A New Approach to Quantising Space-Time: III. State Vestors of Functions on Arrows
Christopher John Isham · Advances in Theoretical and Mathematical Physics · 2004
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.