Gravitational $\theta$ States and the Wave Function of the Universe

The naive no-boundary wave function of the universe is shown to be invariant under diffeomorphisms only for the simplest spacetime topologies. A more general construction which does give an invariant wave function of the universe is exhibited. Similar problems, some familiar, some not, are encountered in a wide range of theories whose physical configuration space is topologically nontrivial. These include the theory of identical particles, Yang-Mills theory, higher-dimensional gravity, and membrane theories. The sum-over-histories formulation of quantum mechanics provides a unified approach to these problems and their resolution.