2003/06/16 by C. J. Isham
Physics and Astronomy · Mathematics · #gr-qc #math-ph #math.MP #quant-ph
published as Adv.Theor.Math.Phys. 8 (2004) 797-811 · Latex2e
arxiv created 2003/06/16 · arxiv updated 2009/11/30
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∈\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.