Duality and the Equivalence Principle of Quantum Mechanics

Following a suggestion by Vafa, we present a quantum–mechanical model for S–duality symmetries observed in the quantum theories of fields, strings and branes. Our formalism may be understood as the topological limit of Berezin’s metric quantisation of the upper half–plane H, in that the metric dependence of Berezin’s method has been removed. Being metric–free, our prescription makes no use of global quantum numbers. Quantum numbers arise only locally, after the choice of a local vacuum to expand around. Our approach may be regarded as a manifestly non–perturbative formulation of quantum mechanics, in that we take no classical phase space and no Poisson brackets as a starting point. Position and momentum operators satisfying the Heisenberg algebra are defined and their spectra are analysed. We provide an explicit construction of the Hilbert space of states. The latter carries no representation of SL(2,R), due to the lifting of the metric dependence. Instead, the reparametrisation invariance of H under SL(2,R) induces a natural SL(2,R) action on the quantum–mechanical operators that implements S–duality. We also link our approach with the equivalence principle of quantum mechanics recently formulated by Faraggi–Matone.