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Irreducible Tensor

Given a general second tensor rank tensor A_(ij) and a metric g_(ij), define

theta=A_(ij)g^(ij)=A_i^i
(1)
omega^i=epsilon^(ijk)A_(jk)
(2)
sigma_(ij)=1/2(A_(ij)+A_(ji))-1/3g_(ij)A_k^k,
(3)

where delta_(ij) is the Kronecker delta and epsilon^(ijk) is the permutation symbol. Then

sigma_(ij)+1/3thetag_(ij)+1/2epsilon_(ijk)omega^k =[1/2(A_(ij)+A_(ji))-1/3g_(ij)A_k^k]+1/3A_k^kg_(ij)+1/2epsilon_(ijk)[epsilon^(lambdamuk)A_(lambdamu)] =1/2(A_(ij)+A_(ji))+1/2(delta_i^lambdadelta_j^mu-delta_i^mudelta_j^lambda)A_(lambdamu) =1/2(A_(ij)+A_(ji))+1/2(A_(ij)-A_(ji))=A_(ij),
(4)

where theta, omega^i, and sigma_(ij) are tensors of tensor rank 0, 1, and 2.

See also

Tensor

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References

Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Irreducible Tensors." Ch. 3 in Quantum Theory of Angular Momentum. Singapore: World Scientific, pp. 61-71, 1988.

Referenced on Wolfram|Alpha

Irreducible Tensor

Cite this as:

Weisstein, Eric W. "Irreducible Tensor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IrreducibleTensor.html

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