M Theory Lesson 287

The categorical diagram calculus for observables and basis structures involves algebra objects, meaning that there are trivalent vertices labelled by a multiplication $m$, since the strings represent the object in the category. There is a dual notion called comultiplication, which we will label by $c$. For the example of finite dimensional Hilbert spaces, we imagine that the strings represent qudits, for some dimension $d$. The parallel inputs stand for the tensor product of qudit spaces. Typical diagrams will therefore contain hexagons, as in Now Andrei Akhvlediani has been giving an excellent series of talks on PROs, PROPS and generalised spider theorems, so yesterday I found myself wondering about an alternative bialgebra morphism, which draws out the paths on the hexagon and looks like: The hexagon has become a little loop. The thing to notice is that all the $c$ labels have moved to the top and all the $m$ labels to the bottom. This process is a lot like what physicists call a normal product in quantum field theory, where all creation operators are put on the left of the annihilation operators. But annihilation acts first, so we should read real time upwards in the diagram, although that is somehow backwards from what is happening in the category.

Soon we will look at the simple ordinal matrices that count the paths on a diagram, and thereby represent a bialgebra morphism. In this case we are considering a $3 \times 3$ matrix, indexed by inputs and outputs, just like the $3 \times 3$ matrices for entanglement. The product of two such matrices forms another path counting matrix of the same kind. These matrices are symmetric, since paths run two ways. Antisymmetry may be introduced by orienting the paths.