Bringing Up a Quantum Baby

Any two innite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘Quantum Mechanics’, then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving vectors in the domain of the Hamiltonian: The \probability densities" (hermitean forms) y for ; in this domain generate an algebra from which the classical conguration space with its topology (and with further renements of the axiom, its C K and C 1 structures) can be reconstructed using Gel’fand - Naimark theory. Classical topology is an attribute of only certain quantum states for these axioms, the conguration space emergent from quantum physics getting progressively less dierentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.