Quantum mechanics with difference operators

A first principle formulation of quantum mechanics on smooth (or discrete) configuration spaces is studied with additive (a- ) or multiplicative (q-) difference operators instead of differential operators. The Borel quantisation is used as a guiding principle, shortly discussed and translated to a framework based on difference operators step by step with certain additional assumptions. q-difference operators are discussed in detail. The framework is explained for the configuration space S¹ and for its N-point discretisation; the connection with q-deformations of the Witt-algebra and its inhomogenisation is elaborated. For a 'natural' choice of the additional assumptions a corresponding q-evolution (Schrödinger-)equation is obtained. The study shows some of the difficulties to generalise a physical theory from a known to a 'new' mathematical formalism.