A Quantum Mechanical Gauge Model and a Possible Dynamical Solution of the StrongCPProblem

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(?−θ_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 rôle of the boundary conditions, and the decomposition intoθsectors already in finite volume. In the infinite volume limit, one essentially recovers the standard picture whenM=0 (“massless fermions”), but one meets substantial differences forM≠0: for generic boundary conditions, independently of the Lagrangean angle of the topological term, the infinite volume limit selects the sector withθ=θ_Mand 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 theθdependence of the free energy density, with a unique minimum atθ=θ_M.