A New Approach to Quantising Space-Time: III. State Vestors of Functions on Arrows

Abstract

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 is in K[A] of Hilbert spaces over Ob(Q). The Hilbert spaces K[A], A are 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 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.