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

In [1], 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 ; K[A] of Hilbert spaces over Ob(Q). The Hilbert spaces K[A], A ∈ 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 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.