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Double Contraction Relation

A Tensor $t$ is said to satisfy the double contraction relation when

{t_{ij}^m}^* t_{ij}^n = \delta_{mn}.

This equation is satisfied by
$\displaystyle \hat t^0$ $\textstyle =$ $\displaystyle {2\hat {\bf z}\hat {\bf z}-\hat {\bf x}\hat {\bf x}-\hat {\bf y}\hat {\bf y}\over \sqrt{6}}$  
$\displaystyle \hat t^{\pm 1}$ $\textstyle =$ $\displaystyle \mp{\textstyle{1\over 2}}(\hat {\bf x}\hat {\bf z}+\hat {\bf z}\h...
...}) - {\textstyle{1\over 2}}i(\hat {\bf y}\hat {\bf z}-\hat {\bf z}\hat {\bf y})$  
$\displaystyle \hat t^{\pm 2}$ $\textstyle =$ $\displaystyle \mp{\textstyle{1\over 2}}(\hat {\bf x}\hat {\bf x}+\hat {\bf y}\h...
...) - {\textstyle{1\over 2}}i(\hat {\bf x}\hat {\bf y}-\hat {\bf y}\hat {\bf x}),$  

where the hat denotes zero trace, symmetric unit Tensors. These Tensors are used to define the Spherical Harmonic Tensor.

See also Spherical Harmonic Tensor, Tensor


Arfken, G. ``Alternating Series.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 140, 1985.

© 1996-9 Eric W. Weisstein