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.