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

$$\hat t^0 = {2\hat {\bf z}\hat {\bf z}-\hat {\bf x}\hat {\bf x}-\hat {\bf y}\hat {\bf y}\over \sqrt{6}}$$

$$\hat t^{\pm 1} = \mp{\textstyle{1\over 2}}(\hat {\bf x}\hat {\bf z}+\hat {\bf z}\h... ...}) - {\textstyle{1\over 2}}i(\hat {\bf y}\hat {\bf z}-\hat {\bf z}\hat {\bf y})$$

$$\hat t^{\pm 2} = \mp{\textstyle{1\over 2}}(\hat {\bf x}\hat {\bf x}+\hat {\bf y}\h... ...) - {\textstyle{1\over 2}}i(\hat {\bf x}\hat {\bf y}-\hat {\bf y}\hat {\bf x}),$$

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

References

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