Quantum mechanics on path space and point interactions on a circle

In this paper we analyze quantum mechanics formulated in terms of wave functions defined on what may be called the path space, rather than the traditional physical space. An explicit theory of quantum mechanics on a circle is given which can be readily applied to describe a superconducting current flowing around a superconducting ring with a Josephson junction. The path space approach provides an elegant and natural interpretation of the current flow across the Josephson junction. A striking feature of the theory is the emergence of a superselection rule inherent in the fundamental structure of the theory, without needing additional ad hoc assumptions. Other point interactions are discussed, including a 6-potential on a circle and the standard Kronig-Penny model of a crystal lattice on the real line.