Quantum norm theory and the quantisation of metric topology

One approach to building a genuine theory of quantum topology would be to construct a quantum theory on the set M(X) of all metrics on a set X. The authors move towards this goal by showing that, for a finite set X, almost all such metrics can be obtained by embedding X into a vector space V and then varying the norm on V. This leads to the subject of 'quantum norm theory' and they give an explicit Fock space representation of such a system. They discuss a model Hamiltonian which can produce a change in metric topology by changing the effective number of points in X.