An antisymmetric tensor
can also be defined from the displacement [u.
Classical" electric and magnetic fields in the vacuum are joined to an antisymmetric tensor
of 2nd rank
where the components of the third-rank material spin (chirality) tensor 3S are herein given via the second-rank antisymmetric tensor
2S as follows:
00] = 0, as for any antisymmetric tensor
of the 2nd rank, after some algebra we obtain the other components of the field tensor [F.
i]) is the three-dimensional antisymmetric tensor
of the space rotation observable angular velocities, which indices can be lifted/lowered by the metric observable tensor so that [D.
This was shown in , by using the simple observation that the only sl(2) singlet at the fourth power of [lambda] is in the sl(2) [direct sum] so(8) representation (0)(0200)--the four-index antisymmetric tensors
(0)(0020) or (0)(0002) do not occur.