Coupling of an antisymmetric tensor
field with the field strength tensor of Yang-Mills is described by these models .
More exactly, [mathematical expression not reproducible], where a tensor [[??].sup.[mu]v] dual to a given antisymmetric tensor
[[psi].sup.[mu]v] is defined as [mathematical expression not reproducible].
where the antisymmetric tensor
[S.sup.[alpha][beta][alpha]] is the contribution of the intrinsic angular momentum.
Note that 6 is coming from the antisymmetric tensor
in 4 x 4 in (47) or [bar.4] x [bar.4] in (48).
where [[epsilon].sub.[mu]v] is antisymmetric tensor
, [a.sub.[mu]] and [b.sub.[mu]] are constant vectors, and a is a constant.
An antisymmetric tensor
can also be defined from the displacement [u.sup.[mu]].
Notice that the tensor [N.sub.[mu]v] is an antisymmetric tensor
. It is evident from Eq.
"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:
Taking into account that [F.sub.00] = [F.sup.00] = 0, as for any antisymmetric tensor
of the 2nd rank, after some algebra we obtain the other components of the field tensor [F.sub.[alpha][beta]]
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.
where A is a scalar and [B.sub.ac], [F.sub.ij] are arbitrary antisymmetric tensors
. Then from (5) it is easy to deduce the expression: