Classical Topology and Quantum States
A. P. Balachandran, Giuseppe Marmo, B.-S. Skagerstam, A. Stern · 1991
Any two infinite-dimensional (separable) Hilbert spaces are unitarilyisomorphic. The sets of all their self-adjoint operators are also thereforeunitarily equivalent. Thus if all self-adjoint operators can be observed, andif there is no further major axiom in quantum physics than those formulated forexample in Dirac's `Quantum Mechanics', then a quantum physicist would not beable to tell a torus from a hole in the ground. We argue that there are indeedsuch axioms involving observables with smooth time evolution: they containcommutative subalgebras from which the spatial slice of spacetime with itstopology (and with further refinements of the axiom, its $C^K-$ and $C^\\infty-$structures) can be reconstructed using Gel'fand - Naimark theory and itsextensions. Classical topology is an attribute of only certain quantumobservables for these axioms, the spatial slice emergent from quantum physicsgetting progressively less differentiable with increasingly higher excitationsof energy and eventually altogether ceasing to exist. After formulating theseaxioms, we apply them to show the possibility of topology change and to discussquantized fuzzy topologies. Fundamental issues concerning the role of time inquantum physics are also addressed.