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, $\Ob\Q$, in a (small) category $\Q$. The quantum states in this approach are cross-sections of a bundle $A\leadsto\K[A]$ of Hilbert spaces over $\Ob\Q$. The Hilbert spaces $\K[A]$, $A\in\Ob\Q$, 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 $\Q$. 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 $\Q$ is a small category of finite sets.