On phase factors, euclidean path integral and QM thetavacua

The ground state of the Hamiltonian H = p22 + W with periodic potential W(x) = W(x + 2π), x ϵ R, is analyzed from the Euclidean point of view. Let Ω0 be the true vacuum which yields a nonregular representation π of the Weyl algebra. ⊕0 ≤ α < 1H(α) the GNS Hilbert space, with sectors such that π(ei2πp) = ei2πα, and dμ the associated path integral measure. It is shown why eiα(x(T) − x(−T))dμ) collapses, for T → + ∞, even after normalization. There is no OS-construction of QM “thetavacuum” functionals (Ωα,.Ωα)∞, except for the case α = 0, see [1–3].