The quantum mechanics of affine variables (Q1602441)

Under duality one understands a transformation of a given quantum theory of field, string or brane into a physically equivalent theory with different variables and parameters. The duality may exchange a field theory with a string theory or may exchange the strong-coupling regime of a given theory with a perturbative regime of its dual theory. This latter form of mapping is known under the name of S-duality. In this paper, a quantum mechanics that naturally incorporates a simple form of S-duality is developed. This form is modelled on the conformal transformation of a complex variable \(z \rightarrow -z^{-1}\). Due to the presence of conformal symmetry, this formalism may also be understood as the appropriate quantum mechanics for the affine variables of quantum gravity, where the affine algebra plays a significant role. The presented formalism may be considered as a topological limit of \textit{F. A. Berezin}'s quantization [Math. USSR, Izv. 8, 1109--1165 (1974); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1116--1175 (1974; Zbl 0312.53049)] on Poincaré's upper half-plane \(\mathbb{H}\). In Berezin's method the metric properties of classical phase space \(\mathcal{M}\) are utilized and quantum numbers arise naturally from the metric on \(\mathcal{M}\). The method used in this paper may be regarded as the Berezin's quantization where the metric dependence has been removed. As a consequence of this topological nature the used quantization exhibits some new features: quantum numbers are not originally present in the prescription and appear only after a local vacuum has been chosen and they are of local nature instead of global. This procedure may be thus considered as a manifestly non-perturbative formulation of quantum mechanics in which no classical phase space and no Poisson brackets are taken as a starting point, i.e., the classical theory is not deformed into its quantum counterpart.