A New Approach to Quantising Space-Time: III. State Vectors as Functions on Arrows

In two recent papers by the author, a new approach was suggested for quantising space-time, or space. This involved developing a procedure for quantising a system whose configuration space–or history-theory analogue–is the set of objects, $\text{Ob}\text{Q}$, in a (small) category $\text{Q}$. The quantum states in this approach are cross-sections of a bundle $A⇝\text{K}[A]$ of Hilbert spaces over $\text{Ob}\text{Q}$. The Hilbert spaces $\text{K}[A]$, $A\in \text{Ob}\text{Q}$, depend strongly on the object $A$, and have to be chosen so as to get an irreducible, faithful, representation of the basic `category quantisation monoid’. In the present paper, we develop a different approach in which the state vectors are complex-valued functions on the set of {\em arrows} in $\text{Q}$. This throws a new light on the Hilbert bundle scheme: in particular, we recover the results of that approach in the, physically important, example when $\text{Q}$ is a small category of finite sets.