the
appropriate contractions are performed. In this case one has
aμ 1 ·μ 1 − 1 λbν 1 ·ν 2 − 1 λcμ 1 μ 2 ···μ 1 − 1 ν 1 ν 2 ···ν 2 − 1.Obviously, this can be generalized to form scalars from the product of three tensors
of ranks 1 , 2 andwith
= 1 + 2 , 1 + 2 − 2 ,...,| 1 − 2 |.