Previously, we saw that density operators generate symmetries in the set of quantum states and that these new symmetries gave non Hermitian quantum states. We then saw that bound states of Feynman diagrams can be represented by matrices of these non Hermitian quantum states. In this post, we use these matrices of non Hermitian quantum states, that is the bound states, to generate symmetries on all the quantum states, bound and free. This is necessary for the recursion that derives E8 as a consequence of treating bound states on an equal footing with free states.
The method of turning a matrix of density matrix states into a symmetry on matrices of density matrix states follows exactly our method of turning a density matrix into a symmetry of density matrices. Let U and V be arbitrary 3×3 matrices whose entries are complex multiples of density matrix states, and let S be an arbitrary 3×3 density matrix whose entries are complex multiples of density matrix states. Since S is a density matrix we have SS = S. The transformation on U is:
and the transformation on V is similar. As in the earlier density matrix case, this transformation preserves addition and multiplication, that is:
since the transformation is linear, that is, , and in the case of multiplication, the exponentials between the products cancel.