  A formulation of quantum mechanics with additive and multiplicative
(q-)difference operators instead of differential operators is studied from
first principles. Borel-quantisation on smooth configuration spaces is used as
guiding quantisation method. After a short discussion this method is translated
step-by-step to a framework based on difference operators. To restrict the
resulting plethora of possible quantisations additional assumptions motivated
by simplicity and plausibility are required. Multiplicative difference
operators and the corresponding q-Borel kinematics are given on the circle and
its N-point discretisation; the connection to q-deformations of the Witt
algebra is discussed. For a "natural" choice of the q-kinematics a
corresponding q-difference evolution equation is obtained. This study shows
general difficulties for a generalisation of a physical theory from a known one
to a "new" framework.

Quantum Physics