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# Advances in Theoretical and Mathematical Physics

## Volume 8 (2004)

### Number 5

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

Pages: 797 – 811

DOI: http://dx.doi.org/10.4310/ATMP.2004.v8.n5.a2

#### Author

#### 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.