The strong CP problem revisited and solved by the gauge group topology

We exploit the non-perturbative result that the $\theta$ angle which defines the vacuum structure is not a $c$-number free parameter, as suggested by the instanton semi-classical approximation, but instead one of the points of the spectrum of the central operator $\tilde{\theta}$ which describes the gauge group topology. Hence, the value of such an angle should not be \textit{a priori} fixed, but rather be determined, as any quantum operator, by the infinite volume limit of the functional integral, where the fermionic mass term uniquely fixes the phase, with $< \tilde{\theta} > = \theta_M$, the mass angle. Such an equality is stable under radiative corrections performed before the infinite volume limit of the functional integral. The mechanisms is carefully controlled in the massive Schwinger model with attention to the infrared problems, to the volume effects induced by boundary terms and with a careful discussion of the infinite volume limit of the functional integral. The extension to the QCD case relies on the choice of modified APS boundary conditions which do not break chiral symmetry, allowing for the crucial role of the fermionic mass term in uniquely determining the phase, with a solution of the strong $CP$ problem.