Quantum rotator and Josephson junction: compact vs. extended phase and dissipative quantum phase transition

The paper reassesses the old dilemma "compact vs. extended phase" in the quantum theory of the rotator and the Josephson junction and the analogy of these two systems with the a particle moving in a periodic potential. This dilemma is in fact the dilemma whether the states with the phases $\varphi$ and $\varphi+2\pi$ are distinguishable, or not. In the past it was widely accepted that in the Josephson junction these states are distinguishable like for a particle moving in a periodic potential. This paper argues that the states with the phases $\varphi$ and $\varphi+2\pi$ are indistinguishable as in a pendulum (a particular example of the quantum rotator). However, this does not lead to revision of the conclusions of the conventional theory predicting the transition from the superconducting to the insulating state in the small Josephson junction.