The $\theta$ vacuum and dissipation

For simple quantum-mechanical models with $\theta$ vacuum (particle on a circle, quantum pendulum) the influence of dissipation on the properties of the ground state is studied. It is shown that weak dissipation does not destroy the coherent vacuum structure (“oscillations” in the parameter $\theta$), but it can lead to the suppression of instantons with a set of quantum numbers that does not agree with the symmetry of the dissipative action. The contribution of dissipation to the tunnel energy shift of the vacuum is found. The obtained results are used to estimate the influence of the “friction on quasiparticles” on the instanton Aharonov–Bohm effect in conductors with charge-density wave and on coherent fluctuations of the charge in Josephson junctions of ultrasmall dimensions.