A.P. Balachandran
Bringing Up a Quantum Baby
(114K, LaTeX)

ABSTRACT.  Any two infinite-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) 
psi^dagger chi for psi, chi in this domain generate an algebra from which 
the classical configuration space with its topology (and with further 
refinements of the axiom, its C^K and C^infinity structures) can be 
reconstructed using Gel'fand - Naimark theory. Classical topology is an 
attribute of only certain quantum states for these axioms, the configuration 
space emergent from quantum physics getting progressively less differentiable 
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.