A model describing a quantum mechanical particle on a circle with minimal electromagnetic interaction with a space independent vector potential, and with a potential &M cos(.&% M ) so that it mimics the massive Schwinger model, is discussed as a prototype of mechanisms and infrared structures which characterize gauge quantum field theories in positive gauges and QCD in particular. The functional integral representation in terms of the field variables which enter in the Lagrangean displays non-standard features, like a complex functional measure (failure of Nelson positivity), a crucial ro^le of the boundary conditions, and the decomposition into % sectors already in finite volume. In the infinite volume limit, one essentially recovers the standard picture when M=0 (massless fermions''), but one meets substantial differences for M{0: for generic boundary conditions, independently of the Lagrangean angle of the topological term, the infinite volume limit selects the sector with %=% M and provides a natural dynamical'' solution of the strong CP problem. In comparison with previous approaches, the strategy discussed here allows us to exploit the consequences of the % dependence of the free energy density, with a unique minimum at %=% M .